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

Percentage Accurate: 97.5% → 98.9%

Time: 7.2s

Alternatives: 15

Speedup: 1.6×

• 33.0% 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: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 86.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

Alternative 2: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+259} \lor \neg \left(t\_1 \leq 10^{+173}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* x y) (/ (* z t) 16.0))))
   (if (or (<= t_1 -2e+259) (not (<= t_1 1e+173)))
     (fma (* 0.0625 z) t (* y x))
     (fma (* -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 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if ((t_1 <= -2e+259) || !(t_1 <= 1e+173)) {
		tmp = fma((0.0625 * z), t, (y * x));
	} else {
		tmp = fma((-0.25 * b), a, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if ((t_1 <= -2e+259) || !(t_1 <= 1e+173))
		tmp = fma(Float64(0.0625 * z), t, Float64(y * x));
	else
		tmp = fma(Float64(-0.25 * b), a, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+259], N[Not[LessEqual[t$95$1, 1e+173]], $MachinePrecision]], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -2e259 or 1e173 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 93.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.1%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, x \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.9%

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

   if -2e259 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1e173

   1. Initial program 99.9%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 88.5%

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

   12. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, c\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 75.1%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -2 \cdot 10^{+259} \lor \neg \left(x \cdot y + \frac{z \cdot t}{16} \leq 10^{+173}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -5e+210)
     t_2
     (if (<= t_1 1e-77)
       (fma y x c)
       (if (<= t_1 4e+164) (* (* t z) 0.0625) 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 * (b * a);
	double tmp;
	if (t_1 <= -5e+210) {
		tmp = t_2;
	} else if (t_1 <= 1e-77) {
		tmp = fma(y, x, c);
	} else if (t_1 <= 4e+164) {
		tmp = (t * z) * 0.0625;
	} 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(b * a))
	tmp = 0.0
	if (t_1 <= -5e+210)
		tmp = t_2;
	elseif (t_1 <= 1e-77)
		tmp = fma(y, x, c);
	elseif (t_1 <= 4e+164)
		tmp = Float64(Float64(t * z) * 0.0625);
	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[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+210], t$95$2, If[LessEqual[t$95$1, 1e-77], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$1, 4e+164], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999998e210 or 4e164 < (/.f64 (*.f64 a b) #s(literal 4 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 a around inf

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

   5. Applied rewrites 81.2%

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

   if -4.9999999999999998e210 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999993e-78

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 91.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.2%

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

   if 9.9999999999999993e-78 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e164

   1. Initial program 97.7%

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

   5. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+210}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;\frac{a \cdot b}{4} \leq 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;\frac{a \cdot b}{4} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \left(-0.25 \cdot a\right) \cdot b\right)\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5e+210)
     (fma (* 0.0625 z) t (* (* -0.25 a) b))
     (if (<= t_1 0.0002)
       (fma y x (fma (* t z) 0.0625 c))
       (fma (* -0.25 b) a (* 0.0625 (* t z)))))))

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 <= -5e+210) {
		tmp = fma((0.0625 * z), t, ((-0.25 * a) * b));
	} else if (t_1 <= 0.0002) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma((-0.25 * b), a, (0.0625 * (t * z)));
	}
	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 <= -5e+210)
		tmp = fma(Float64(0.0625 * z), t, Float64(Float64(-0.25 * a) * b));
	elseif (t_1 <= 0.0002)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(Float64(-0.25 * b), a, Float64(0.0625 * Float64(t * z)));
	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, -5e+210], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(N[(-0.25 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999998e210

   1. Initial program 90.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in a around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, \frac{-1}{4} \cdot \left(a \cdot b\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 90.2%

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

   if -4.9999999999999998e210 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000001e-4

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 90.6%

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

   if 2.0000000000000001e-4 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.7%

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

   12. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 84.6%

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

Alternative 5: 84.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \mathbf{elif}\;t\_1 \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, 0.0625 \cdot \left(t \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5e+210)
     (fma (* -0.25 b) a c)
     (if (<= t_1 0.0002)
       (fma y x (fma (* t z) 0.0625 c))
       (fma (* -0.25 b) a (* 0.0625 (* t z)))))))

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 <= -5e+210) {
		tmp = fma((-0.25 * b), a, c);
	} else if (t_1 <= 0.0002) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else {
		tmp = fma((-0.25 * b), a, (0.0625 * (t * z)));
	}
	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 <= -5e+210)
		tmp = fma(Float64(-0.25 * b), a, c);
	elseif (t_1 <= 0.0002)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	else
		tmp = fma(Float64(-0.25 * b), a, Float64(0.0625 * Float64(t * z)));
	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, -5e+210], N[(N[(-0.25 * b), $MachinePrecision] * a + c), $MachinePrecision], If[LessEqual[t$95$1, 0.0002], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999998e210

   1. Initial program 90.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 94.0%

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

   12. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, c\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 87.8%

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

   if -4.9999999999999998e210 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000001e-4

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 90.6%

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

   if 2.0000000000000001e-4 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 90.7%

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

   12. Taylor expanded in z around inf

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, \frac{1}{16} \cdot \left(t \cdot z\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 84.6%

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

Alternative 6: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+164}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+210) (not (<= t_1 4e+164)))
     (fma (* -0.25 b) a c)
     (fma y x (fma (* t z) 0.0625 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 <= -5e+210) || !(t_1 <= 4e+164)) {
		tmp = fma((-0.25 * b), a, c);
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, 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 <= -5e+210) || !(t_1 <= 4e+164))
		tmp = fma(Float64(-0.25 * b), a, c);
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, 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[Or[LessEqual[t$95$1, -5e+210], N[Not[LessEqual[t$95$1, 4e+164]], $MachinePrecision]], N[(N[(-0.25 * b), $MachinePrecision] * a + c), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999998e210 or 4e164 < (/.f64 (*.f64 a b) #s(literal 4 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 b around inf

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 95.6%

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

   12. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, c\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 86.9%

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

   if -4.9999999999999998e210 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e164

   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 a around 0

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

   5. Applied rewrites 87.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+210} \lor \neg \left(\frac{a \cdot b}{4} \leq 4 \cdot 10^{+164}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+259} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+151}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -2e+259) (not (<= t_1 2e+151)))
     (* (* t z) 0.0625)
     (fma (* -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+259) || !(t_1 <= 2e+151)) {
		tmp = (t * z) * 0.0625;
	} else {
		tmp = fma((-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+259) || !(t_1 <= 2e+151))
		tmp = Float64(Float64(t * z) * 0.0625);
	else
		tmp = fma(Float64(-0.25 * 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+259], N[Not[LessEqual[t$95$1, 2e+151]], $MachinePrecision]], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], N[(N[(-0.25 * b), $MachinePrecision] * a + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e259 or 2.00000000000000003e151 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 91.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 83.7%

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

   if -2e259 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.00000000000000003e151

   1. Initial program 99.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 74.3%

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

   12. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, c\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 63.7%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -2 \cdot 10^{+259} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+151}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+259}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -2e+259)
     (* (* t z) 0.0625)
     (if (<= t_1 5e+58) (fma (* -0.25 b) a c) (fma (* 0.0625 z) t 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+259) {
		tmp = (t * z) * 0.0625;
	} else if (t_1 <= 5e+58) {
		tmp = fma((-0.25 * b), a, c);
	} else {
		tmp = fma((0.0625 * z), t, 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+259)
		tmp = Float64(Float64(t * z) * 0.0625);
	elseif (t_1 <= 5e+58)
		tmp = fma(Float64(-0.25 * b), a, c);
	else
		tmp = fma(Float64(0.0625 * z), t, 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[LessEqual[t$95$1, -2e+259], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], If[LessEqual[t$95$1, 5e+58], N[(N[(-0.25 * b), $MachinePrecision] * a + c), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e259

   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

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

   5. Applied rewrites 86.3%

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

   if -2e259 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.99999999999999986e58

   1. Initial program 99.4%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 72.9%

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

   12. Taylor expanded in z around 0

       \[\leadsto \mathsf{fma}\left(\frac{-1}{4} \cdot b, a, c\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 64.3%

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

   if 4.99999999999999986e58 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.8%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in c around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, c\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 77.9%

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

Alternative 9: 59.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+210} \lor \neg \left(t\_1 \leq 0.0002\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+210) (not (<= t_1 0.0002)))
     (* -0.25 (* b a))
     (fma 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 <= -5e+210) || !(t_1 <= 0.0002)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = fma(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 <= -5e+210) || !(t_1 <= 0.0002))
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = fma(y, x, 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[Or[LessEqual[t$95$1, -5e+210], N[Not[LessEqual[t$95$1, 0.0002]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.9999999999999998e210 or 2.0000000000000001e-4 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 93.5%

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

   5. Applied rewrites 68.6%

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

   if -4.9999999999999998e210 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000001e-4

   1. Initial program 99.4%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 57.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+210} \lor \neg \left(\frac{a \cdot b}{4} \leq 0.0002\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -100000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \mathsf{fma}\left(0.0625 \cdot z, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -100000000000.0)
   (fma y x (fma (* t z) 0.0625 c))
   (if (<= (* x y) 1e+177)
     (fma (* -0.25 b) a (fma (* 0.0625 z) t c))
     (fma (* 0.0625 z) t (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -100000000000.0) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else if ((x * y) <= 1e+177) {
		tmp = fma((-0.25 * b), a, fma((0.0625 * z), t, c));
	} else {
		tmp = fma((0.0625 * z), t, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -100000000000.0)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	elseif (Float64(x * y) <= 1e+177)
		tmp = fma(Float64(-0.25 * b), a, fma(Float64(0.0625 * z), t, c));
	else
		tmp = fma(Float64(0.0625 * z), t, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -100000000000.0], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+177], N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e11

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.0%

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

   if -1e11 < (*.f64 x y) < 1e177

   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 b around inf

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 95.0%

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

   if 1e177 < (*.f64 x y)

   1. Initial program 87.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, x \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.0%

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

Alternative 11: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -100000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(0.0625, z \cdot t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -100000000000.0)
   (fma y x (fma (* t z) 0.0625 c))
   (if (<= (* x y) 1e+177)
     (fma (* -0.25 a) b (fma 0.0625 (* z t) c))
     (fma (* 0.0625 z) t (* y x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -100000000000.0) {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	} else if ((x * y) <= 1e+177) {
		tmp = fma((-0.25 * a), b, fma(0.0625, (z * t), c));
	} else {
		tmp = fma((0.0625 * z), t, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -100000000000.0)
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	elseif (Float64(x * y) <= 1e+177)
		tmp = fma(Float64(-0.25 * a), b, fma(0.0625, Float64(z * t), c));
	else
		tmp = fma(Float64(0.0625 * z), t, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -100000000000.0], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+177], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(0.0625 * N[(z * t), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e11

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.0%

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

   if -1e11 < (*.f64 x y) < 1e177

   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 b around inf

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

   5. Applied rewrites 87.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 94.9%

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

   if 1e177 < (*.f64 x y)

   1. Initial program 87.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, x \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.0%

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

Alternative 12: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;x \cdot y \leq -100000000000:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+177}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t z) 0.0625 c)))
   (if (<= (* x y) -100000000000.0)
     (fma y x t_1)
     (if (<= (* x y) 1e+177)
       (fma -0.25 (* b a) t_1)
       (fma (* 0.0625 z) t (* y x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * z), 0.0625, c);
	double tmp;
	if ((x * y) <= -100000000000.0) {
		tmp = fma(y, x, t_1);
	} else if ((x * y) <= 1e+177) {
		tmp = fma(-0.25, (b * a), t_1);
	} else {
		tmp = fma((0.0625 * z), t, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (Float64(x * y) <= -100000000000.0)
		tmp = fma(y, x, t_1);
	elseif (Float64(x * y) <= 1e+177)
		tmp = fma(-0.25, Float64(b * a), t_1);
	else
		tmp = fma(Float64(0.0625 * z), t, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -100000000000.0], N[(y * x + t$95$1), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+177], N[(-0.25 * N[(b * a), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1e11

   1. Initial program 93.5%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 85.0%

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

   if -1e11 < (*.f64 x y) < 1e177

   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 0

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

   5. Applied rewrites 94.9%

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

   if 1e177 < (*.f64 x y)

   1. Initial program 87.5%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{16} \cdot z, t, x \cdot y\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 94.0%

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

Alternative 13: 40.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1000 \lor \neg \left(x \cdot y \leq 10^{+173}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -1000.0) (not (<= (* x y) 1e+173))) (* 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) <= -1000.0) || !((x * y) <= 1e+173)) {
		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) <= (-1000.0d0)) .or. (.not. ((x * y) <= 1d+173))) 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) <= -1000.0) || !((x * y) <= 1e+173)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -1000.0) or not ((x * y) <= 1e+173):
		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) <= -1000.0) || !(Float64(x * y) <= 1e+173))
		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) <= -1000.0) || ~(((x * y) <= 1e+173)))
		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], -1000.0], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+173]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e3 or 1e173 < (*.f64 x y)

   1. Initial program 91.4%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 60.5%

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

   if -1e3 < (*.f64 x y) < 1e173

   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 c around inf

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

   5. Applied rewrites 25.7%

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

4. Final simplification 36.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1000 \lor \neg \left(x \cdot y \leq 10^{+173}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, x, c\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

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

3. Taylor expanded in a around 0

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

5. Applied rewrites 71.3%

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

6. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
7. Step-by-step derivation

8. Applied rewrites 42.6%

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]

9. Final simplification 42.6%

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
10. Add Preprocessing

Alternative 15: 22.3% 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 97.2%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 20.9%

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

Reproduce

herbie shell --seed 2025025 
(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))