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

Percentage Accurate: 97.7% → 97.7%

Time: 4.1s

Alternatives: 11

Speedup: 1.0×

• 32.2% 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: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

Alternative 2: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+124} \lor \neg \left(t\_1 \leq 10^{+181}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \end{array} \]

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+124) (not (<= t_1 1e+181)))
     (fma (* 0.0625 t) z (* y x))
     (+ (* -0.25 (* b a)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if ((t_1 <= -2e+124) || !(t_1 <= 1e+181)) {
		tmp = fma((0.0625 * t), z, (y * x));
	} else {
		tmp = (-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+124) || !(t_1 <= 1e+181))
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	else
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+124], N[Not[LessEqual[t$95$1, 1e+181]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.9999999999999999e124 or 9.9999999999999992e180 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   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 a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.4%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 93.3

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

   8. Applied rewrites 93.3%

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

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 85.4

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

   11. Applied rewrites 85.4%

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

   if -1.9999999999999999e124 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 9.9999999999999992e180

   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 a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -2 \cdot 10^{+124} \lor \neg \left(x \cdot y + \frac{z \cdot t}{16} \leq 10^{+181}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+135}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-146}:\\ \;\;\;\;c\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* t z) 0.0625)))
   (if (<= t_1 -1e+135)
     t_2
     (if (<= t_1 2e-146) c (if (<= t_1 5e+30) (* y x) t_2)))))

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 t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = t_2;
	} else if (t_1 <= 2e-146) {
		tmp = c;
	} else if (t_1 <= 5e+30) {
		tmp = y * x;
	} 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 = (z * t) / 16.0d0
    t_2 = (t * z) * 0.0625d0
    if (t_1 <= (-1d+135)) then
        tmp = t_2
    else if (t_1 <= 2d-146) then
        tmp = c
    else if (t_1 <= 5d+30) then
        tmp = y * x
    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 = (z * t) / 16.0;
	double t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -1e+135) {
		tmp = t_2;
	} else if (t_1 <= 2e-146) {
		tmp = c;
	} else if (t_1 <= 5e+30) {
		tmp = y * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (z * t) / 16.0
	t_2 = (t * z) * 0.0625
	tmp = 0
	if t_1 <= -1e+135:
		tmp = t_2
	elif t_1 <= 2e-146:
		tmp = c
	elif t_1 <= 5e+30:
		tmp = y * x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(t * z) * 0.0625)
	tmp = 0.0
	if (t_1 <= -1e+135)
		tmp = t_2;
	elseif (t_1 <= 2e-146)
		tmp = c;
	elseif (t_1 <= 5e+30)
		tmp = Float64(y * x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) / 16.0;
	t_2 = (t * z) * 0.0625;
	tmp = 0.0;
	if (t_1 <= -1e+135)
		tmp = t_2;
	elseif (t_1 <= 2e-146)
		tmp = c;
	elseif (t_1 <= 5e+30)
		tmp = y * x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+135], t$95$2, If[LessEqual[t$95$1, 2e-146], c, If[LessEqual[t$95$1, 5e+30], N[(y * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -9.99999999999999962e134 or 4.9999999999999998e30 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 63.0

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

   5. Applied rewrites 63.0%

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

   if -9.99999999999999962e134 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.00000000000000005e-146

   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 38.7%

      \[\leadsto \color{blue}{c} \]

   if 2.00000000000000005e-146 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.9999999999999998e30

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

Alternative 4: 89.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -3.5 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_2\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, t\_2\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -3.5e+139)
     (fma (* 0.0625 t) z (fma y x c))
     (if (<= t_1 5e+30) (+ (fma y x t_2) c) (+ (fma (* 0.0625 t) z t_2) 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 t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -3.5e+139) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if (t_1 <= 5e+30) {
		tmp = fma(y, x, t_2) + c;
	} else {
		tmp = fma((0.0625 * t), z, t_2) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -3.5e+139)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (t_1 <= 5e+30)
		tmp = Float64(fma(y, x, t_2) + c);
	else
		tmp = Float64(fma(Float64(0.0625 * t), z, t_2) + 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]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -3.5e+139], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+30], N[(N[(y * x + t$95$2), $MachinePrecision] + c), $MachinePrecision], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + t$95$2), $MachinePrecision] + c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -3.49999999999999978e139

   1. Initial program 98.1%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 98.1

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

   8. Applied rewrites 98.1%

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

   if -3.49999999999999978e139 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.9999999999999998e30

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 94.3

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

   5. Applied rewrites 94.3%

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

   if 4.9999999999999998e30 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 96.0%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 90.4

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

   5. Applied rewrites 90.4%

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

Alternative 5: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+134} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -2e+134) (not (<= t_1 5e+198)))
     (+ (fma y x (* -0.25 (* b a))) c)
     (fma (* 0.0625 t) z (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 <= -2e+134) || !(t_1 <= 5e+198)) {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	} else {
		tmp = fma((0.0625 * t), z, 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 <= -2e+134) || !(t_1 <= 5e+198))
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, 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, -2e+134], N[Not[LessEqual[t$95$1, 5e+198]], $MachinePrecision]], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999984e134 or 5.00000000000000049e198 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

   if -1.99999999999999984e134 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000049e198

   1. Initial program 99.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 92.6

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

   8. Applied rewrites 92.6%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -2 \cdot 10^{+134} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+198}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+134} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+212}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -2e+134) (not (<= t_1 4e+212)))
     (+ (* -0.25 (* b a)) c)
     (fma (* 0.0625 t) z (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 <= -2e+134) || !(t_1 <= 4e+212)) {
		tmp = (-0.25 * (b * a)) + c;
	} else {
		tmp = fma((0.0625 * t), z, 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 <= -2e+134) || !(t_1 <= 4e+212))
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, 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, -2e+134], N[Not[LessEqual[t$95$1, 4e+212]], $MachinePrecision]], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999984e134 or 3.9999999999999996e212 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 98.3%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -1.99999999999999984e134 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 3.9999999999999996e212

   1. Initial program 99.0%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -2 \cdot 10^{+134} \lor \neg \left(\frac{a \cdot b}{4} \leq 4 \cdot 10^{+212}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-36}:\\ \;\;\;\;\left(\frac{t\_1}{a} - 0.25 \cdot b\right) \cdot a\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* 0.0625 t) z (fma y x c))))
   (if (<= a -3.5e-36)
     (* (- (/ t_1 a) (* 0.25 b)) a)
     (if (<= a 6.8e-32) t_1 (+ (fma y x (* -0.25 (* b a))) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((0.0625 * t), z, fma(y, x, c));
	double tmp;
	if (a <= -3.5e-36) {
		tmp = ((t_1 / a) - (0.25 * b)) * a;
	} else if (a <= 6.8e-32) {
		tmp = t_1;
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, fma(y, x, c))
	tmp = 0.0
	if (a <= -3.5e-36)
		tmp = Float64(Float64(Float64(t_1 / a) - Float64(0.25 * b)) * a);
	elseif (a <= 6.8e-32)
		tmp = t_1;
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -3.5e-36], N[(N[(N[(t$95$1 / a), $MachinePrecision] - N[(0.25 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[a, 6.8e-32], t$95$1, N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.5e-36

   1. Initial program 98.8%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.6%

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

   if -3.5e-36 < a < 6.79999999999999956e-32

   1. Initial program 98.3%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 91.8

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

   8. Applied rewrites 91.8%

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

   if 6.79999999999999956e-32 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 72.7

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

   5. Applied rewrites 72.7%

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

Alternative 8: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -1e+67) (not (<= (* x y) 5e+100)))
   (fma (* 0.0625 t) z (fma y x c))
   (+ (fma (* -0.25 b) a (* (* t z) 0.0625)) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -1e+67) || !((x * y) <= 5e+100)) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = fma((-0.25 * b), a, ((t * z) * 0.0625)) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -1e+67) || !(Float64(x * y) <= 5e+100))
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = Float64(fma(Float64(-0.25 * b), a, Float64(Float64(t * z) * 0.0625)) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -1e+67], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+100]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999983e66 or 4.9999999999999999e100 < (*.f64 x y)

   1. Initial program 96.9%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 91.9

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

   8. Applied rewrites 91.9%

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

   if -9.99999999999999983e66 < (*.f64 x y) < 4.9999999999999999e100

   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

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 95.4

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

   7. Applied rewrites 95.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+67} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+100}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+236} \lor \neg \left(x \cdot y \leq 10^{+179}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -2e+236) (not (<= (* x y) 1e+179)))
   (* y x)
   (fma (* 0.0625 t) z c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -2e+236) || !((x * y) <= 1e+179)) {
		tmp = y * x;
	} else {
		tmp = fma((0.0625 * t), z, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -2e+236) || !(Float64(x * y) <= 1e+179))
		tmp = Float64(y * x);
	else
		tmp = fma(Float64(0.0625 * t), z, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -2e+236], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+179]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000011e236 or 9.9999999999999998e178 < (*.f64 x y)

   1. Initial program 96.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.00000000000000011e236 < (*.f64 x y) < 9.9999999999999998e178

   1. Initial program 99.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-+r+ N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 75.9

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

   8. Applied rewrites 75.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+236} \lor \neg \left(x \cdot y \leq 10^{+179}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+67} \lor \neg \left(x \cdot y \leq 10^{+126}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -1e+67) (not (<= (* x y) 1e+126))) (* 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) <= -1e+67) || !((x * y) <= 1e+126)) {
		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) <= (-1d+67)) .or. (.not. ((x * y) <= 1d+126))) 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) <= -1e+67) || !((x * y) <= 1e+126)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -1e+67) or not ((x * y) <= 1e+126):
		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) <= -1e+67) || !(Float64(x * y) <= 1e+126))
		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) <= -1e+67) || ~(((x * y) <= 1e+126)))
		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], -1e+67], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+126]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -9.99999999999999983e66 or 9.99999999999999925e125 < (*.f64 x y)

   1. Initial program 96.7%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if -9.99999999999999983e66 < (*.f64 x y) < 9.99999999999999925e125

   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 36.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+67} \lor \neg \left(x \cdot y \leq 10^{+126}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 22.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 25.5%

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

Reproduce

herbie shell --seed 2025057 
(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))