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

Percentage Accurate: 97.5% → 97.5%

Time: 7.8s

Alternatives: 15

Speedup: 1.0×

• 32.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

Alternative 2: 67.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+151}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* 0.0625 t) z c)))
   (if (<= t_1 -4e+151)
     t_2
     (if (<= t_1 -2e-116)
       (fma y x c)
       (if (<= t_1 2e+30) (fma -0.25 (* b a) (* 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 = fma((0.0625 * t), z, c);
	double tmp;
	if (t_1 <= -4e+151) {
		tmp = t_2;
	} else if (t_1 <= -2e-116) {
		tmp = fma(y, x, c);
	} else if (t_1 <= 2e+30) {
		tmp = fma(-0.25, (b * a), (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 = fma(Float64(0.0625 * t), z, c)
	tmp = 0.0
	if (t_1 <= -4e+151)
		tmp = t_2;
	elseif (t_1 <= -2e-116)
		tmp = fma(y, x, c);
	elseif (t_1 <= 2e+30)
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	else
		tmp = t_2;
	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[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+151], t$95$2, If[LessEqual[t$95$1, -2e-116], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+30], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.00000000000000007e151 or 2e30 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

   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 80.7%

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

   if -4.00000000000000007e151 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2e-116

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.7%

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

   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 87.6

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

   8. Applied rewrites 87.6%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 73.6

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

   11. Applied rewrites 73.6%

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

   if -2e-116 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e30

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 74.6

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in z around 0

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

       1. lower-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 71.5

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

   11. Applied rewrites 71.5%

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

Alternative 3: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+191} \lor \neg \left(t\_1 \leq 10^{+147}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* x y) (/ (* z t) 16.0))))
   (if (or (<= t_1 -4e+191) (not (<= t_1 1e+147)))
     (fma (* t z) 0.0625 (* 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 <= -4e+191) || !(t_1 <= 1e+147)) {
		tmp = fma((t * z), 0.0625, (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 <= -4e+191) || !(t_1 <= 1e+147))
		tmp = fma(Float64(t * z), 0.0625, 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, -4e+191], N[Not[LessEqual[t$95$1, 1e+147]], $MachinePrecision]], N[(N[(t * z), $MachinePrecision] * 0.0625 + 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))) < -4.00000000000000029e191 or 9.9999999999999998e146 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.8%

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

   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%

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

   9. Taylor expanded in c around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 88.5

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

   11. Applied rewrites 88.5%

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

   if -4.00000000000000029e191 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 9.9999999999999998e146

   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 71.2

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

   5. Applied rewrites 71.2%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -4 \cdot 10^{+191} \lor \neg \left(x \cdot y + \frac{z \cdot t}{16} \leq 10^{+147}\right):\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_2\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, t\_2\right) + c\\ \end{array} \end{array} \]

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 -5e-75)
     (fma (* 0.0625 t) z (fma y x c))
     (if (<= t_1 2e+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 <= -5e-75) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if (t_1 <= 2e+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 <= -5e-75)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (t_1 <= 2e+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, -5e-75], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+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)) < -4.99999999999999979e-75

   1. Initial program 96.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   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 89.9

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

   8. Applied rewrites 89.9%

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

   if -4.99999999999999979e-75 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e30

   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 96.4

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

   5. Applied rewrites 96.4%

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

   if 2e30 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.3%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

Alternative 5: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-75} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+27}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e-75) (not (<= t_1 2e+27)))
     (fma (* 0.0625 t) z (fma y x c))
     (+ (fma y x (* -0.25 (* b a))) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e-75) || !(t_1 <= 2e+27)) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e-75) || !(t_1 <= 2e+27))
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-75], N[Not[LessEqual[t$95$1, 2e+27]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.99999999999999979e-75 or 2e27 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   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 89.4

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

   8. Applied rewrites 89.4%

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

   if -4.99999999999999979e-75 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e27

   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 96.4

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

   5. Applied rewrites 96.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{-75} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+27}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-75}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -5e-75)
     (fma (* 0.0625 t) z (fma y x c))
     (if (<= t_1 2e+27)
       (+ (fma y x (* -0.25 (* b a))) c)
       (+ (fma y x (* (* 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 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -5e-75) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if (t_1 <= 2e+27) {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	} else {
		tmp = fma(y, x, ((t * z) * 0.0625)) + 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 <= -5e-75)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (t_1 <= 2e+27)
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	else
		tmp = Float64(fma(y, x, Float64(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[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-75], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+27], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.99999999999999979e-75

   1. Initial program 96.2%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.6%

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

   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 89.9

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

   8. Applied rewrites 89.9%

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

   if -4.99999999999999979e-75 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e27

   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 96.4

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

   5. Applied rewrites 96.4%

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

   if 2e27 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 89.0

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

   7. Applied rewrites 89.0%

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

Alternative 7: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1.5 \cdot 10^{+83} \lor \neg \left(t\_1 \leq 10^{+197}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1.5e+83) (not (<= t_1 1e+197)))
     (fma -0.25 (* b a) (* y x))
     (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 <= -1.5e+83) || !(t_1 <= 1e+197)) {
		tmp = fma(-0.25, (b * a), (y * x));
	} 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 <= -1.5e+83) || !(t_1 <= 1e+197))
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	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, -1.5e+83], N[Not[LessEqual[t$95$1, 1e+197]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $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)) < -1.5e83 or 9.9999999999999995e196 < (/.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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.0%

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

   6. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in z around 0

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

       1. lower-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lift-*.f64 83.4

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

   11. Applied rewrites 83.4%

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

   if -1.5e83 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999995e196

   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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.3%

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

   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 91.5

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

   8. Applied rewrites 91.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1.5 \cdot 10^{+83} \lor \neg \left(\frac{a \cdot b}{4} \leq 10^{+197}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+151} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -4e+151) (not (<= t_1 2e+30)))
     (fma (* 0.0625 t) z c)
     (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -4e+151) || !(t_1 <= 2e+30)) {
		tmp = fma((0.0625 * t), z, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -4e+151) || !(t_1 <= 2e+30))
		tmp = fma(Float64(0.0625 * t), z, c);
	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[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+151], N[Not[LessEqual[t$95$1, 2e+30]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.00000000000000007e151 or 2e30 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.9%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.5%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

   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 80.7%

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

   if -4.00000000000000007e151 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2e30

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.1%

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

   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 71.6

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

   8. Applied rewrites 71.6%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 65.7

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

   11. Applied rewrites 65.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -4 \cdot 10^{+151} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+30}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+151} \lor \neg \left(t\_1 \leq 10^{+108}\right):\\ \;\;\;\;\left(0.0625 \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -4e+151) (not (<= t_1 1e+108)))
     (* (* 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 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -4e+151) || !(t_1 <= 1e+108)) {
		tmp = (0.0625 * t) * z;
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -4e+151) || !(t_1 <= 1e+108))
		tmp = Float64(Float64(0.0625 * t) * z);
	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[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e+151], N[Not[LessEqual[t$95$1, 1e+108]], $MachinePrecision]], N[(N[(0.0625 * t), $MachinePrecision] * z), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.00000000000000007e151 or 1e108 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 92.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{1}{16} \cdot \left(t \cdot z\right)} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if -4.00000000000000007e151 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1e108

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.6%

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

   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 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 65.4

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

   11. Applied rewrites 65.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -4 \cdot 10^{+151} \lor \neg \left(\frac{z \cdot t}{16} \leq 10^{+108}\right):\\ \;\;\;\;\left(0.0625 \cdot t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+33} \lor \neg \left(t\_1 \leq 10^{+197}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+33) (not (<= t_1 1e+197)))
     (* -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+33) || !(t_1 <= 1e+197)) {
		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+33) || !(t_1 <= 1e+197))
		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+33], N[Not[LessEqual[t$95$1, 1e+197]], $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)) < -4.99999999999999973e33 or 9.9999999999999995e196 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if -4.99999999999999973e33 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999995e196

   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 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.9%

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

   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 94.0

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

   8. Applied rewrites 94.0%

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

   9. Taylor expanded in z around 0

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

       1. *-commutative N/A

          \[\leadsto c + y \cdot x \]

       2. +-commutative N/A

          \[\leadsto y \cdot x + c \]

       3. lift-fma.f64 59.0

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

   11. Applied rewrites 59.0%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+33} \lor \neg \left(\frac{a \cdot b}{4} \leq 10^{+197}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-105} \lor \neg \left(b \leq 1.5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{b} - 0.25 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= b -1.6e-105) (not (<= b 1.5e-60)))
   (* (- (/ (fma (* 0.0625 t) z (fma y x c)) b) (* 0.25 a)) b)
   (+ (fma y x (* (* 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 ((b <= -1.6e-105) || !(b <= 1.5e-60)) {
		tmp = ((fma((0.0625 * t), z, fma(y, x, c)) / b) - (0.25 * a)) * b;
	} else {
		tmp = fma(y, x, ((t * z) * 0.0625)) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((b <= -1.6e-105) || !(b <= 1.5e-60))
		tmp = Float64(Float64(Float64(fma(Float64(0.0625 * t), z, fma(y, x, c)) / b) - Float64(0.25 * a)) * b);
	else
		tmp = Float64(fma(y, x, Float64(Float64(t * z) * 0.0625)) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[b, -1.6e-105], N[Not[LessEqual[b, 1.5e-60]], $MachinePrecision]], N[(N[(N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(0.25 * a), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.59999999999999991e-105 or 1.50000000000000009e-60 < b

   1. Initial program 96.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.5%

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

   if -1.59999999999999991e-105 < b < 1.50000000000000009e-60

   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 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.6

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

   5. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 94.6

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

   7. Applied rewrites 94.6%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-105} \lor \neg \left(b \leq 1.5 \cdot 10^{-60}\right):\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{b} - 0.25 \cdot a\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{elif}\;c \leq 7.3 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -3.7e+99)
   (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)
   (if (<= c 7.3e+71)
     (fma (* 0.0625 t) z (fma -0.25 (* b a) (* y x)))
     (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 tmp;
	if (c <= -3.7e+99) {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	} else if (c <= 7.3e+71) {
		tmp = fma((0.0625 * t), z, fma(-0.25, (b * a), (y * x)));
	} else {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= -3.7e+99)
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	elseif (c <= 7.3e+71)
		tmp = fma(Float64(0.0625 * t), z, fma(-0.25, Float64(b * a), Float64(y * x)));
	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_] := If[LessEqual[c, -3.7e+99], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[c, 7.3e+71], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.7000000000000001e99

   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 92.1

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

   5. Applied rewrites 92.1%

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

   if -3.7000000000000001e99 < c < 7.29999999999999996e71

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. *-commutative N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 93.8

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

   8. Applied rewrites 93.8%

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

   if 7.29999999999999996e71 < c

   1. Initial program 91.8%

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.7%

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

   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 89.7

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

   8. Applied rewrites 89.7%

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

Alternative 13: 41.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+153} \lor \neg \left(x \cdot y \leq 10^{+147}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -1e+153) (not (<= (* x y) 1e+147))) (* 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+153) || !((x * y) <= 1e+147)) {
		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+153)) .or. (.not. ((x * y) <= 1d+147))) 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+153) || !((x * y) <= 1e+147)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -1e+153) or not ((x * y) <= 1e+147):
		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+153) || !(Float64(x * y) <= 1e+147))
		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+153) || ~(((x * y) <= 1e+147)))
		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+153], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+147]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1e153 or 9.9999999999999998e146 < (*.f64 x y)

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -1e153 < (*.f64 x y) < 9.9999999999999998e146

   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 c around inf

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

   5. Applied rewrites 29.8%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+153} \lor \neg \left(x \cdot y \leq 10^{+147}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.1% 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 97.7%

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

3. Taylor expanded in 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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 81.3%

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

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 78.4

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

8. Applied rewrites 78.4%

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

9. Taylor expanded in z around 0

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

    1. *-commutative N/A

       \[\leadsto c + y \cdot x \]

    2. +-commutative N/A

       \[\leadsto y \cdot x + c \]

    3. lift-fma.f64 49.0

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

11. Applied rewrites 49.0%

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

Alternative 15: 22.4% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.7%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 22.2%

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

Reproduce

herbie shell --seed 2025051 
(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))