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

Percentage Accurate: 97.4% → 98.9%

Time: 5.5s

Alternatives: 10

Speedup: 1.1×

• 33.2% 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: 97.4% 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: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(a \cdot -0.25, b, \mathsf{fma}\left(y, x, c\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (fma (* z 0.0625) t (fma (* a -0.25) b (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma((z * 0.0625), t, fma((a * -0.25), b, fma(y, x, c)));
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(Float64(z * 0.0625), t, fma(Float64(a * -0.25), b, fma(y, x, c)))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(z * 0.0625), $MachinePrecision] * t + N[(N[(a * -0.25), $MachinePrecision] * b + N[(y * x + c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 97.4%

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

3. Applied rewrites 98.9%

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

Alternative 2: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(a \cdot -0.25, b, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(a \cdot b\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+34)
     (fma (* z 0.0625) t (fma (* a -0.25) b c))
     (if (<= t_1 20.0)
       (fma (* z 0.0625) t (fma y x c))
       (+ (fma y x (* -0.25 (* a b))) 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+34) {
		tmp = fma((z * 0.0625), t, fma((a * -0.25), b, c));
	} else if (t_1 <= 20.0) {
		tmp = fma((z * 0.0625), t, fma(y, x, c));
	} else {
		tmp = fma(y, x, (-0.25 * (a * b))) + 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+34)
		tmp = fma(Float64(z * 0.0625), t, fma(Float64(a * -0.25), b, c));
	elseif (t_1 <= 20.0)
		tmp = fma(Float64(z * 0.0625), t, fma(y, x, c));
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(a * b))) + 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+34], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(N[(a * -0.25), $MachinePrecision] * b + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 20.0], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999946e33

   1. Initial program 97.4%

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 73.7%

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

   if -9.99999999999999946e33 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 20

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

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

   if 20 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

Alternative 3: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(y, x, -0.25 \cdot \left(a \cdot b\right)\right) + c\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (+ (fma y x (* -0.25 (* a b))) c)))
   (if (<= t_1 -1e+34)
     t_2
     (if (<= t_1 20.0) (fma (* z 0.0625) t (fma y x c)) t_2))))

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 t_2 = fma(y, x, (-0.25 * (a * b))) + c;
	double tmp;
	if (t_1 <= -1e+34) {
		tmp = t_2;
	} else if (t_1 <= 20.0) {
		tmp = fma((z * 0.0625), t, fma(y, x, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(fma(y, x, Float64(-0.25 * Float64(a * b))) + c)
	tmp = 0.0
	if (t_1 <= -1e+34)
		tmp = t_2;
	elseif (t_1 <= 20.0)
		tmp = fma(Float64(z * 0.0625), t, fma(y, x, c));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y * x + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+34], t$95$2, If[LessEqual[t$95$1, 20.0], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999946e33 or 20 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   if -9.99999999999999946e33 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 20

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, \color{blue}{t}, \mathsf{fma}\left(y, x, c\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(a \cdot b\right)\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+140}:\\ \;\;\;\;t\_1 + c\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* a b))) (t_2 (/ (* a b) 4.0)))
   (if (<= t_2 -4e+140)
     (+ t_1 c)
     (if (<= t_2 2e+168) (fma (* z 0.0625) t (fma y x c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -0.25 * (a * b);
	double t_2 = (a * b) / 4.0;
	double tmp;
	if (t_2 <= -4e+140) {
		tmp = t_1 + c;
	} else if (t_2 <= 2e+168) {
		tmp = fma((z * 0.0625), t, fma(y, x, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-0.25 * Float64(a * b))
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_2 <= -4e+140)
		tmp = Float64(t_1 + c);
	elseif (t_2 <= 2e+168)
		tmp = fma(Float64(z * 0.0625), t, fma(y, x, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+140], N[(t$95$1 + c), $MachinePrecision], If[LessEqual[t$95$2, 2e+168], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.00000000000000024e140

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.3%

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

   if -4.00000000000000024e140 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.9999999999999999e168

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

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

   if 1.9999999999999999e168 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.4%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 28.3%

      \[\leadsto \color{blue}{-0.25 \cdot \left(a \cdot b\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+210}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (* x y))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -2e+210) t_1 (if (<= t_2 2e+141) (+ (* -0.25 (* a b)) c) t_1))))

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), (x * y));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -2e+210) {
		tmp = t_1;
	} else if (t_2 <= 2e+141) {
		tmp = (-0.25 * (a * b)) + c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+210)
		tmp = t_1;
	elseif (t_2 <= 2e+141)
		tmp = Float64(Float64(-0.25 * Float64(a * b)) + c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+210], t$95$1, If[LessEqual[t$95$2, 2e+141], N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.99999999999999985e210 or 2.00000000000000003e141 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 48.1%

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

   10. Taylor expanded in c around 0

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

   12. Applied rewrites 53.2%

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

   if -1.99999999999999985e210 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 2.00000000000000003e141

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.3%

      \[\leadsto -0.25 \cdot \color{blue}{\left(a \cdot b\right)} + c \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 63.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{elif}\;x \cdot y \leq 7.2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1.45e+152)
   (* x y)
   (if (<= (* x y) 1.8e-6)
     (+ (* -0.25 (* a b)) c)
     (if (<= (* x y) 7.2e+168) (fma (* z 0.0625) t c) (* x y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1.45e+152) {
		tmp = x * y;
	} else if ((x * y) <= 1.8e-6) {
		tmp = (-0.25 * (a * b)) + c;
	} else if ((x * y) <= 7.2e+168) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1.45e+152)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 1.8e-6)
		tmp = Float64(Float64(-0.25 * Float64(a * b)) + c);
	elseif (Float64(x * y) <= 7.2e+168)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.45e+152], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1.8e-6], N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 7.2e+168], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], N[(x * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.4499999999999999e152 or 7.1999999999999999e168 < (*.f64 x y)

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 28.8%

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

   if -1.4499999999999999e152 < (*.f64 x y) < 1.79999999999999992e-6

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 48.3%

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

   if 1.79999999999999992e-6 < (*.f64 x y) < 7.1999999999999999e168

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1.3 \cdot 10^{+119}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 7.2 \cdot 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, c\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1.3e+119)
   (* x y)
   (if (<= (* x y) 7.2e+168) (fma (* z 0.0625) t c) (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1.3e+119) {
		tmp = x * y;
	} else if ((x * y) <= 7.2e+168) {
		tmp = fma((z * 0.0625), t, c);
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1.3e+119)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 7.2e+168)
		tmp = fma(Float64(z * 0.0625), t, c);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1.3e+119], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 7.2e+168], N[(N[(z * 0.0625), $MachinePrecision] * t + c), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.3e119 or 7.1999999999999999e168 < (*.f64 x y)

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 28.8%

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

   if -1.3e119 < (*.f64 x y) < 7.1999999999999999e168

   1. Initial program 97.4%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 73.4%

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

   6. Applied rewrites 73.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 48.1%

      \[\leadsto \mathsf{fma}\left(z \cdot 0.0625, t, c\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 44.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.25 \cdot \left(a \cdot b\right)\\ t_2 := \frac{z \cdot t}{16}\\ t_3 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-233}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* a b)))
        (t_2 (/ (* z t) 16.0))
        (t_3 (* 0.0625 (* t z))))
   (if (<= t_2 -1e+21)
     t_3
     (if (<= t_2 -5e-233)
       t_1
       (if (<= t_2 2e-14) (* x y) (if (<= t_2 5e+202) 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 * (a * b);
	double t_2 = (z * t) / 16.0;
	double t_3 = 0.0625 * (t * z);
	double tmp;
	if (t_2 <= -1e+21) {
		tmp = t_3;
	} else if (t_2 <= -5e-233) {
		tmp = t_1;
	} else if (t_2 <= 2e-14) {
		tmp = x * y;
	} else if (t_2 <= 5e+202) {
		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) * (a * b)
    t_2 = (z * t) / 16.0d0
    t_3 = 0.0625d0 * (t * z)
    if (t_2 <= (-1d+21)) then
        tmp = t_3
    else if (t_2 <= (-5d-233)) then
        tmp = t_1
    else if (t_2 <= 2d-14) then
        tmp = x * y
    else if (t_2 <= 5d+202) 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 * (a * b);
	double t_2 = (z * t) / 16.0;
	double t_3 = 0.0625 * (t * z);
	double tmp;
	if (t_2 <= -1e+21) {
		tmp = t_3;
	} else if (t_2 <= -5e-233) {
		tmp = t_1;
	} else if (t_2 <= 2e-14) {
		tmp = x * y;
	} else if (t_2 <= 5e+202) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = -0.25 * (a * b)
	t_2 = (z * t) / 16.0
	t_3 = 0.0625 * (t * z)
	tmp = 0
	if t_2 <= -1e+21:
		tmp = t_3
	elif t_2 <= -5e-233:
		tmp = t_1
	elif t_2 <= 2e-14:
		tmp = x * y
	elif t_2 <= 5e+202:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-0.25 * Float64(a * b))
	t_2 = Float64(Float64(z * t) / 16.0)
	t_3 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (t_2 <= -1e+21)
		tmp = t_3;
	elseif (t_2 <= -5e-233)
		tmp = t_1;
	elseif (t_2 <= 2e-14)
		tmp = Float64(x * y);
	elseif (t_2 <= 5e+202)
		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 * (a * b);
	t_2 = (z * t) / 16.0;
	t_3 = 0.0625 * (t * z);
	tmp = 0.0;
	if (t_2 <= -1e+21)
		tmp = t_3;
	elseif (t_2 <= -5e-233)
		tmp = t_1;
	elseif (t_2 <= 2e-14)
		tmp = x * y;
	elseif (t_2 <= 5e+202)
		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[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$3 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+21], t$95$3, If[LessEqual[t$95$2, -5e-233], t$95$1, If[LessEqual[t$95$2, 2e-14], N[(x * y), $MachinePrecision], If[LessEqual[t$95$2, 5e+202], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1e21 or 4.9999999999999999e202 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 28.8%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 28.1%

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

   if -1e21 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.00000000000000012e-233 or 2e-14 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.9999999999999999e202

   1. Initial program 97.4%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 28.3%

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

   if -5.00000000000000012e-233 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e-14

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 28.8%

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

Alternative 9: 43.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+168}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* a b))))
   (if (<= t_1 -2e+55) t_2 (if (<= t_1 2e+168) (* x y) t_2))))

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 t_2 = -0.25 * (a * b);
	double tmp;
	if (t_1 <= -2e+55) {
		tmp = t_2;
	} else if (t_1 <= 2e+168) {
		tmp = x * y;
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = (a * b) / 4.0d0
    t_2 = (-0.25d0) * (a * b)
    if (t_1 <= (-2d+55)) then
        tmp = t_2
    else if (t_1 <= 2d+168) then
        tmp = x * y
    else
        tmp = t_2
    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 t_2 = -0.25 * (a * b);
	double tmp;
	if (t_1 <= -2e+55) {
		tmp = t_2;
	} else if (t_1 <= 2e+168) {
		tmp = x * y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) / 4.0
	t_2 = -0.25 * (a * b)
	tmp = 0
	if t_1 <= -2e+55:
		tmp = t_2
	elif t_1 <= 2e+168:
		tmp = x * y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(a * b))
	tmp = 0.0
	if (t_1 <= -2e+55)
		tmp = t_2;
	elseif (t_1 <= 2e+168)
		tmp = Float64(x * y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) / 4.0;
	t_2 = -0.25 * (a * b);
	tmp = 0.0;
	if (t_1 <= -2e+55)
		tmp = t_2;
	elseif (t_1 <= 2e+168)
		tmp = x * y;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+55], t$95$2, If[LessEqual[t$95$1, 2e+168], N[(x * y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000002e55 or 1.9999999999999999e168 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.4%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 28.3%

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

   if -2.00000000000000002e55 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.9999999999999999e168

   1. Initial program 97.4%

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

   2. 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. Applied rewrites 73.8%

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

   6. Applied rewrites 74.1%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 28.8%

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

Alternative 10: 28.8% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.4%

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

2. 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. Applied rewrites 73.8%

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

6. Applied rewrites 74.1%

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

7. Taylor expanded in x around inf

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

9. Applied rewrites 28.8%

   \[\leadsto \color{blue}{x \cdot y} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(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))