Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, C

Percentage Accurate: 97.5% → 97.5%

Time: 2.7s

Alternatives: 14

Speedup: 1.0×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, c)
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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, c)
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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

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, c)
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), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 64.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(b \cdot a\right) + c\\ t_2 := \frac{z \cdot t}{16}\\ t_3 := \left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+130}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot x + c\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+235}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* -0.25 (* b a)) c))
        (t_2 (/ (* z t) 16.0))
        (t_3 (+ (* (* t z) 0.0625) c)))
   (if (<= t_2 -1e+130)
     t_3
     (if (<= t_2 -5e-222)
       t_1
       (if (<= t_2 2e+45) (+ (* y x) c) (if (<= t_2 5e+235) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-0.25 * (b * a)) + c;
	double t_2 = (z * t) / 16.0;
	double t_3 = ((t * z) * 0.0625) + c;
	double tmp;
	if (t_2 <= -1e+130) {
		tmp = t_3;
	} else if (t_2 <= -5e-222) {
		tmp = t_1;
	} else if (t_2 <= 2e+45) {
		tmp = (y * x) + c;
	} else if (t_2 <= 5e+235) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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, c)
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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((-0.25d0) * (b * a)) + c
    t_2 = (z * t) / 16.0d0
    t_3 = ((t * z) * 0.0625d0) + c
    if (t_2 <= (-1d+130)) then
        tmp = t_3
    else if (t_2 <= (-5d-222)) then
        tmp = t_1
    else if (t_2 <= 2d+45) then
        tmp = (y * x) + c
    else if (t_2 <= 5d+235) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-0.25 * (b * a)) + c;
	double t_2 = (z * t) / 16.0;
	double t_3 = ((t * z) * 0.0625) + c;
	double tmp;
	if (t_2 <= -1e+130) {
		tmp = t_3;
	} else if (t_2 <= -5e-222) {
		tmp = t_1;
	} else if (t_2 <= 2e+45) {
		tmp = (y * x) + c;
	} else if (t_2 <= 5e+235) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (-0.25 * (b * a)) + c
	t_2 = (z * t) / 16.0
	t_3 = ((t * z) * 0.0625) + c
	tmp = 0
	if t_2 <= -1e+130:
		tmp = t_3
	elif t_2 <= -5e-222:
		tmp = t_1
	elif t_2 <= 2e+45:
		tmp = (y * x) + c
	elif t_2 <= 5e+235:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-0.25 * Float64(b * a)) + c)
	t_2 = Float64(Float64(z * t) / 16.0)
	t_3 = Float64(Float64(Float64(t * z) * 0.0625) + c)
	tmp = 0.0
	if (t_2 <= -1e+130)
		tmp = t_3;
	elseif (t_2 <= -5e-222)
		tmp = t_1;
	elseif (t_2 <= 2e+45)
		tmp = Float64(Float64(y * x) + c);
	elseif (t_2 <= 5e+235)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (-0.25 * (b * a)) + c;
	t_2 = (z * t) / 16.0;
	t_3 = ((t * z) * 0.0625) + c;
	tmp = 0.0;
	if (t_2 <= -1e+130)
		tmp = t_3;
	elseif (t_2 <= -5e-222)
		tmp = t_1;
	elseif (t_2 <= 2e+45)
		tmp = (y * x) + c;
	elseif (t_2 <= 5e+235)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+130], t$95$3, If[LessEqual[t$95$2, -5e-222], t$95$1, If[LessEqual[t$95$2, 2e+45], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$2, 5e+235], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e130 or 5.00000000000000027e235 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      3. lower-*.f64 82.6

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 + c \]

   5. Applied rewrites 82.6%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c \]

   if -1.0000000000000001e130 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.00000000000000008e-222 or 1.9999999999999999e45 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.00000000000000027e235

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) + c \]

      3. lower-*.f64 68.3

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) + c \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} + c \]

   if -5.00000000000000008e-222 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.9999999999999999e45

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

Alternative 3: 62.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(b \cdot a\right) + c\\ t_2 := \frac{z \cdot t}{16}\\ t_3 := \left(0.0625 \cdot t\right) \cdot z\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+195}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+45}:\\ \;\;\;\;y \cdot x + c\\ \mathbf{elif}\;t\_2 \leq 1.5 \cdot 10^{+243}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* -0.25 (* b a)) c))
        (t_2 (/ (* z t) 16.0))
        (t_3 (* (* 0.0625 t) z)))
   (if (<= t_2 -5e+195)
     t_3
     (if (<= t_2 -5e-222)
       t_1
       (if (<= t_2 2e+45) (+ (* y x) c) (if (<= t_2 1.5e+243) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-0.25 * (b * a)) + c;
	double t_2 = (z * t) / 16.0;
	double t_3 = (0.0625 * t) * z;
	double tmp;
	if (t_2 <= -5e+195) {
		tmp = t_3;
	} else if (t_2 <= -5e-222) {
		tmp = t_1;
	} else if (t_2 <= 2e+45) {
		tmp = (y * x) + c;
	} else if (t_2 <= 1.5e+243) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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, c)
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), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ((-0.25d0) * (b * a)) + c
    t_2 = (z * t) / 16.0d0
    t_3 = (0.0625d0 * t) * z
    if (t_2 <= (-5d+195)) then
        tmp = t_3
    else if (t_2 <= (-5d-222)) then
        tmp = t_1
    else if (t_2 <= 2d+45) then
        tmp = (y * x) + c
    else if (t_2 <= 1.5d+243) then
        tmp = t_1
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (-0.25 * (b * a)) + c;
	double t_2 = (z * t) / 16.0;
	double t_3 = (0.0625 * t) * z;
	double tmp;
	if (t_2 <= -5e+195) {
		tmp = t_3;
	} else if (t_2 <= -5e-222) {
		tmp = t_1;
	} else if (t_2 <= 2e+45) {
		tmp = (y * x) + c;
	} else if (t_2 <= 1.5e+243) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (-0.25 * (b * a)) + c
	t_2 = (z * t) / 16.0
	t_3 = (0.0625 * t) * z
	tmp = 0
	if t_2 <= -5e+195:
		tmp = t_3
	elif t_2 <= -5e-222:
		tmp = t_1
	elif t_2 <= 2e+45:
		tmp = (y * x) + c
	elif t_2 <= 1.5e+243:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(-0.25 * Float64(b * a)) + c)
	t_2 = Float64(Float64(z * t) / 16.0)
	t_3 = Float64(Float64(0.0625 * t) * z)
	tmp = 0.0
	if (t_2 <= -5e+195)
		tmp = t_3;
	elseif (t_2 <= -5e-222)
		tmp = t_1;
	elseif (t_2 <= 2e+45)
		tmp = Float64(Float64(y * x) + c);
	elseif (t_2 <= 1.5e+243)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (-0.25 * (b * a)) + c;
	t_2 = (z * t) / 16.0;
	t_3 = (0.0625 * t) * z;
	tmp = 0.0;
	if (t_2 <= -5e+195)
		tmp = t_3;
	elseif (t_2 <= -5e-222)
		tmp = t_1;
	elseif (t_2 <= 2e+45)
		tmp = (y * x) + c;
	elseif (t_2 <= 1.5e+243)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.0625 * t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+195], t$95$3, If[LessEqual[t$95$2, -5e-222], t$95$1, If[LessEqual[t$95$2, 2e+45], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$2, 1.5e+243], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.9999999999999998e195 or 1.49999999999999992e243 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 91.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. lower-*.f64 81.6

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 \]

   5. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      6. lift-*.f64 82.8

         \[\leadsto \left(0.0625 \cdot t\right) \cdot z \]

   7. Applied rewrites 82.8%

      \[\leadsto \left(0.0625 \cdot t\right) \cdot \color{blue}{z} \]

   if -4.9999999999999998e195 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.00000000000000008e-222 or 1.9999999999999999e45 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.49999999999999992e243

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} + c \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} + c \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) + c \]

      3. lower-*.f64 65.8

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) + c \]

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} + c \]

   if -5.00000000000000008e-222 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.9999999999999999e45

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 77.4

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

   5. Applied rewrites 77.4%

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

Alternative 4: 95.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-14} \lor \neg \left(x \cdot y \leq 10^{+16}\right):\\ \;\;\;\;\left(y + \frac{t\_1}{x}\right) \cdot x + c\\ \mathbf{else}:\\ \;\;\;\;t\_1 + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (* -0.25 (* b a)))))
   (if (or (<= (* x y) -5e-14) (not (<= (* x y) 1e+16)))
     (+ (* (+ y (/ t_1 x)) x) c)
     (+ t_1 c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * t), z, (-0.25 * (b * a)));
	double tmp;
	if (((x * y) <= -5e-14) || !((x * y) <= 1e+16)) {
		tmp = ((y + (t_1 / x)) * x) + c;
	} else {
		tmp = t_1 + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a)))
	tmp = 0.0
	if ((Float64(x * y) <= -5e-14) || !(Float64(x * y) <= 1e+16))
		tmp = Float64(Float64(Float64(y + Float64(t_1 / x)) * x) + c);
	else
		tmp = Float64(t_1 + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e-14], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+16]], $MachinePrecision]], N[(N[(N[(y + N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] + c), $MachinePrecision], N[(t$95$1 + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000002e-14 or 1e16 < (*.f64 x y)

   1. Initial program 96.1%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(\left(y + \frac{1}{16} \cdot \frac{t \cdot z}{x}\right) - \frac{1}{4} \cdot \frac{a \cdot b}{x}\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(y + \frac{1}{16} \cdot \frac{t \cdot z}{x}\right) - \frac{1}{4} \cdot \frac{a \cdot b}{x}\right) \cdot \color{blue}{x} + c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(y + \frac{1}{16} \cdot \frac{t \cdot z}{x}\right) - \frac{1}{4} \cdot \frac{a \cdot b}{x}\right) \cdot \color{blue}{x} + c \]

   5. Applied rewrites 95.6%

      \[\leadsto \color{blue}{\left(y + \frac{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)}{x}\right) \cdot x} + c \]

   if -5.0000000000000002e-14 < (*.f64 x y) < 1e16

   1. Initial program 98.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 97.0

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-14} \lor \neg \left(x \cdot y \leq 10^{+16}\right):\\ \;\;\;\;\left(y + \frac{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)}{x}\right) \cdot x + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+216} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+235}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e+216) (not (<= t_1 5e+235)))
     (+ (* (* t z) 0.0625) c)
     (+ (fma (* -0.25 b) a (* y x)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e+216) || !(t_1 <= 5e+235)) {
		tmp = ((t * z) * 0.0625) + c;
	} else {
		tmp = fma((-0.25 * b), a, (y * x)) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e+216) || !(t_1 <= 5e+235))
		tmp = Float64(Float64(Float64(t * z) * 0.0625) + c);
	else
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(y * x)) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+216], N[Not[LessEqual[t$95$1, 5e+235]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e216 or 5.00000000000000027e235 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      3. lower-*.f64 89.8

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 + c \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c \]

   if -2e216 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.00000000000000027e235

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 88.4

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y \cdot x}\right) + c \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y} \cdot x\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + y \cdot x\right) + c \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{y} \cdot x\right) + c \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + x \cdot \color{blue}{y}\right) + c \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, x \cdot y\right) + c \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, x \cdot y\right) + c \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, y \cdot x\right) + c \]

      10. lift-*.f64 88.4

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c \]

   7. Applied rewrites 88.4%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, y \cdot x\right) + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{+216} \lor \neg \left(\frac{z \cdot t}{16} \leq 5 \cdot 10^{+235}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+216} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+235}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e+216) (not (<= t_1 5e+235)))
     (+ (* (* t z) 0.0625) c)
     (+ (fma y x (* -0.25 (* b a))) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -2e+216) || !(t_1 <= 5e+235)) {
		tmp = ((t * z) * 0.0625) + c;
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -2e+216) || !(t_1 <= 5e+235))
		tmp = Float64(Float64(Float64(t * z) * 0.0625) + c);
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+216], N[Not[LessEqual[t$95$1, 5e+235]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e216 or 5.00000000000000027e235 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} + c \]

      3. lower-*.f64 89.8

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 + c \]

   5. Applied rewrites 89.8%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} + c \]

   if -2e216 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.00000000000000027e235

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 88.4

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{+216} \lor \neg \left(\frac{z \cdot t}{16} \leq 5 \cdot 10^{+235}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+93)
     (+ (fma (* -0.25 b) a (* y x)) c)
     (if (<= t_1 2e+46)
       (+ (fma (* t z) 0.0625 (* y x)) c)
       (+ (fma y x (* -0.25 (* b a))) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -1e+93) {
		tmp = fma((-0.25 * b), a, (y * x)) + c;
	} else if (t_1 <= 2e+46) {
		tmp = fma((t * z), 0.0625, (y * x)) + c;
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -1e+93)
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(y * x)) + c);
	elseif (t_1 <= 2e+46)
		tmp = Float64(fma(Float64(t * z), 0.0625, Float64(y * x)) + c);
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+93], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+46], N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.00000000000000004e93

   1. Initial program 88.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 85.9

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y \cdot x}\right) + c \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y} \cdot x\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + y \cdot x\right) + c \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{y} \cdot x\right) + c \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + x \cdot \color{blue}{y}\right) + c \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, x \cdot y\right) + c \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, x \cdot y\right) + c \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, y \cdot x\right) + c \]

      10. lift-*.f64 88.1

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c \]

   7. Applied rewrites 88.1%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, y \cdot x\right) + c \]

   if -1.00000000000000004e93 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e46

   1. Initial program 99.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + \color{blue}{x} \cdot y\right) + c \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{\frac{1}{16}}, x \cdot y\right) + c \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, x \cdot y\right) + c \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, y \cdot x\right) + c \]

      5. lower-*.f64 95.8

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c \]

   5. Applied rewrites 95.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)} + c \]

   if 2e46 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 98.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 88.7

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+196} \lor \neg \left(t\_1 \leq 1.5 \cdot 10^{+243}\right):\\ \;\;\;\;\left(0.0625 \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -1e+196) (not (<= t_1 1.5e+243)))
     (* (* 0.0625 t) z)
     (+ (* y x) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -1e+196) || !(t_1 <= 1.5e+243)) {
		tmp = (0.0625 * t) * z;
	} else {
		tmp = (y * x) + c;
	}
	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, c)
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), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * t) / 16.0d0
    if ((t_1 <= (-1d+196)) .or. (.not. (t_1 <= 1.5d+243))) then
        tmp = (0.0625d0 * t) * z
    else
        tmp = (y * x) + c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -1e+196) || !(t_1 <= 1.5e+243)) {
		tmp = (0.0625 * t) * z;
	} else {
		tmp = (y * x) + c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (z * t) / 16.0
	tmp = 0
	if (t_1 <= -1e+196) or not (t_1 <= 1.5e+243):
		tmp = (0.0625 * t) * z
	else:
		tmp = (y * x) + c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -1e+196) || !(t_1 <= 1.5e+243))
		tmp = Float64(Float64(0.0625 * t) * z);
	else
		tmp = Float64(Float64(y * x) + c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) / 16.0;
	tmp = 0.0;
	if ((t_1 <= -1e+196) || ~((t_1 <= 1.5e+243)))
		tmp = (0.0625 * t) * z;
	else
		tmp = (y * x) + c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+196], N[Not[LessEqual[t$95$1, 1.5e+243]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z), $MachinePrecision], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -9.9999999999999995e195 or 1.49999999999999992e243 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      3. lower-*.f64 82.8

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 \]

   5. Applied rewrites 82.8%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \color{blue}{\frac{1}{16}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(t \cdot z\right) \cdot \frac{1}{16} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{16} \cdot \color{blue}{\left(t \cdot z\right)} \]

      4. associate-*r* N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{16} \cdot t\right) \cdot \color{blue}{z} \]

      6. lift-*.f64 84.0

         \[\leadsto \left(0.0625 \cdot t\right) \cdot z \]

   7. Applied rewrites 84.0%

      \[\leadsto \left(0.0625 \cdot t\right) \cdot \color{blue}{z} \]

   if -9.9999999999999995e195 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.49999999999999992e243

   1. Initial program 98.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -1 \cdot 10^{+196} \lor \neg \left(\frac{z \cdot t}{16} \leq 1.5 \cdot 10^{+243}\right):\\ \;\;\;\;\left(0.0625 \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+166} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+159}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1e+166) (not (<= t_1 2e+159)))
     (* -0.25 (* b a))
     (+ (* y x) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+166) || !(t_1 <= 2e+159)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = (y * x) + c;
	}
	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, c)
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), intent (in) :: c
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a * b) / 4.0d0
    if ((t_1 <= (-1d+166)) .or. (.not. (t_1 <= 2d+159))) then
        tmp = (-0.25d0) * (b * a)
    else
        tmp = (y * x) + c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+166) || !(t_1 <= 2e+159)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = (y * x) + c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) / 4.0
	tmp = 0
	if (t_1 <= -1e+166) or not (t_1 <= 2e+159):
		tmp = -0.25 * (b * a)
	else:
		tmp = (y * x) + c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -1e+166) || !(t_1 <= 2e+159))
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = Float64(Float64(y * x) + c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) / 4.0;
	tmp = 0.0;
	if ((t_1 <= -1e+166) || ~((t_1 <= 2e+159)))
		tmp = -0.25 * (b * a);
	else
		tmp = (y * x) + c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+166], N[Not[LessEqual[t$95$1, 2e+159]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.9999999999999994e165 or 1.9999999999999999e159 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 89.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \color{blue}{\left(a \cdot b\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{4} \cdot \left(b \cdot \color{blue}{a}\right) \]

      3. lower-*.f64 79.7

         \[\leadsto -0.25 \cdot \left(b \cdot \color{blue}{a}\right) \]

   5. Applied rewrites 79.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]

   if -9.9999999999999994e165 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.9999999999999999e159

   1. Initial program 99.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1 \cdot 10^{+166} \lor \neg \left(\frac{a \cdot b}{4} \leq 2 \cdot 10^{+159}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{elif}\;t \leq 3.65 \cdot 10^{-113}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z, \frac{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)}{t}\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= t -1.15e-80)
   (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)
   (if (<= t 3.65e-113)
     (+ (fma (* -0.25 b) a (* y x)) c)
     (* (fma 0.0625 z (/ (- (fma y x c) (* 0.25 (* b a))) t)) t))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (t <= -1.15e-80) {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	} else if (t <= 3.65e-113) {
		tmp = fma((-0.25 * b), a, (y * x)) + c;
	} else {
		tmp = fma(0.0625, z, ((fma(y, x, c) - (0.25 * (b * a))) / t)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (t <= -1.15e-80)
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	elseif (t <= 3.65e-113)
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(y * x)) + c);
	else
		tmp = Float64(fma(0.0625, z, Float64(Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a))) / t)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[t, -1.15e-80], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t, 3.65e-113], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * z + N[(N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.1499999999999999e-80

   1. Initial program 97.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 81.7

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]

   if -1.1499999999999999e-80 < t < 3.65000000000000006e-113

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(x \cdot y - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(x \cdot y + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(y \cdot x + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{x}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, x, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      7. lower-*.f64 92.5

         \[\leadsto \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 92.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
   6. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(y \cdot x + \color{blue}{\frac{-1}{4} \cdot \left(b \cdot a\right)}\right) + c \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y \cdot x}\right) + c \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + \color{blue}{y} \cdot x\right) + c \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\frac{-1}{4} \cdot \left(b \cdot a\right) + y \cdot x\right) + c \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + \color{blue}{y} \cdot x\right) + c \]

      6. *-commutative N/A

         \[\leadsto \left(\left(\frac{-1}{4} \cdot b\right) \cdot a + x \cdot \color{blue}{y}\right) + c \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, \color{blue}{a}, x \cdot y\right) + c \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, x \cdot y\right) + c \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, y \cdot x\right) + c \]

      10. lift-*.f64 92.6

          \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, a, y \cdot x\right) + c \]

   7. Applied rewrites 92.6%

      \[\leadsto \mathsf{fma}\left(-0.25 \cdot b, \color{blue}{a}, y \cdot x\right) + c \]

   if 3.65000000000000006e-113 < t

   1. Initial program 94.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\left(\frac{1}{16} \cdot z + \left(\frac{c}{t} + \frac{x \cdot y}{t}\right)\right) - \frac{1}{4} \cdot \frac{a \cdot b}{t}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot z + \left(\frac{c}{t} + \frac{x \cdot y}{t}\right)\right) - \frac{1}{4} \cdot \frac{a \cdot b}{t}\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot z + \left(\frac{c}{t} + \frac{x \cdot y}{t}\right)\right) - \frac{1}{4} \cdot \frac{a \cdot b}{t}\right) \cdot \color{blue}{t} \]

   5. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625, z, \frac{\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)}{t}\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+115} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -2e+115) (not (<= (* x y) 5e+73)))
   (+ (fma (* t z) 0.0625 (* y x)) c)
   (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -2e+115) || !((x * y) <= 5e+73)) {
		tmp = fma((t * z), 0.0625, (y * x)) + c;
	} else {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+115) || !(Float64(x * y) <= 5e+73))
		tmp = Float64(fma(Float64(t * z), 0.0625, Float64(y * x)) + c);
	else
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+115], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+73]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + N[(y * x), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2e115 or 4.99999999999999976e73 < (*.f64 x y)

   1. Initial program 95.7%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} + c \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(t \cdot z\right) \cdot \frac{1}{16} + \color{blue}{x} \cdot y\right) + c \]

      2. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \color{blue}{\frac{1}{16}}, x \cdot y\right) + c \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, x \cdot y\right) + c \]

      4. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(t \cdot z, \frac{1}{16}, y \cdot x\right) + c \]

      5. lower-*.f64 90.4

         \[\leadsto \mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c \]

   5. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)} + c \]

   if -2e115 < (*.f64 x y) < 4.99999999999999976e73

   1. Initial program 98.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{1}{16} \cdot \left(t \cdot z\right) - \frac{1}{4} \cdot \left(a \cdot b\right)\right)} + c \]
   4. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(\frac{1}{16} \cdot \left(t \cdot z\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)}\right) + c \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)} \cdot \left(a \cdot b\right)\right) + c \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(\frac{1}{16} \cdot t\right) \cdot z + \frac{-1}{4} \cdot \left(\color{blue}{a} \cdot b\right)\right) + c \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, \color{blue}{z}, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) + c \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot t, z, \frac{-1}{4} \cdot \left(b \cdot a\right)\right) + c \]

      8. lower-*.f64 95.1

         \[\leadsto \mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c \]

   5. Applied rewrites 95.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right)} + c \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+115} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+229} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+73}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -5e+229) (not (<= (* x y) 5e+73))) (* y x) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+229) || !((x * y) <= 5e+73)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	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, c)
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), intent (in) :: c
    real(8) :: tmp
    if (((x * y) <= (-5d+229)) .or. (.not. ((x * y) <= 5d+73))) then
        tmp = y * x
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+229) || !((x * y) <= 5e+73)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -5e+229) or not ((x * y) <= 5e+73):
		tmp = y * x
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+229) || !(Float64(x * y) <= 5e+73))
		tmp = Float64(y * x);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((x * y) <= -5e+229) || ~(((x * y) <= 5e+73)))
		tmp = y * x;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+229], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+73]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000005e229 or 4.99999999999999976e73 < (*.f64 x y)

   1. Initial program 94.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -5.0000000000000005e229 < (*.f64 x y) < 4.99999999999999976e73

   1. Initial program 98.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 31.9%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+229} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+73}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ y \cdot x + c \end{array} \]

FPCore

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

C

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

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, c)
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), intent (in) :: c
    code = (y * x) + c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 48.9

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

5. Applied rewrites 48.9%

   \[\leadsto \color{blue}{y \cdot x} + c \]
6. Add Preprocessing

Alternative 14: 22.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

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, c)
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), intent (in) :: c
    code = c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.3%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing

3. Taylor expanded in c around inf

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

5. Applied rewrites 25.4%

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

Reproduce

herbie shell --seed 2025083 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))