Graphics.Rasterific.CubicBezier:cachedBezierAt from Rasterific-0.6.1

Percentage Accurate: 92.5% → 96.7%

Time: 9.4s

Alternatives: 10

Speedup: 0.5×

• 32.3% 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: 92.5% 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: 96.7% 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 99.1%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   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 y around 0

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

      1. distribute-lft-in N/A

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

      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 83.7

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

   5. Applied rewrites 83.7%

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

Alternative 2: 38.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{-94}:\\ \;\;\;\;a \cdot t\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-224}:\\ \;\;\;\;z \cdot y\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+53}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;a \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -4.2e-94)
   (* a t)
   (if (<= a -1.1e-224)
     (* z y)
     (if (<= a 4.5e-143) x (if (<= a 1.2e+53) (* z y) (* a t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.2e-94) {
		tmp = a * t;
	} else if (a <= -1.1e-224) {
		tmp = z * y;
	} else if (a <= 4.5e-143) {
		tmp = x;
	} else if (a <= 1.2e+53) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-4.2d-94)) then
        tmp = a * t
    else if (a <= (-1.1d-224)) then
        tmp = z * y
    else if (a <= 4.5d-143) then
        tmp = x
    else if (a <= 1.2d+53) then
        tmp = z * y
    else
        tmp = a * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -4.2e-94) {
		tmp = a * t;
	} else if (a <= -1.1e-224) {
		tmp = z * y;
	} else if (a <= 4.5e-143) {
		tmp = x;
	} else if (a <= 1.2e+53) {
		tmp = z * y;
	} else {
		tmp = a * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -4.2e-94:
		tmp = a * t
	elif a <= -1.1e-224:
		tmp = z * y
	elif a <= 4.5e-143:
		tmp = x
	elif a <= 1.2e+53:
		tmp = z * y
	else:
		tmp = a * t
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -4.2e-94)
		tmp = Float64(a * t);
	elseif (a <= -1.1e-224)
		tmp = Float64(z * y);
	elseif (a <= 4.5e-143)
		tmp = x;
	elseif (a <= 1.2e+53)
		tmp = Float64(z * y);
	else
		tmp = Float64(a * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -4.2e-94)
		tmp = a * t;
	elseif (a <= -1.1e-224)
		tmp = z * y;
	elseif (a <= 4.5e-143)
		tmp = x;
	elseif (a <= 1.2e+53)
		tmp = z * y;
	else
		tmp = a * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -4.2e-94], N[(a * t), $MachinePrecision], If[LessEqual[a, -1.1e-224], N[(z * y), $MachinePrecision], If[LessEqual[a, 4.5e-143], x, If[LessEqual[a, 1.2e+53], N[(z * y), $MachinePrecision], N[(a * t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.2000000000000002e-94 or 1.2e53 < a

   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 46.4

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

   5. Applied rewrites 46.4%

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

   if -4.2000000000000002e-94 < a < -1.1e-224 or 4.5e-143 < a < 1.2e53

   1. Initial program 97.0%

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

   3. Taylor expanded in 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 43.7

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

   5. Applied rewrites 43.7%

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

   if -1.1e-224 < a < 4.5e-143

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 64.3%

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

Alternative 3: 87.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-99} \lor \neg \left(a \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -5e-99) (not (<= a 3.5e-7)))
   (fma (fma b z t) a 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 ((a <= -5e-99) || !(a <= 3.5e-7)) {
		tmp = fma(fma(b, z, t), a, 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 ((a <= -5e-99) || !(a <= 3.5e-7))
		tmp = fma(fma(b, z, t), a, 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[a, -5e-99], N[Not[LessEqual[a, 3.5e-7]], $MachinePrecision]], N[(N[(b * z + t), $MachinePrecision] * a + x), $MachinePrecision], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.99999999999999969e-99 or 3.49999999999999984e-7 < a

   1. Initial program 87.7%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      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 88.5

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

   5. Applied rewrites 88.5%

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

   if -4.99999999999999969e-99 < a < 3.49999999999999984e-7

   1. Initial program 98.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{-99} \lor \neg \left(a \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(b, z, t\right), a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.4 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(a, t, \mathsf{fma}\left(z, y, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot z, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.4e+102)
   (* (fma b a y) z)
   (if (<= b 1.65e+147) (fma a t (fma z y x)) (fma (* b z) a x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.4e+102) {
		tmp = fma(b, a, y) * z;
	} else if (b <= 1.65e+147) {
		tmp = fma(a, t, fma(z, y, x));
	} else {
		tmp = fma((b * z), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.4e+102)
		tmp = Float64(fma(b, a, y) * z);
	elseif (b <= 1.65e+147)
		tmp = fma(a, t, fma(z, y, x));
	else
		tmp = fma(Float64(b * z), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.4e+102], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[b, 1.65e+147], N[(a * t + N[(z * y + x), $MachinePrecision]), $MachinePrecision], N[(N[(b * z), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.4e102

   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 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. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-out N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. associate-*l* N/A

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

   5. Applied rewrites 79.9%

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

   if -3.4e102 < b < 1.65000000000000012e147

   1. Initial program 95.0%

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

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

      2. +-commutative N/A

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

      3. associate-+r+ N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 89.6

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

   5. Applied rewrites 89.6%

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

   if 1.65000000000000012e147 < b

   1. Initial program 81.7%

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

   3. Taylor expanded in y around 0

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

      1. distribute-lft-in N/A

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

      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 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 82.8

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

   8. Applied rewrites 82.8%

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

Alternative 5: 75.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-11} \lor \neg \left(z \leq 1.4 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -4.7e-11) (not (<= z 1.4e+25))) (* (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 <= -4.7e-11) || !(z <= 1.4e+25)) {
		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 <= -4.7e-11) || !(z <= 1.4e+25))
		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, -4.7e-11], N[Not[LessEqual[z, 1.4e+25]], $MachinePrecision]], N[(N[(b * a + y), $MachinePrecision] * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.69999999999999993e-11 or 1.4000000000000001e25 < z

   1. Initial program 85.8%

      \[\left(\left(x + y \cdot z\right) + t \cdot a\right) + \left(a \cdot z\right) \cdot b \]
   2. 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. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-out N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. mul-1-neg N/A

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

      16. distribute-lft-out N/A

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

      17. *-commutative N/A

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

      18. mul-1-neg N/A

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

      19. associate-*l* N/A

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

   5. Applied rewrites 76.7%

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

   if -4.69999999999999993e-11 < z < 1.4000000000000001e25

   1. Initial program 97.7%

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

   3. Taylor expanded in 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 79.3

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

   5. Applied rewrites 79.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{-11} \lor \neg \left(z \leq 1.4 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(b, a, y\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.46 \cdot 10^{+108} \lor \neg \left(b \leq 1.48 \cdot 10^{+147}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.46e+108) (not (<= b 1.48e+147))) (* (* b a) z) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.46e+108) || !(b <= 1.48e+147)) {
		tmp = (b * a) * z;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.46e+108) || !(b <= 1.48e+147))
		tmp = Float64(Float64(b * a) * z);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.46e+108], N[Not[LessEqual[b, 1.48e+147]], $MachinePrecision]], N[(N[(b * a), $MachinePrecision] * z), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.45999999999999991e108 or 1.48000000000000002e147 < b

   1. Initial program 86.2%

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

   3. Taylor expanded in b around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.45999999999999991e108 < b < 1.48000000000000002e147

   1. Initial program 94.5%

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

   3. 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 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.46 \cdot 10^{+108} \lor \neg \left(b \leq 1.48 \cdot 10^{+147}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+23} \lor \neg \left(t \leq 2.9 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -5e+23) (not (<= t 2.9e-18))) (fma a t x) (fma z y x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -5e+23) || !(t <= 2.9e-18)) {
		tmp = fma(a, t, x);
	} else {
		tmp = fma(z, y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -5e+23) || !(t <= 2.9e-18))
		tmp = fma(a, t, x);
	else
		tmp = fma(z, y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -5e+23], N[Not[LessEqual[t, 2.9e-18]], $MachinePrecision]], N[(a * t + x), $MachinePrecision], N[(z * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.9999999999999999e23 or 2.9e-18 < t

   1. Initial program 89.4%

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

   3. Taylor expanded in 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 67.9

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

   5. Applied rewrites 67.9%

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

   if -4.9999999999999999e23 < t < 2.9e-18

   1. Initial program 95.1%

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

   3. Taylor expanded in 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 65.1

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

   5. Applied rewrites 65.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+23} \lor \neg \left(t \leq 2.9 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-141} \lor \neg \left(a \leq 9.5 \cdot 10^{-55}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= a -7.5e-141) (not (<= a 9.5e-55))) (* a t) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((a <= -7.5e-141) || !(a <= 9.5e-55)) {
		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 ((a <= (-7.5d-141)) .or. (.not. (a <= 9.5d-55))) 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 ((a <= -7.5e-141) || !(a <= 9.5e-55)) {
		tmp = a * t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (a <= -7.5e-141) or not (a <= 9.5e-55):
		tmp = a * t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((a <= -7.5e-141) || !(a <= 9.5e-55))
		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 ((a <= -7.5e-141) || ~((a <= 9.5e-55)))
		tmp = a * t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[a, -7.5e-141], N[Not[LessEqual[a, 9.5e-55]], $MachinePrecision]], N[(a * t), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if a < -7.50000000000000046e-141 or 9.5000000000000006e-55 < a

   1. Initial program 88.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 42.0

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

   5. Applied rewrites 42.0%

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

   if -7.50000000000000046e-141 < a < 9.5000000000000006e-55

   1. Initial program 98.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 52.1%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{-141} \lor \neg \left(a \leq 9.5 \cdot 10^{-55}\right):\\ \;\;\;\;a \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+112}:\\ \;\;\;\;z \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -3.3e+112) (* z y) (fma a t x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -3.3e+112) {
		tmp = z * y;
	} else {
		tmp = fma(a, t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -3.3e+112)
		tmp = Float64(z * y);
	else
		tmp = fma(a, t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -3.3e+112], N[(z * y), $MachinePrecision], N[(a * t + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2999999999999999e112

   1. Initial program 88.1%

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

   3. Taylor expanded in y around 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 58.2

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

   5. Applied rewrites 58.2%

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

   if -3.2999999999999999e112 < y

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 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 60.8

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

   5. 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 10: 27.1% 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 92.2%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 25.9%

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

Developer Target 1: 97.5% 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 2025026 
(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)))