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

Percentage Accurate: 97.5% → 98.7%

Time: 4.1s

Alternatives: 13

Speedup: 0.5×

• 33.5% 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: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \leq 10^{+304}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)}{a} - 0.25 \cdot b\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c) 1e+304)
   (- (fma z (/ t 16.0) (* y x)) (fma (* 0.25 a) b (- c)))
   (* (- (/ (fma (* 0.0625 t) z (fma y x c)) a) (* 0.25 b)) a)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c) <= 1e+304) {
		tmp = fma(z, (t / 16.0), (y * x)) - fma((0.25 * a), b, -c);
	} else {
		tmp = ((fma((0.0625 * t), z, fma(y, x, c)) / a) - (0.25 * b)) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c) <= 1e+304)
		tmp = Float64(fma(z, Float64(t / 16.0), Float64(y * x)) - fma(Float64(0.25 * a), b, Float64(-c)));
	else
		tmp = Float64(Float64(Float64(fma(Float64(0.0625 * t), z, fma(y, x, c)) / a) - Float64(0.25 * b)) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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], 1e+304], N[(N[(z * N[(t / 16.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * a), $MachinePrecision] * b + (-c)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] - N[(0.25 * b), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c) < 9.9999999999999994e303

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \color{blue}{\mathsf{fma}\left(0.25 \cdot a, b, -c\right)} \]

   if 9.9999999999999994e303 < (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c)

   1. Initial program 83.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.9%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c) INFINITY)
   (- (fma z (/ t 16.0) (* y x)) (fma (* 0.25 a) b (- c)))
   (- (fma z (* 0.0625 t) (* 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) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c) <= ((double) INFINITY)) {
		tmp = fma(z, (t / 16.0), (y * x)) - fma((0.25 * a), b, -c);
	} else {
		tmp = fma(z, (0.0625 * t), (y * x)) - -c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c) <= Inf)
		tmp = Float64(fma(z, Float64(t / 16.0), Float64(y * x)) - fma(Float64(0.25 * a), b, Float64(-c)));
	else
		tmp = Float64(fma(z, Float64(0.0625 * t), Float64(y * x)) - Float64(-c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[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], Infinity], N[(N[(z * N[(t / 16.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(0.25 * a), $MachinePrecision] * b + (-c)), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(0.0625 * t), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c) < +inf.0

   1. Initial program 100.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right) \]

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \color{blue}{\mathsf{fma}\left(0.25 \cdot a, b, -c\right)} \]

   if +inf.0 < (+.f64 (-.f64 (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) (/.f64 (*.f64 a b) #s(literal 4 binary64))) c)

   1. Initial program 0.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 25.0

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

   4. Applied rewrites 25.0%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 75.0

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 75.0

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

   10. Applied rewrites 75.0%

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

Alternative 3: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+108} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+113}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e+108) (not (<= t_1 5e+113)))
     (- (fma z (* 0.0625 t) (* y x)) (- c))
     (- (* x y) (fma (* 0.25 a) b (- c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e+108) || !(t_1 <= 5e+113)) {
		tmp = fma(z, (0.0625 * t), (y * x)) - -c;
	} else {
		tmp = (x * y) - fma((0.25 * a), b, -c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e+108) || !(t_1 <= 5e+113))
		tmp = Float64(fma(z, Float64(0.0625 * t), Float64(y * x)) - Float64(-c));
	else
		tmp = Float64(Float64(x * y) - fma(Float64(0.25 * a), b, Float64(-c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+108], N[Not[LessEqual[t$95$1, 5e+113]], $MachinePrecision]], N[(N[(z * N[(0.0625 * t), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - (-c)), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - N[(N[(0.25 * a), $MachinePrecision] * b + (-c)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.99999999999999991e108 or 5e113 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 93.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.5

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 85.6

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

   7. Applied rewrites 85.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 85.6

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

   10. Applied rewrites 85.6%

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

   if -4.99999999999999991e108 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 5e113

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 98.8

         \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right) \]

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \color{blue}{\mathsf{fma}\left(0.25 \cdot a, b, -c\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 95.4

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

   10. Applied rewrites 95.4%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{+108} \lor \neg \left(\frac{z \cdot t}{16} \leq 5 \cdot 10^{+113}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+213} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+214}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e+213) (not (<= t_1 2e+214)))
     (+ (* (* t z) 0.0625) c)
     (- (* x y) (fma (* 0.25 a) b (- c))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e+213) || !(t_1 <= 2e+214)) {
		tmp = ((t * z) * 0.0625) + c;
	} else {
		tmp = (x * y) - fma((0.25 * a), b, -c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e+213) || !(t_1 <= 2e+214))
		tmp = Float64(Float64(Float64(t * z) * 0.0625) + c);
	else
		tmp = Float64(Float64(x * y) - fma(Float64(0.25 * a), b, Float64(-c)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+213], N[Not[LessEqual[t$95$1, 2e+214]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - N[(N[(0.25 * a), $MachinePrecision] * b + (-c)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.9999999999999998e213 or 1.9999999999999999e214 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -4.9999999999999998e213 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.9999999999999999e214

   1. Initial program 99.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 99.0

         \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right) \]

   7. Applied rewrites 99.0%

      \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \color{blue}{\mathsf{fma}\left(0.25 \cdot a, b, -c\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 88.9

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

   10. Applied rewrites 88.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{+213} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+214}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \mathsf{fma}\left(0.25 \cdot a, b, -c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+213} \lor \neg \left(t\_1 \leq 2 \cdot 10^{+214}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e+213) (not (<= t_1 2e+214)))
     (+ (* (* t z) 0.0625) c)
     (+ (fma y x (* -0.25 (* b a))) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e+213) || !(t_1 <= 2e+214)) {
		tmp = ((t * z) * 0.0625) + c;
	} else {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e+213) || !(t_1 <= 2e+214))
		tmp = Float64(Float64(Float64(t * z) * 0.0625) + c);
	else
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+213], N[Not[LessEqual[t$95$1, 2e+214]], $MachinePrecision]], N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -4.9999999999999998e213 or 1.9999999999999999e214 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 88.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

   if -4.9999999999999998e213 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.9999999999999999e214

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{+213} \lor \neg \left(\frac{z \cdot t}{16} \leq 2 \cdot 10^{+214}\right):\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+72} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+212}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -5e+72) (not (<= t_1 5e+212)))
     (* -0.25 (* b a))
     (+ (* y x) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e+72) || !(t_1 <= 5e+212)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = (y * x) + 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) :: t_1
    real(8) :: tmp
    t_1 = (a * b) / 4.0d0
    if ((t_1 <= (-5d+72)) .or. (.not. (t_1 <= 5d+212))) then
        tmp = (-0.25d0) * (b * a)
    else
        tmp = (y * x) + 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 t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -5e+72) || !(t_1 <= 5e+212)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = (y * x) + c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	t_1 = (a * b) / 4.0
	tmp = 0
	if (t_1 <= -5e+72) or not (t_1 <= 5e+212):
		tmp = -0.25 * (b * a)
	else:
		tmp = (y * x) + c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -5e+72) || !(t_1 <= 5e+212))
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = Float64(Float64(y * x) + c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	t_1 = (a * b) / 4.0;
	tmp = 0.0;
	if ((t_1 <= -5e+72) || ~((t_1 <= 5e+212)))
		tmp = -0.25 * (b * a);
	else
		tmp = (y * x) + c;
	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]}, If[Or[LessEqual[t$95$1, -5e+72], N[Not[LessEqual[t$95$1, 5e+212]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.99999999999999992e72 or 4.99999999999999992e212 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if -4.99999999999999992e72 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.99999999999999992e212

   1. Initial program 98.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -5 \cdot 10^{+72} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+212}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x + c\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 - \left(\left(a \cdot b\right) \cdot 0.25 - c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -5e+74) (not (<= (* x y) 1e+106)))
   (- (fma z (* 0.0625 t) (* y x)) (- c))
   (- (* (* t z) 0.0625) (- (* (* a b) 0.25) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+74) || !((x * y) <= 1e+106)) {
		tmp = fma(z, (0.0625 * t), (y * x)) - -c;
	} else {
		tmp = ((t * z) * 0.0625) - (((a * b) * 0.25) - c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+74) || !(Float64(x * y) <= 1e+106))
		tmp = Float64(fma(z, Float64(0.0625 * t), Float64(y * x)) - Float64(-c));
	else
		tmp = Float64(Float64(Float64(t * z) * 0.0625) - Float64(Float64(Float64(a * b) * 0.25) - c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+74], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+106]], $MachinePrecision]], N[(N[(z * N[(0.0625 * t), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - (-c)), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] - N[(N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999963e74 or 1.00000000000000009e106 < (*.f64 x y)

   1. Initial program 93.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.6

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 88.6

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

   10. Applied rewrites 88.6%

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

   if -4.99999999999999963e74 < (*.f64 x y) < 1.00000000000000009e106

   1. Initial program 98.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 95.4

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 - \left(\frac{b \cdot a}{4} - c\right) \]

   7. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\left(t \cdot z\right) \cdot 0.0625} - \left(\frac{b \cdot a}{4} - c\right) \]

   8. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 95.4

         \[\leadsto \left(t \cdot z\right) \cdot 0.0625 - \left(\left(a \cdot b\right) \cdot 0.25 - c\right) \]

   10. Applied rewrites 95.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625 - \left(\left(a \cdot b\right) \cdot 0.25 - c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -5e+74) (not (<= (* x y) 1e+106)))
   (- (fma z (* 0.0625 t) (* y x)) (- c))
   (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+74) || !((x * y) <= 1e+106)) {
		tmp = fma(z, (0.0625 * t), (y * x)) - -c;
	} else {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+74) || !(Float64(x * y) <= 1e+106))
		tmp = Float64(fma(z, Float64(0.0625 * t), Float64(y * x)) - Float64(-c));
	else
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+74], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+106]], $MachinePrecision]], N[(N[(z * N[(0.0625 * t), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - (-c)), $MachinePrecision], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999963e74 or 1.00000000000000009e106 < (*.f64 x y)

   1. Initial program 93.0%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.3

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

   4. Applied rewrites 95.3%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 88.6

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

   7. Applied rewrites 88.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-*.f64 88.6

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

   10. Applied rewrites 88.6%

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

   if -4.99999999999999963e74 < (*.f64 x y) < 1.00000000000000009e106

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 95.4

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

   5. Applied rewrites 95.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+74} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;\mathsf{fma}\left(z, 0.0625 \cdot t, y \cdot x\right) - \left(-c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+71}:\\ \;\;\;\;y \cdot x + c\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+69}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{else}:\\ \;\;\;\;x \cdot y - \left(a \cdot b\right) \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -5e+71)
   (+ (* y x) c)
   (if (<= (* x y) 5e+69)
     (+ (* -0.25 (* b a)) c)
     (- (* x y) (* (* a b) 0.25)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -5e+71) {
		tmp = (y * x) + c;
	} else if ((x * y) <= 5e+69) {
		tmp = (-0.25 * (b * a)) + c;
	} else {
		tmp = (x * y) - ((a * b) * 0.25);
	}
	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) <= (-5d+71)) then
        tmp = (y * x) + c
    else if ((x * y) <= 5d+69) then
        tmp = ((-0.25d0) * (b * a)) + c
    else
        tmp = (x * y) - ((a * b) * 0.25d0)
    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) <= -5e+71) {
		tmp = (y * x) + c;
	} else if ((x * y) <= 5e+69) {
		tmp = (-0.25 * (b * a)) + c;
	} else {
		tmp = (x * y) - ((a * b) * 0.25);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -5e+71:
		tmp = (y * x) + c
	elif (x * y) <= 5e+69:
		tmp = (-0.25 * (b * a)) + c
	else:
		tmp = (x * y) - ((a * b) * 0.25)
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -5e+71)
		tmp = Float64(Float64(y * x) + c);
	elseif (Float64(x * y) <= 5e+69)
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	else
		tmp = Float64(Float64(x * y) - Float64(Float64(a * b) * 0.25));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -5e+71)
		tmp = (y * x) + c;
	elseif ((x * y) <= 5e+69)
		tmp = (-0.25 * (b * a)) + c;
	else
		tmp = (x * y) - ((a * b) * 0.25);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -5e+71], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+69], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(x * y), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -4.99999999999999972e71

   1. Initial program 95.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -4.99999999999999972e71 < (*.f64 x y) < 5.00000000000000036e69

   1. Initial program 98.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if 5.00000000000000036e69 < (*.f64 x y)

   1. Initial program 91.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower--.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 93.6

         \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \mathsf{fma}\left(0.25 \cdot a, b, -c\right) \]

   7. Applied rewrites 93.6%

      \[\leadsto \mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \color{blue}{\mathsf{fma}\left(0.25 \cdot a, b, -c\right)} \]

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 83.4

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

   10. Applied rewrites 83.4%

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

   11. Taylor expanded in a around inf

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

       1. *-commutative N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 79.2

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

   13. Applied rewrites 79.2%

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

Alternative 10: 66.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+71} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;y \cdot x + c\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -5e+71) (not (<= (* x y) 1e+106)))
   (+ (* y x) c)
   (+ (* -0.25 (* b a)) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+71) || !((x * y) <= 1e+106)) {
		tmp = (y * x) + c;
	} else {
		tmp = (-0.25 * (b * a)) + c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((x * y) <= (-5d+71)) .or. (.not. ((x * y) <= 1d+106))) then
        tmp = (y * x) + c
    else
        tmp = ((-0.25d0) * (b * a)) + c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+71) || !((x * y) <= 1e+106)) {
		tmp = (y * x) + c;
	} else {
		tmp = (-0.25 * (b * a)) + c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -5e+71) or not ((x * y) <= 1e+106):
		tmp = (y * x) + c
	else:
		tmp = (-0.25 * (b * a)) + c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+71) || !(Float64(x * y) <= 1e+106))
		tmp = Float64(Float64(y * x) + c);
	else
		tmp = Float64(Float64(-0.25 * Float64(b * a)) + c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((x * y) <= -5e+71) || ~(((x * y) <= 1e+106)))
		tmp = (y * x) + c;
	else
		tmp = (-0.25 * (b * a)) + c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+71], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+106]], $MachinePrecision]], N[(N[(y * x), $MachinePrecision] + c), $MachinePrecision], N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -4.99999999999999972e71 or 1.00000000000000009e106 < (*.f64 x y)

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.99999999999999972e71 < (*.f64 x y) < 1.00000000000000009e106

   1. Initial program 98.8%

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

   3. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+71} \lor \neg \left(x \cdot y \leq 10^{+106}\right):\\ \;\;\;\;y \cdot x + c\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right) + c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+45} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+103}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -5e+45) (not (<= (* x y) 5e+103))) (* 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+45) || !((x * y) <= 5e+103)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (((x * y) <= (-5d+45)) .or. (.not. ((x * y) <= 5d+103))) then
        tmp = y * x
    else
        tmp = c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((x * y) <= -5e+45) || !((x * y) <= 5e+103)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -5e+45) or not ((x * y) <= 5e+103):
		tmp = y * x
	else:
		tmp = c
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((Float64(x * y) <= -5e+45) || !(Float64(x * y) <= 5e+103))
		tmp = Float64(y * x);
	else
		tmp = c;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if (((x * y) <= -5e+45) || ~(((x * y) <= 5e+103)))
		tmp = y * x;
	else
		tmp = c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[N[(x * y), $MachinePrecision], -5e+45], N[Not[LessEqual[N[(x * y), $MachinePrecision], 5e+103]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e45 or 5e103 < (*.f64 x y)

   1. Initial program 93.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -5e45 < (*.f64 x y) < 5e103

   1. Initial program 98.8%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 29.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{+45} \lor \neg \left(x \cdot y \leq 5 \cdot 10^{+103}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ y \cdot x + c \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (y * x) + 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 = (y * x) + c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 46.0

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

5. Applied rewrites 46.0%

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

Alternative 13: 21.9% accurate, 47.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 22.4%

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

Reproduce

herbie shell --seed 2025085 
(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))