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

Percentage Accurate: 97.7% → 97.7%

Time: 11.4s

Alternatives: 12

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

Alternative 2: 75.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+229} \lor \neg \left(t\_1 \leq 4 \cdot 10^{+113}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* x y) (/ (* z t) 16.0))))
   (if (or (<= t_1 -1e+229) (not (<= t_1 4e+113)))
     (fma y x (* (* t z) 0.0625))
     (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 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if ((t_1 <= -1e+229) || !(t_1 <= 4e+113)) {
		tmp = fma(y, x, ((t * z) * 0.0625));
	} else {
		tmp = fma((-0.25 * a), b, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if ((t_1 <= -1e+229) || !(t_1 <= 4e+113))
		tmp = fma(y, x, Float64(Float64(t * z) * 0.0625));
	else
		tmp = fma(Float64(-0.25 * a), b, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+229], N[Not[LessEqual[t$95$1, 4e+113]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -9.9999999999999999e228 or 4e113 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   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 a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 86.8

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

   8. Applied rewrites 86.8%

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

   if -9.9999999999999999e228 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 4e113

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 88.8

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 77.5%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{z \cdot t}{16} \leq -1 \cdot 10^{+229} \lor \neg \left(x \cdot y + \frac{z \cdot t}{16} \leq 4 \cdot 10^{+113}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1e+88) (not (<= t_1 5e+34)))
     (fma (* -0.25 a) b (fma x y c))
     (fma (* 0.0625 z) t (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34)) {
		tmp = fma((-0.25 * a), b, fma(x, y, c));
	} else {
		tmp = fma((0.0625 * z), t, fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34))
		tmp = fma(Float64(-0.25 * a), b, fma(x, y, c));
	else
		tmp = fma(Float64(0.0625 * z), t, fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+88], N[Not[LessEqual[t$95$1, 5e+34]], $MachinePrecision]], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * z), $MachinePrecision] * t + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999959e87 or 4.9999999999999998e34 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   if -9.99999999999999959e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.9999999999999998e34

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 96.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 96.7

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

   7. Applied rewrites 96.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1 \cdot 10^{+88} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot z, t, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1e+88) (not (<= t_1 5e+34)))
     (fma (* -0.25 a) b (fma x y c))
     (fma y x (fma (* t z) 0.0625 c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34)) {
		tmp = fma((-0.25 * a), b, fma(x, y, c));
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34))
		tmp = fma(Float64(-0.25 * a), b, fma(x, y, c));
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+88], N[Not[LessEqual[t$95$1, 5e+34]], $MachinePrecision]], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999959e87 or 4.9999999999999998e34 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 90.8

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

   8. Applied rewrites 90.8%

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

   if -9.99999999999999959e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.9999999999999998e34

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1 \cdot 10^{+88} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1e+88) (not (<= t_1 5e+34)))
     (fma -0.25 (* b a) (fma y x c))
     (fma y x (fma (* t z) 0.0625 c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34)) {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	} else {
		tmp = fma(y, x, fma((t * z), 0.0625, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+34))
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	else
		tmp = fma(y, x, fma(Float64(t * z), 0.0625, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+88], N[Not[LessEqual[t$95$1, 5e+34]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999959e87 or 4.9999999999999998e34 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

   if -9.99999999999999959e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.9999999999999998e34

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1 \cdot 10^{+88} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+34}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+200} \lor \neg \left(t\_1 \leq 10^{+144}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (or (<= t_1 -5e+200) (not (<= t_1 1e+144)))
     (fma y x (* (* t z) 0.0625))
     (fma -0.25 (* b a) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if ((t_1 <= -5e+200) || !(t_1 <= 1e+144)) {
		tmp = fma(y, x, ((t * z) * 0.0625));
	} else {
		tmp = fma(-0.25, (b * a), fma(y, x, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if ((t_1 <= -5e+200) || !(t_1 <= 1e+144))
		tmp = fma(y, x, Float64(Float64(t * z) * 0.0625));
	else
		tmp = fma(-0.25, Float64(b * a), fma(y, x, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e+200], N[Not[LessEqual[t$95$1, 1e+144]], $MachinePrecision]], N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -5.00000000000000019e200 or 1.00000000000000002e144 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 91.7

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 87.7

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

   8. Applied rewrites 87.7%

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

   if -5.00000000000000019e200 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.00000000000000002e144

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.8

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

   5. Applied rewrites 90.8%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot t}{16} \leq -5 \cdot 10^{+200} \lor \neg \left(\frac{z \cdot t}{16} \leq 10^{+144}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (or (<= t_1 -1e+88) (not (<= t_1 5e+87)))
     (* -0.25 (* b a))
     (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+87)) {
		tmp = -0.25 * (b * a);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if ((t_1 <= -1e+88) || !(t_1 <= 5e+87))
		tmp = Float64(-0.25 * Float64(b * a));
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+88], N[Not[LessEqual[t$95$1, 5e+87]], $MachinePrecision]], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.99999999999999959e87 or 4.9999999999999998e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.9%

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

   3. Taylor expanded in 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 68.2

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

   5. Applied rewrites 68.2%

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

   if -9.99999999999999959e87 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4.9999999999999998e87

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 94.5

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

   5. Applied rewrites 94.5%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 67.8%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq -1 \cdot 10^{+88} \lor \neg \left(\frac{a \cdot b}{4} \leq 5 \cdot 10^{+87}\right):\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-40} \lor \neg \left(z \leq 6.2 \cdot 10^{-101}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625, t, \frac{\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)}{z}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= z -6.8e-40) (not (<= z 6.2e-101)))
   (* (fma 0.0625 t (/ (fma -0.25 (* b a) (fma y x c)) z)) z)
   (fma (* -0.25 a) b (fma x y c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((z <= -6.8e-40) || !(z <= 6.2e-101)) {
		tmp = fma(0.0625, t, (fma(-0.25, (b * a), fma(y, x, c)) / z)) * z;
	} else {
		tmp = fma((-0.25 * a), b, fma(x, y, c));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if ((z <= -6.8e-40) || !(z <= 6.2e-101))
		tmp = Float64(fma(0.0625, t, Float64(fma(-0.25, Float64(b * a), fma(y, x, c)) / z)) * z);
	else
		tmp = fma(Float64(-0.25 * a), b, fma(x, y, c));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[Or[LessEqual[z, -6.8e-40], N[Not[LessEqual[z, 6.2e-101]], $MachinePrecision]], N[(N[(0.0625 * t + N[(N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x + c), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(x * y + c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.79999999999999968e-40 or 6.19999999999999946e-101 < z

   1. Initial program 98.6%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.8%

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

   if -6.79999999999999968e-40 < z < 6.19999999999999946e-101

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 95.2

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

   8. Applied rewrites 95.2%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.8 \cdot 10^{-40} \lor \neg \left(z \leq 6.2 \cdot 10^{-101}\right):\\ \;\;\;\;\mathsf{fma}\left(0.0625, t, \frac{\mathsf{fma}\left(-0.25, b \cdot a, \mathsf{fma}\left(y, x, c\right)\right)}{z}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \mathsf{fma}\left(x, y, c\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq -5 \cdot 10^{-94}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -4e+165)
   (fma y x c)
   (if (<= (* x y) -5e-94)
     (fma (* t z) 0.0625 c)
     (if (<= (* x y) 1e+47) (fma (* -0.25 a) b c) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -4e+165) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= -5e-94) {
		tmp = fma((t * z), 0.0625, c);
	} else if ((x * y) <= 1e+47) {
		tmp = fma((-0.25 * a), b, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -4e+165)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= -5e-94)
		tmp = fma(Float64(t * z), 0.0625, c);
	elseif (Float64(x * y) <= 1e+47)
		tmp = fma(Float64(-0.25 * a), b, c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -4e+165], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], -5e-94], N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+47], N[(N[(-0.25 * a), $MachinePrecision] * b + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -3.9999999999999996e165 or 1e47 < (*.f64 x y)

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.1%

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

   if -3.9999999999999996e165 < (*.f64 x y) < -4.9999999999999995e-94

   1. Initial program 97.6%

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

   3. Taylor expanded in a around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 73.3

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

   8. Applied rewrites 73.3%

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

   if -4.9999999999999995e-94 < (*.f64 x y) < 1e47

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in z around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 76.7

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

   8. Applied rewrites 76.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 73.4%

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

Alternative 10: 41.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+165} \lor \neg \left(x \cdot y \leq 10^{+47}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (or (<= (* x y) -4e+165) (not (<= (* x y) 1e+47))) (* 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) <= -4e+165) || !((x * y) <= 1e+47)) {
		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) <= (-4d+165)) .or. (.not. ((x * y) <= 1d+47))) 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) <= -4e+165) || !((x * y) <= 1e+47)) {
		tmp = y * x;
	} else {
		tmp = c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if ((x * y) <= -4e+165) or not ((x * y) <= 1e+47):
		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) <= -4e+165) || !(Float64(x * y) <= 1e+47))
		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) <= -4e+165) || ~(((x * y) <= 1e+47)))
		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], -4e+165], N[Not[LessEqual[N[(x * y), $MachinePrecision], 1e+47]], $MachinePrecision]], N[(y * x), $MachinePrecision], c]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -3.9999999999999996e165 or 1e47 < (*.f64 x y)

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -3.9999999999999996e165 < (*.f64 x y) < 1e47

   1. Initial program 99.4%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 31.7%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq -4 \cdot 10^{+165} \lor \neg \left(x \cdot y \leq 10^{+47}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;c\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.3% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in a around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 72.1

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

5. Applied rewrites 72.1%

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

6. Taylor expanded in z around 0

   \[\leadsto \mathsf{fma}\left(y, x, c\right) \]
7. Step-by-step derivation

8. Applied rewrites 52.3%

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

Alternative 12: 22.5% 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 99.2%

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

3. Taylor expanded in c around inf

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

5. Applied rewrites 23.8%

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

Reproduce

herbie shell --seed 2025026 
(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))