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

Percentage Accurate: 97.5% → 98.8%

Time: 5.3s

Alternatives: 12

Speedup: 1.1×

• 32.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 1.1× speedup

Math

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

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 97.5%

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

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

   18. add-flip-rev N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. sub-negate-rev N/A

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

   22. 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 2: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \mathsf{fma}\left(z \cdot 0.0625, t, \mathsf{fma}\left(b \cdot -0.25, a, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (fma (* z 0.0625) t (fma (* b -0.25) a c))))
   (if (<= t_1 -4e+95)
     t_2
     (if (<= t_1 1e+68) (fma (* 0.0625 z) t (fma y x c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = fma((z * 0.0625), t, fma((b * -0.25), a, c));
	double tmp;
	if (t_1 <= -4e+95) {
		tmp = t_2;
	} else if (t_1 <= 1e+68) {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = fma(Float64(z * 0.0625), t, fma(Float64(b * -0.25), a, c))
	tmp = 0.0
	if (t_1 <= -4e+95)
		tmp = t_2;
	elseif (t_1 <= 1e+68)
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * 0.0625), $MachinePrecision] * t + N[(N[(b * -0.25), $MachinePrecision] * a + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+95], t$95$2, If[LessEqual[t$95$1, 1e+68], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.00000000000000008e95 or 9.99999999999999953e67 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.5%

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

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

      18. add-flip-rev N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. sub-negate-rev N/A

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

      22. 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. Taylor expanded in x around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 74.2

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

   6. Applied rewrites 74.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.2

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip-rev N/A

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

      7. sub-flip N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-*r* N/A

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

      12. remove-double-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if -4.00000000000000008e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999953e67

   1. Initial program 97.5%

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

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

      18. add-flip-rev N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. sub-negate-rev N/A

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

      22. 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. Taylor expanded in a around 0

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

   6. Applied rewrites 74.8%

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

Alternative 3: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := \left(x \cdot y - 0.25 \cdot \left(a \cdot b\right)\right) + c\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+53}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (+ (- (* x y) (* 0.25 (* a b))) c)))
   (if (<= t_1 -5e+53)
     t_2
     (if (<= t_1 5e+83) (fma (* 0.0625 z) t (fma y x c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = ((x * y) - (0.25 * (a * b))) + c;
	double tmp;
	if (t_1 <= -5e+53) {
		tmp = t_2;
	} else if (t_1 <= 5e+83) {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(Float64(Float64(x * y) - Float64(0.25 * Float64(a * b))) + c)
	tmp = 0.0
	if (t_1 <= -5e+53)
		tmp = t_2;
	elseif (t_1 <= 5e+83)
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] - N[(0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+53], t$95$2, If[LessEqual[t$95$1, 5e+83], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5.0000000000000004e53 or 5.00000000000000029e83 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.5%

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

   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 73.6

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

   4. Applied rewrites 73.6%

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

   if -5.0000000000000004e53 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000029e83

   1. Initial program 97.5%

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

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

      18. add-flip-rev N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. sub-negate-rev N/A

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

      22. 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. Taylor expanded in a around 0

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

   6. Applied rewrites 74.8%

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

Alternative 4: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (+ (* -0.25 (* a b)) c)))
   (if (<= t_1 -4e+95)
     t_2
     (if (<= t_1 5e+120) (fma (* 0.0625 z) t (fma y x c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = (-0.25 * (a * b)) + c;
	double tmp;
	if (t_1 <= -4e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+120) {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(Float64(-0.25 * Float64(a * b)) + c)
	tmp = 0.0
	if (t_1 <= -4e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+120)
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+95], t$95$2, If[LessEqual[t$95$1, 5e+120], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.00000000000000008e95 or 5.00000000000000019e120 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.5%

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

   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 73.6

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

   4. Applied rewrites 73.6%

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

   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 48.6

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

   7. Applied rewrites 48.6%

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

   if -4.00000000000000008e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000019e120

   1. Initial program 97.5%

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

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

      18. add-flip-rev N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. sub-negate-rev N/A

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

      22. 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. Taylor expanded in a around 0

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

   6. Applied rewrites 74.8%

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

Alternative 5: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (* x y))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -2e+129) t_1 (if (<= t_2 5e+153) (+ (* -0.25 (* a b)) c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), (x * y));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -2e+129) {
		tmp = t_1;
	} else if (t_2 <= 5e+153) {
		tmp = (-0.25 * (a * b)) + c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -2e+129)
		tmp = t_1;
	elseif (t_2 <= 5e+153)
		tmp = Float64(Float64(-0.25 * Float64(a * b)) + c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+129], t$95$1, If[LessEqual[t$95$2, 5e+153], N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -2e129 or 5.00000000000000018e153 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.9%

       \[\leadsto c \]

   11. Taylor expanded in c around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 53.1

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

   13. Applied rewrites 53.1%

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

   if -2e129 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.00000000000000018e153

   1. Initial program 97.5%

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

   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 73.6

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

   4. Applied rewrites 73.6%

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

   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 48.6

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

   7. Applied rewrites 48.6%

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

Alternative 6: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+96}:\\ \;\;\;\;-0.25 \cdot \left(a \cdot b\right) + c\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (+ c (* 0.0625 (* t z)))))
   (if (<= t_1 -2e+124) t_2 (if (<= t_1 5e+96) (+ (* -0.25 (* a b)) 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 = c + (0.0625 * (t * z));
	double tmp;
	if (t_1 <= -2e+124) {
		tmp = t_2;
	} else if (t_1 <= 5e+96) {
		tmp = (-0.25 * (a * b)) + c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * t) / 16.0d0
    t_2 = c + (0.0625d0 * (t * z))
    if (t_1 <= (-2d+124)) then
        tmp = t_2
    else if (t_1 <= 5d+96) then
        tmp = ((-0.25d0) * (a * b)) + c
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = c + (0.0625 * (t * z));
	double tmp;
	if (t_1 <= -2e+124) {
		tmp = t_2;
	} else if (t_1 <= 5e+96) {
		tmp = (-0.25 * (a * b)) + c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (z * t) / 16.0
	t_2 = c + (0.0625 * (t * z))
	tmp = 0
	if t_1 <= -2e+124:
		tmp = t_2
	elif t_1 <= 5e+96:
		tmp = (-0.25 * (a * b)) + 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(c + Float64(0.0625 * Float64(t * z)))
	tmp = 0.0
	if (t_1 <= -2e+124)
		tmp = t_2;
	elseif (t_1 <= 5e+96)
		tmp = Float64(Float64(-0.25 * Float64(a * b)) + c);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (z * t) / 16.0;
	t_2 = c + (0.0625 * (t * z));
	tmp = 0.0;
	if (t_1 <= -2e+124)
		tmp = t_2;
	elseif (t_1 <= 5e+96)
		tmp = (-0.25 * (a * b)) + c;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(c + N[(0.0625 * N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+124], t$95$2, If[LessEqual[t$95$1, 5e+96], N[(N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.9999999999999999e124 or 5.0000000000000004e96 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.3

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

   7. Applied rewrites 49.3%

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

   if -1.9999999999999999e124 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.0000000000000004e96

   1. Initial program 97.5%

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

   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 73.6

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

   4. Applied rewrites 73.6%

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

   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 48.6

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

   7. Applied rewrites 48.6%

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

Alternative 7: 63.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+177}:\\ \;\;\;\;c + 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* a b))))
   (if (<= t_1 -4e+95) t_2 (if (<= t_1 5e+177) (+ 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);
	double tmp;
	if (t_1 <= -4e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+177) {
		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)
    if (t_1 <= (-4d+95)) then
        tmp = t_2
    else if (t_1 <= 5d+177) 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);
	double tmp;
	if (t_1 <= -4e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+177) {
		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)
	tmp = 0
	if t_1 <= -4e+95:
		tmp = t_2
	elif t_1 <= 5e+177:
		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(-0.25 * Float64(a * b))
	tmp = 0.0
	if (t_1 <= -4e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+177)
		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);
	tmp = 0.0;
	if (t_1 <= -4e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+177)
		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[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+95], t$95$2, If[LessEqual[t$95$1, 5e+177], 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)) < -4.00000000000000008e95 or 5.0000000000000003e177 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 27.8

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

   4. Applied rewrites 27.8%

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

   if -4.00000000000000008e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.0000000000000003e177

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.3

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

   7. Applied rewrites 49.3%

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

Alternative 8: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := 0.0625 \cdot \left(t \cdot z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* 0.0625 (* t z))))
   (if (<= t_1 -2e+124) t_2 (if (<= t_1 5e+83) (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+124) {
		tmp = t_2;
	} else if (t_1 <= 5e+83) {
		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+124)
		tmp = t_2;
	elseif (t_1 <= 5e+83)
		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+124], t$95$2, If[LessEqual[t$95$1, 5e+83], 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)) < -1.9999999999999999e124 or 5.00000000000000029e83 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.9%

       \[\leadsto c \]

   11. Taylor expanded in z around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 28.4

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

   13. Applied rewrites 28.4%

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

   if -1.9999999999999999e124 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5.00000000000000029e83

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

         \[\leadsto x \cdot y - \left(\mathsf{neg}\left(c\right)\right) \]

      5. sub-flip N/A

         \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 49.2

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

   9. Applied rewrites 49.2%

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

Alternative 9: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(a \cdot b\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+95}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+120}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* a b))))
   (if (<= t_1 -4e+95) t_2 (if (<= t_1 5e+120) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (a * b);
	double tmp;
	if (t_1 <= -4e+95) {
		tmp = t_2;
	} else if (t_1 <= 5e+120) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(a * b))
	tmp = 0.0
	if (t_1 <= -4e+95)
		tmp = t_2;
	elseif (t_1 <= 5e+120)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+95], t$95$2, If[LessEqual[t$95$1, 5e+120], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.00000000000000008e95 or 5.00000000000000019e120 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.5%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 27.8

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

   4. Applied rewrites 27.8%

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

   if -4.00000000000000008e95 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 5.00000000000000019e120

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

         \[\leadsto x \cdot y - \left(\mathsf{neg}\left(c\right)\right) \]

      5. sub-flip N/A

         \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto y \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      7. remove-double-neg N/A

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

      8. lower-fma.f64 49.2

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

   9. Applied rewrites 49.2%

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

Alternative 10: 49.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

2. Taylor expanded in a around 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 74.3

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

4. Applied rewrites 74.3%

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

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 49.2

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

7. Applied rewrites 49.2%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. add-flip N/A

      \[\leadsto x \cdot y - \left(\mathsf{neg}\left(c\right)\right) \]

   5. sub-flip N/A

      \[\leadsto x \cdot y + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto y \cdot x + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   7. remove-double-neg N/A

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

   8. lower-fma.f64 49.2

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

9. Applied rewrites 49.2%

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

Alternative 11: 38.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+34}:\\ \;\;\;\;c\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+109}:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= c -2.4e+34) c (if (<= c 5.5e+109) (* x y) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -2.4e+34) {
		tmp = c;
	} else if (c <= 5.5e+109) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (c <= (-2.4d+34)) then
        tmp = c
    else if (c <= 5.5d+109) then
        tmp = x * y
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (c <= -2.4e+34) {
		tmp = c;
	} else if (c <= 5.5e+109) {
		tmp = x * y;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if c <= -2.4e+34:
		tmp = c
	elif c <= 5.5e+109:
		tmp = x * y
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (c <= -2.4e+34)
		tmp = c;
	elseif (c <= 5.5e+109)
		tmp = Float64(x * y);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (c <= -2.4e+34)
		tmp = c;
	elseif (c <= 5.5e+109)
		tmp = x * y;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[c, -2.4e+34], c, If[LessEqual[c, 5.5e+109], N[(x * y), $MachinePrecision], c]]

Derivation

1. Split input into 2 regimes

2. if c < -2.39999999999999987e34 or 5.4999999999999998e109 < c

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.9%

       \[\leadsto c \]

   if -2.39999999999999987e34 < c < 5.4999999999999998e109

   1. Initial program 97.5%

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

   2. Taylor expanded in a around 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 74.3

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

   4. Applied rewrites 74.3%

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

   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 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 22.9%

       \[\leadsto c \]

   11. Taylor expanded in x around inf

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

       1. lower-*.f64 28.4

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

   13. Applied rewrites 28.4%

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

Alternative 12: 22.9% accurate, 24.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.5%

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

2. Taylor expanded in a around 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 74.3

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

4. Applied rewrites 74.3%

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

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 49.2

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

7. Applied rewrites 49.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 22.9%

    \[\leadsto c \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))