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

Percentage Accurate: 97.7% → 97.7%

Time: 4.2s

Alternatives: 14

Speedup: 1.0×

• 32.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

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

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(a * Float64(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[(a * N[(b / 4.0), $MachinePrecision]), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 97.7

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

3. Applied rewrites 97.7%

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

Alternative 2: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(-0.25 \cdot a, b, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (- (fma y x c) (* 0.25 (* b a)))))
   (if (<= (* x y) -2e+54)
     t_1
     (if (<= (* x y) 1e+170) (fma (* 0.0625 t) z (fma (* -0.25 a) b c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, c) - (0.25 * (b * a));
	double tmp;
	if ((x * y) <= -2e+54) {
		tmp = t_1;
	} else if ((x * y) <= 1e+170) {
		tmp = fma((0.0625 * t), z, fma((-0.25 * a), b, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)))
	tmp = 0.0
	if (Float64(x * y) <= -2e+54)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+170)
		tmp = fma(Float64(0.0625 * t), z, fma(Float64(-0.25 * a), b, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+54], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+170], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000002e54 or 1.00000000000000003e170 < (*.f64 x y)

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.3

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

   4. Applied rewrites 85.3%

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

   if -2.0000000000000002e54 < (*.f64 x y) < 1.00000000000000003e170

   1. Initial program 99.2%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.2

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

   4. Applied rewrites 91.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 91.6

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

   6. Applied rewrites 91.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 91.6

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

   8. Applied rewrites 91.6%

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

Alternative 3: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+115}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (fma y x c))))
   (if (<= t_1 -1e+115)
     t_2
     (if (<= t_1 5e-35) (- (fma y x c) (* 0.25 (* b a))) 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, fma(y, x, c));
	double tmp;
	if (t_1 <= -1e+115) {
		tmp = t_2;
	} else if (t_1 <= 5e-35) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} 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, fma(y, x, c))
	tmp = 0.0
	if (t_1 <= -1e+115)
		tmp = t_2;
	elseif (t_1 <= 5e-35)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	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 + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+115], t$95$2, If[LessEqual[t$95$1, 5e-35], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1e115 or 4.99999999999999964e-35 < (/.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. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{b} + \left(\frac{c}{b} + \frac{x \cdot y}{b}\right)\right) - \frac{1}{4} \cdot a\right)} \]
   3. 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

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

   4. Applied rewrites 78.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. 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 82.9

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

   7. Applied rewrites 82.9%

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

   if -1e115 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.99999999999999964e-35

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 93.5

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

   4. Applied rewrites 93.5%

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

Alternative 4: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+114)
     (fma (* -0.25 a) b (* y x))
     (if (<= t_1 2e+153)
       (fma (* 0.0625 t) z (fma y x c))
       (fma -0.25 (* b a) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = fma((-0.25 * a), b, (y * x));
	} else if (t_1 <= 2e+153) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else {
		tmp = fma(-0.25, (b * a), c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = fma(Float64(-0.25 * a), b, Float64(y * x));
	elseif (t_1 <= 2e+153)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	else
		tmp = fma(-0.25, Float64(b * a), c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1e114

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in c around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if -1e114 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e153

   1. Initial program 99.1%

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

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(\left(\frac{1}{16} \cdot \frac{t \cdot z}{b} + \left(\frac{c}{b} + \frac{x \cdot y}{b}\right)\right) - \frac{1}{4} \cdot a\right)} \]
   3. 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

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{c + \left(\frac{1}{16} \cdot \left(t \cdot z\right) + x \cdot y\right)} \]
   6. 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 90.8

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

   7. Applied rewrites 90.8%

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

   if 2e153 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

Alternative 5: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25 \cdot a, b, y \cdot x\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+54}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* -0.25 a) b (* y x))))
   (if (<= (* x y) -2e+54)
     t_1
     (if (<= (* x y) -5e-170)
       (fma -0.25 (* b a) c)
       (if (<= (* x y) 1e+28) (fma (* 0.0625 t) z c) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((-0.25 * a), b, (y * x));
	double tmp;
	if ((x * y) <= -2e+54) {
		tmp = t_1;
	} else if ((x * y) <= -5e-170) {
		tmp = fma(-0.25, (b * a), c);
	} else if ((x * y) <= 1e+28) {
		tmp = fma((0.0625 * t), z, c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(-0.25 * a), b, Float64(y * x))
	tmp = 0.0
	if (Float64(x * y) <= -2e+54)
		tmp = t_1;
	elseif (Float64(x * y) <= -5e-170)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (Float64(x * y) <= 1e+28)
		tmp = fma(Float64(0.0625 * t), z, c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(-0.25 * a), $MachinePrecision] * b + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+54], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], -5e-170], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+28], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000002e54 or 9.99999999999999958e27 < (*.f64 x y)

   1. Initial program 95.8%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.4

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

   4. Applied rewrites 83.4%

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

   5. Taylor expanded in c around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -2.0000000000000002e54 < (*.f64 x y) < -5.0000000000000001e-170

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 61.8

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

   7. Applied rewrites 61.8%

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

   if -5.0000000000000001e-170 < (*.f64 x y) < 9.99999999999999958e27

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 64.8%

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

Alternative 6: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-170}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \cdot y \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+220}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma -0.25 (* b a) c)))
   (if (<= (* x y) -2e+54)
     (fma y x c)
     (if (<= (* x y) -5e-170)
       t_1
       (if (<= (* x y) 1e+28)
         (fma (* 0.0625 t) z c)
         (if (<= (* x y) 1e+220) t_1 (fma y x c)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(-0.25, (b * a), c);
	double tmp;
	if ((x * y) <= -2e+54) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= -5e-170) {
		tmp = t_1;
	} else if ((x * y) <= 1e+28) {
		tmp = fma((0.0625 * t), z, c);
	} else if ((x * y) <= 1e+220) {
		tmp = t_1;
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(-0.25, Float64(b * a), c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+54)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= -5e-170)
		tmp = t_1;
	elseif (Float64(x * y) <= 1e+28)
		tmp = fma(Float64(0.0625 * t), z, c);
	elseif (Float64(x * y) <= 1e+220)
		tmp = t_1;
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -2e+54], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-170], t$95$1, If[LessEqual[N[(x * y), $MachinePrecision], 1e+28], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+220], t$95$1, N[(y * x + c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2.0000000000000002e54 or 1e220 < (*.f64 x y)

   1. Initial program 94.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 74.6

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

   7. Applied rewrites 74.6%

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

   if -2.0000000000000002e54 < (*.f64 x y) < -5.0000000000000001e-170 or 9.99999999999999958e27 < (*.f64 x y) < 1e220

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.2

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

   4. Applied rewrites 73.2%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 54.1

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

   7. Applied rewrites 54.1%

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

   if -5.0000000000000001e-170 < (*.f64 x y) < 9.99999999999999958e27

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-*.f64 96.8

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

   6. Applied rewrites 96.8%

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

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 64.8%

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

Alternative 7: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+220}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+54)
   (fma y x c)
   (if (<= (* x y) 1e-182)
     (fma (* -0.25 a) b c)
     (if (<= (* x y) 1e+28)
       (fma (* t z) 0.0625 c)
       (if (<= (* x y) 1e+220) (fma -0.25 (* b a) c) (fma 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) <= -2e+54) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 1e-182) {
		tmp = fma((-0.25 * a), b, c);
	} else if ((x * y) <= 1e+28) {
		tmp = fma((t * z), 0.0625, c);
	} else if ((x * y) <= 1e+220) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+54)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 1e-182)
		tmp = fma(Float64(-0.25 * a), b, c);
	elseif (Float64(x * y) <= 1e+28)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (Float64(x * y) <= 1e+220)
		tmp = fma(-0.25, Float64(b * a), c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+54], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e-182], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+28], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+220], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.0000000000000002e54 or 1e220 < (*.f64 x y)

   1. Initial program 94.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.6

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

   4. Applied rewrites 85.6%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 74.6

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

   7. Applied rewrites 74.6%

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

   if -2.0000000000000002e54 < (*.f64 x y) < 1e-182

   1. Initial program 99.2%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.1

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

   4. Applied rewrites 67.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 63.8

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

   7. Applied rewrites 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 63.9

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

   9. Applied rewrites 63.9%

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

   if 1e-182 < (*.f64 x y) < 9.99999999999999958e27

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 60.0

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

   7. Applied rewrites 60.0%

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

   if 9.99999999999999958e27 < (*.f64 x y) < 1e220

   1. Initial program 99.3%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 77.8

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

   4. Applied rewrites 77.8%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 8: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+114)
     (fma (* -0.25 a) b c)
     (if (<= t_1 2e+153) (fma y x c) (fma -0.25 (* b a) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = fma((-0.25 * a), b, c);
	} else if (t_1 <= 2e+153) {
		tmp = fma(y, x, c);
	} else {
		tmp = fma(-0.25, (b * a), c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = fma(Float64(-0.25 * a), b, c);
	elseif (t_1 <= 2e+153)
		tmp = fma(y, x, c);
	else
		tmp = fma(-0.25, Float64(b * a), c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(y * x + c), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1e114

   1. Initial program 94.8%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.5

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

   4. Applied rewrites 85.5%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 74.1

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

   7. Applied rewrites 74.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 74.1

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

   9. Applied rewrites 74.1%

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

   if -1e114 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e153

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 60.5

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

   7. Applied rewrites 60.5%

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

   if 2e153 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.4%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 87.1

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 77.3

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

   7. Applied rewrites 77.3%

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

Alternative 9: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma -0.25 (* b a) c)))
   (if (<= t_1 -1e+114) t_2 (if (<= t_1 2e+153) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma(-0.25, (b * a), c);
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = t_2;
	} else if (t_1 <= 2e+153) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(-0.25, Float64(b * a), c)
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = t_2;
	elseif (t_1 <= 2e+153)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], t$95$2, If[LessEqual[t$95$1, 2e+153], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1e114 or 2e153 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.6%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto c - \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right) \cdot \left(a \cdot b\right) \]

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if -1e114 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e153

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.1

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 60.5

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

   7. Applied rewrites 60.5%

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

Alternative 10: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;\left(-0.25 \cdot b\right) \cdot a\\ \mathbf{elif}\;t\_1 \leq 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+114)
     (* (* -0.25 b) a)
     (if (<= t_1 1e+168) (fma y x c) (* -0.25 (* b a))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = (-0.25 * b) * a;
	} else if (t_1 <= 1e+168) {
		tmp = fma(y, x, c);
	} else {
		tmp = -0.25 * (b * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = Float64(Float64(-0.25 * b) * a);
	elseif (t_1 <= 1e+168)
		tmp = fma(y, x, c);
	else
		tmp = Float64(-0.25 * Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], N[(N[(-0.25 * b), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 1e+168], N[(y * x + c), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1e114

   1. Initial program 94.8%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 67.1

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

   4. Applied rewrites 67.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(b \cdot a\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 67.1

         \[\leadsto \left(-0.25 \cdot b\right) \cdot a \]

   6. Applied rewrites 67.1%

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

   if -1e114 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999993e167

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 60.4

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

   7. Applied rewrites 60.4%

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

   if 9.9999999999999993e167 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.0%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 73.2

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

   4. Applied rewrites 73.2%

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

Alternative 11: 63.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+168}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -1e+114) t_2 (if (<= t_1 1e+168) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -1e+114) {
		tmp = t_2;
	} else if (t_1 <= 1e+168) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -1e+114)
		tmp = t_2;
	elseif (t_1 <= 1e+168)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+114], t$95$2, If[LessEqual[t$95$1, 1e+168], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1e114 or 9.9999999999999993e167 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.5%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 69.8

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

   4. Applied rewrites 69.8%

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

   if -1e114 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999993e167

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 69.3

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

   4. Applied rewrites 69.3%

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

   5. Taylor expanded in a around 0

      \[\leadsto c + \color{blue}{x \cdot y} \]
   6. 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 60.4

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

   7. Applied rewrites 60.4%

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

Alternative 12: 49.4% accurate, 4.1× 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. Taylor expanded in z around 0

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 74.4

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

4. Applied rewrites 74.4%

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

5. Taylor expanded in a around 0

   \[\leadsto c + \color{blue}{x \cdot y} \]
6. 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 49.4

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

7. Applied rewrites 49.4%

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

Alternative 13: 42.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+54}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 20000000000000:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+54) (* y x) (if (<= (* x y) 20000000000000.0) c (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -2e+54) {
		tmp = y * x;
	} else if ((x * y) <= 20000000000000.0) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	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) <= (-2d+54)) then
        tmp = y * x
    else if ((x * y) <= 20000000000000.0d0) then
        tmp = c
    else
        tmp = y * x
    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) <= -2e+54) {
		tmp = y * x;
	} else if ((x * y) <= 20000000000000.0) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -2e+54:
		tmp = y * x
	elif (x * y) <= 20000000000000.0:
		tmp = c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+54)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 20000000000000.0)
		tmp = c;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -2e+54)
		tmp = y * x;
	elseif ((x * y) <= 20000000000000.0)
		tmp = c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+54], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 20000000000000.0], c, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.0000000000000002e54 or 2e13 < (*.f64 x y)

   1. Initial program 95.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 56.5

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

   4. Applied rewrites 56.5%

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

   if -2.0000000000000002e54 < (*.f64 x y) < 2e13

   1. Initial program 99.2%

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

   2. Taylor expanded in c around inf

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

   4. Applied rewrites 30.8%

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

Alternative 14: 22.7% accurate, 24.7× 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. Taylor expanded in c around inf

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

4. Applied rewrites 22.7%

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

Reproduce

herbie shell --seed 2025110 
(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))