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

Percentage Accurate: 97.7% → 98.9%

Time: 4.6s

Alternatives: 13

Speedup: 1.1×

• 65.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. add-flip-rev N/A

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

   9. lower-fma.f64 N/A

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

   10. add-flip-rev N/A

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

   11. associate--r- N/A

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

   12. sub-flip N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

3. Applied rewrites 98.9%

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

Alternative 2: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. add-flip-rev N/A

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

   9. lower-fma.f64 N/A

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

   10. add-flip-rev N/A

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

   11. associate--r- N/A

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

   12. sub-flip N/A

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

   13. lift-/.f64 N/A

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

   14. mult-flip N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. associate-*l* N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

3. Applied rewrites 98.9%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. associate-+l+ N/A

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

   6. +-commutative N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 98.8%

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 98.8%

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

   17. lower-fma.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. associate-*r* N/A

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

   20. lower-fma.f64 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f64 98.8%

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

5. Applied rewrites 98.8%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift--.f64 N/A

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

   3. associate-+l- N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. associate--l+ N/A

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

   7. lift-/.f64 N/A

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

   8. mult-flip N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. sub-flip N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-fma.f64 N/A

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

3. Applied rewrites 98.8%

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

Alternative 4: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, \mathsf{fma}\left(-0.25, a \cdot b, x \cdot y\right)\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(0.0625 \cdot z, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot -0.25, a, c\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -0.02)
     (fma (* t 0.0625) z (fma -0.25 (* a b) (* x y)))
     (if (<= t_1 5e+47)
       (fma y x (fma (* 0.0625 z) t c))
       (fma y x (fma (* b -0.25) 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 <= -0.02) {
		tmp = fma((t * 0.0625), z, fma(-0.25, (a * b), (x * y)));
	} else if (t_1 <= 5e+47) {
		tmp = fma(y, x, fma((0.0625 * z), t, c));
	} else {
		tmp = fma(y, x, fma((b * -0.25), 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 <= -0.02)
		tmp = fma(Float64(t * 0.0625), z, fma(-0.25, Float64(a * b), Float64(x * y)));
	elseif (t_1 <= 5e+47)
		tmp = fma(y, x, fma(Float64(0.0625 * z), t, c));
	else
		tmp = fma(y, x, fma(Float64(b * -0.25), 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, -0.02], N[(N[(t * 0.0625), $MachinePrecision] * z + N[(-0.25 * N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+47], N[(y * x + N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -0.0200000000000000004

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 74.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 74.1%

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

   8. Applied rewrites 74.1%

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

   9. Taylor expanded in c around 0

      \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + x \cdot y}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 77.3%

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

   11. Applied rewrites 77.3%

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

   if -0.0200000000000000004 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000022e47

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 74.2%

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

   if 5.00000000000000022e47 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.4%

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

   6. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.5%

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

   8. Applied rewrites 74.5%

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

Alternative 5: 90.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot -0.25, a, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(0.0625 \cdot z, t, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma y x (fma (* b -0.25) a c))))
   (if (<= t_1 -2e+24)
     t_2
     (if (<= t_1 5e+47) (fma y x (fma (* 0.0625 z) t c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma(y, x, fma((b * -0.25), a, c));
	double tmp;
	if (t_1 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5e+47) {
		tmp = fma(y, x, fma((0.0625 * z), t, 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(y, x, fma(Float64(b * -0.25), a, c))
	tmp = 0.0
	if (t_1 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5e+47)
		tmp = fma(y, x, fma(Float64(0.0625 * z), t, 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[(y * x + N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+24], t$95$2, If[LessEqual[t$95$1, 5e+47], N[(y * x + N[(N[(0.0625 * z), $MachinePrecision] * t + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2e24 or 5.00000000000000022e47 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.4%

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

   6. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.5%

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

   8. Applied rewrites 74.5%

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

   if -2e24 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000022e47

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in a around 0

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

   6. Applied rewrites 74.2%

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

Alternative 6: 90.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot -0.25, a, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma y x (fma (* b -0.25) a c))))
   (if (<= t_1 -2e+24)
     t_2
     (if (<= t_1 5e+47) (fma (* z 0.0625) t (fma x y c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma(y, x, fma((b * -0.25), a, c));
	double tmp;
	if (t_1 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5e+47) {
		tmp = fma((z * 0.0625), t, fma(x, y, 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(y, x, fma(Float64(b * -0.25), a, c))
	tmp = 0.0
	if (t_1 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5e+47)
		tmp = fma(Float64(z * 0.0625), t, fma(x, y, 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[(y * x + N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+24], t$95$2, If[LessEqual[t$95$1, 5e+47], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2e24 or 5.00000000000000022e47 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.4%

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

   6. Applied rewrites 74.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.5%

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

   8. Applied rewrites 74.5%

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

   if -2e24 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000022e47

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 74.1%

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

   6. Applied rewrites 74.1%

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

Alternative 7: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := -0.25 \cdot \left(a \cdot b\right)\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+147}:\\ \;\;\;\;t\_1 + c\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+201}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* -0.25 (* a b))) (t_2 (/ (* a b) 4.0)))
   (if (<= t_2 -5e+147)
     (+ t_1 c)
     (if (<= t_2 4e+201) (fma (* z 0.0625) t (fma x y c)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = -0.25 * (a * b);
	double t_2 = (a * b) / 4.0;
	double tmp;
	if (t_2 <= -5e+147) {
		tmp = t_1 + c;
	} else if (t_2 <= 4e+201) {
		tmp = fma((z * 0.0625), t, fma(x, y, c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(-0.25 * Float64(a * b))
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_2 <= -5e+147)
		tmp = Float64(t_1 + c);
	elseif (t_2 <= 4e+201)
		tmp = fma(Float64(z * 0.0625), t, fma(x, y, c));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+147], N[(t$95$1 + c), $MachinePrecision], If[LessEqual[t$95$2, 4e+201], N[(N[(z * 0.0625), $MachinePrecision] * t + N[(x * y + c), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000002e147

   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

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.1%

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

   if -5.0000000000000002e147 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.00000000000000015e201

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

      1. lift-+.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. +-commutative N/A

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

      14. associate-+l+ N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 74.1%

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

   6. Applied rewrites 74.1%

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

   if 4.00000000000000015e201 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.7%

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

   2. 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. lower-*.f64 28.8%

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

   4. Applied rewrites 28.8%

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

Alternative 8: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+24}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+47}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (+ (* -0.25 (* a b)) c)))
   (if (<= t_1 -2e+24) t_2 (if (<= t_1 5e+47) (+ c (* 0.0625 (* t z))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = (-0.25 * (a * b)) + c;
	double tmp;
	if (t_1 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5e+47) {
		tmp = c + (0.0625 * (t * z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (a * b) / 4.0d0
    t_2 = ((-0.25d0) * (a * b)) + c
    if (t_1 <= (-2d+24)) then
        tmp = t_2
    else if (t_1 <= 5d+47) then
        tmp = c + (0.0625d0 * (t * z))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = (-0.25 * (a * b)) + c;
	double tmp;
	if (t_1 <= -2e+24) {
		tmp = t_2;
	} else if (t_1 <= 5e+47) {
		tmp = c + (0.0625 * (t * z));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) / 4.0
	t_2 = (-0.25 * (a * b)) + c
	tmp = 0
	if t_1 <= -2e+24:
		tmp = t_2
	elif t_1 <= 5e+47:
		tmp = c + (0.0625 * (t * z))
	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(Float64(-0.25 * Float64(a * b)) + c)
	tmp = 0.0
	if (t_1 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5e+47)
		tmp = Float64(c + Float64(0.0625 * Float64(t * z)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) / 4.0;
	t_2 = (-0.25 * (a * b)) + c;
	tmp = 0.0;
	if (t_1 <= -2e+24)
		tmp = t_2;
	elseif (t_1 <= 5e+47)
		tmp = c + (0.0625 * (t * z));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+24], t$95$2, If[LessEqual[t$95$1, 5e+47], N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2e24 or 5.00000000000000022e47 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.7%

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

   2. Taylor expanded in z around 0

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

      1. lower--.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 74.1%

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 49.3%

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

   7. Applied rewrites 49.3%

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

   if -2e24 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000022e47

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 48.7%

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

   7. Applied rewrites 48.7%

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

Alternative 9: 65.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+102}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+152)
   (fma y x c)
   (if (<= (* x y) 2e+102) (+ c (* 0.0625 (* t z))) (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+152) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 2e+102) {
		tmp = c + (0.0625 * (t * z));
	} 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) <= -5e+152)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 2e+102)
		tmp = Float64(c + Float64(0.0625 * Float64(t * z)));
	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], -5e+152], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+102], N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e152 or 1.99999999999999995e102 < (*.f64 x y)

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 48.7%

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

   7. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 48.7%

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

   9. Applied rewrites 48.7%

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

   if -5e152 < (*.f64 x y) < 1.99999999999999995e102

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 48.7%

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

   7. Applied rewrites 48.7%

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

Alternative 10: 62.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+246}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* 0.0625 (* t z))))
   (if (<= t_1 -2e+246) t_2 (if (<= t_1 2e+121) (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 = (z * t) / 16.0;
	double t_2 = 0.0625 * (t * z);
	double tmp;
	if (t_1 <= -2e+246) {
		tmp = t_2;
	} else if (t_1 <= 2e+121) {
		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(z * t) / 16.0)
	t_2 = Float64(0.0625 * Float64(t * z))
	tmp = 0.0
	if (t_1 <= -2e+246)
		tmp = t_2;
	elseif (t_1 <= 2e+121)
		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[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+246], t$95$2, If[LessEqual[t$95$1, 2e+121], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.00000000000000014e246 or 2.00000000000000007e121 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-+l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.8%

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 98.8%

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

      17. lower-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. associate-*r* N/A

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

      20. lower-fma.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 98.8%

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 28.2%

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

   8. Applied rewrites 28.2%

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

   if -2.00000000000000014e246 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 2.00000000000000007e121

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 48.7%

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

   7. Applied rewrites 48.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 48.7%

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

   9. Applied rewrites 48.7%

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

Alternative 11: 48.7% accurate, 4.1× speedup

Math

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

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 a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 73.6%

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

4. Applied rewrites 73.6%

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

5. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 48.7%

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

7. Applied rewrites 48.7%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 48.7%

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

9. Applied rewrites 48.7%

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

Alternative 12: 41.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \cdot y \leq -5.4 \cdot 10^{+152}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \cdot y \leq 4.5 \cdot 10^{+110}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5.4e+152) (* x y) (if (<= (* x y) 4.5e+110) c (* x y))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5.4e+152) {
		tmp = x * y;
	} else if ((x * y) <= 4.5e+110) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	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) <= (-5.4d+152)) then
        tmp = x * y
    else if ((x * y) <= 4.5d+110) then
        tmp = c
    else
        tmp = x * y
    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) <= -5.4e+152) {
		tmp = x * y;
	} else if ((x * y) <= 4.5e+110) {
		tmp = c;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -5.4e+152:
		tmp = x * y
	elif (x * y) <= 4.5e+110:
		tmp = c
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5.4e+152)
		tmp = Float64(x * y);
	elseif (Float64(x * y) <= 4.5e+110)
		tmp = c;
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -5.4e+152)
		tmp = x * y;
	elseif ((x * y) <= 4.5e+110)
		tmp = c;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5.4e+152], N[(x * y), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 4.5e+110], c, N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.4000000000000003e152 or 4.5000000000000003e110 < (*.f64 x y)

   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

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

      2. lift--.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. add-flip-rev N/A

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

      9. lower-fma.f64 N/A

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

      10. add-flip-rev N/A

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

      11. associate--r- N/A

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

      12. sub-flip N/A

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

      13. lift-/.f64 N/A

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

      14. mult-flip N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.4%

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

   6. Applied rewrites 74.4%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 28.2%

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

   9. Applied rewrites 28.2%

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

   if -5.4000000000000003e152 < (*.f64 x y) < 4.5000000000000003e110

   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 a around 0

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

      1. lower-+.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 73.6%

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 48.7%

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

   7. Applied rewrites 48.7%

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

   8. Taylor expanded in x around 0

      \[\leadsto c \]
   9. Step-by-step derivation

   10. Applied rewrites 22.5%

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

Alternative 13: 22.5% accurate, 24.7× speedup

Math

\[c \]

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 a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 73.6%

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

4. Applied rewrites 73.6%

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

5. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 48.7%

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

7. Applied rewrites 48.7%

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

8. Taylor expanded in x around 0

   \[\leadsto c \]
9. Step-by-step derivation

10. Applied rewrites 22.5%

    \[\leadsto c \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))