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

Percentage Accurate: 98.1% → 98.1%

Time: 4.3s

Alternatives: 16

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: 98.1% 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: 98.1% 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 98.1%

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

Alternative 2: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625 + c\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* (* t z) 0.0625) c)) (t_2 (/ (* a b) 4.0)))
   (if (<= t_2 -5e+51)
     (fma y x (* -0.25 (* b a)))
     (if (<= t_2 1e-171)
       t_1
       (if (<= t_2 4e-30)
         (fma y x c)
         (if (<= t_2 1e+126) t_1 (fma -0.25 (* b a) (* y x))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t * z) * 0.0625) + c;
	double t_2 = (a * b) / 4.0;
	double tmp;
	if (t_2 <= -5e+51) {
		tmp = fma(y, x, (-0.25 * (b * a)));
	} else if (t_2 <= 1e-171) {
		tmp = t_1;
	} else if (t_2 <= 4e-30) {
		tmp = fma(y, x, c);
	} else if (t_2 <= 1e+126) {
		tmp = t_1;
	} else {
		tmp = fma(-0.25, (b * a), (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t * z) * 0.0625) + c)
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_2 <= -5e+51)
		tmp = fma(y, x, Float64(-0.25 * Float64(b * a)));
	elseif (t_2 <= 1e-171)
		tmp = t_1;
	elseif (t_2 <= 4e-30)
		tmp = fma(y, x, c);
	elseif (t_2 <= 1e+126)
		tmp = t_1;
	else
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+51], N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-171], t$95$1, If[LessEqual[t$95$2, 4e-30], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$2, 1e+126], t$95$1, N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e51

   1. Initial program 96.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in c around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 73.0

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

   8. Applied rewrites 73.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 73.7

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

   10. Applied rewrites 73.7%

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

   if -5e51 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999998e-172 or 4e-30 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999925e125

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   if 9.9999999999999998e-172 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e-30

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

   if 9.99999999999999925e125 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 86.4

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

   5. Applied rewrites 86.4%

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

   6. Taylor expanded in c around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 80.6

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

   8. Applied rewrites 80.6%

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

Alternative 3: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625 + c\\ t_2 := \frac{a \cdot b}{4}\\ t_3 := \mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* (* t z) 0.0625) c))
        (t_2 (/ (* a b) 4.0))
        (t_3 (fma -0.25 (* b a) (* y x))))
   (if (<= t_2 -5e+51)
     t_3
     (if (<= t_2 1e-171)
       t_1
       (if (<= t_2 4e-30) (fma y x c) (if (<= t_2 1e+126) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t * z) * 0.0625) + c;
	double t_2 = (a * b) / 4.0;
	double t_3 = fma(-0.25, (b * a), (y * x));
	double tmp;
	if (t_2 <= -5e+51) {
		tmp = t_3;
	} else if (t_2 <= 1e-171) {
		tmp = t_1;
	} else if (t_2 <= 4e-30) {
		tmp = fma(y, x, c);
	} else if (t_2 <= 1e+126) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t * z) * 0.0625) + c)
	t_2 = Float64(Float64(a * b) / 4.0)
	t_3 = fma(-0.25, Float64(b * a), Float64(y * x))
	tmp = 0.0
	if (t_2 <= -5e+51)
		tmp = t_3;
	elseif (t_2 <= 1e-171)
		tmp = t_1;
	elseif (t_2 <= 4e-30)
		tmp = fma(y, x, c);
	elseif (t_2 <= 1e+126)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+51], t$95$3, If[LessEqual[t$95$2, 1e-171], t$95$1, If[LessEqual[t$95$2, 4e-30], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$2, 1e+126], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e51 or 9.99999999999999925e125 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.9%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

   6. Taylor expanded in c around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if -5e51 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999998e-172 or 4e-30 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999925e125

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 62.5

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

   5. Applied rewrites 62.5%

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

   if 9.9999999999999998e-172 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e-30

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

Alternative 4: 64.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625 + c\\ t_2 := \frac{a \cdot b}{4}\\ t_3 := -0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-171}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (+ (* (* t z) 0.0625) c))
        (t_2 (/ (* a b) 4.0))
        (t_3 (+ (* -0.25 (* b a)) c)))
   (if (<= t_2 -2e+223)
     t_3
     (if (<= t_2 1e-171)
       t_1
       (if (<= t_2 4e-30) (fma y x c) (if (<= t_2 2e+64) t_1 t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = ((t * z) * 0.0625) + c;
	double t_2 = (a * b) / 4.0;
	double t_3 = (-0.25 * (b * a)) + c;
	double tmp;
	if (t_2 <= -2e+223) {
		tmp = t_3;
	} else if (t_2 <= 1e-171) {
		tmp = t_1;
	} else if (t_2 <= 4e-30) {
		tmp = fma(y, x, c);
	} else if (t_2 <= 2e+64) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(Float64(t * z) * 0.0625) + c)
	t_2 = Float64(Float64(a * b) / 4.0)
	t_3 = Float64(Float64(-0.25 * Float64(b * a)) + c)
	tmp = 0.0
	if (t_2 <= -2e+223)
		tmp = t_3;
	elseif (t_2 <= 1e-171)
		tmp = t_1;
	elseif (t_2 <= 4e-30)
		tmp = fma(y, x, c);
	elseif (t_2 <= 2e+64)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+223], t$95$3, If[LessEqual[t$95$2, 1e-171], t$95$1, If[LessEqual[t$95$2, 4e-30], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$2, 2e+64], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000009e223 or 2.00000000000000004e64 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 95.4%

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

   3. Taylor expanded in a around 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 74.7

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

   5. Applied rewrites 74.7%

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

   if -2.00000000000000009e223 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.9999999999999998e-172 or 4e-30 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.00000000000000004e64

   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 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 59.5

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

   5. Applied rewrites 59.5%

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

   if 9.9999999999999998e-172 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e-30

   1. Initial program 99.8%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 62.2

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

   8. Applied rewrites 62.2%

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

Alternative 5: 61.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right) + c\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -600000000:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (+ (* -0.25 (* b a)) c)))
   (if (<= t_1 -2e+223)
     t_2
     (if (<= t_1 -600000000.0)
       (* (* t z) 0.0625)
       (if (<= t_1 2e+50) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = (-0.25 * (b * a)) + c;
	double tmp;
	if (t_1 <= -2e+223) {
		tmp = t_2;
	} else if (t_1 <= -600000000.0) {
		tmp = (t * z) * 0.0625;
	} else if (t_1 <= 2e+50) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(Float64(-0.25 * Float64(b * a)) + c)
	tmp = 0.0
	if (t_1 <= -2e+223)
		tmp = t_2;
	elseif (t_1 <= -600000000.0)
		tmp = Float64(Float64(t * z) * 0.0625);
	elseif (t_1 <= 2e+50)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+223], t$95$2, If[LessEqual[t$95$1, -600000000.0], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], If[LessEqual[t$95$1, 2e+50], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000009e223 or 2.0000000000000002e50 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   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 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 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.00000000000000009e223 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -6e8

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   if -6e8 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.0000000000000002e50

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 63.0

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

   8. Applied rewrites 63.0%

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

Alternative 6: 60.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+223}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -600000000:\\ \;\;\;\;\left(t \cdot z\right) \cdot 0.0625\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -2e+223)
     t_2
     (if (<= t_1 -600000000.0)
       (* (* t z) 0.0625)
       (if (<= t_1 2e+140) (fma y x c) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -2e+223) {
		tmp = t_2;
	} else if (t_1 <= -600000000.0) {
		tmp = (t * z) * 0.0625;
	} else if (t_1 <= 2e+140) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -2e+223)
		tmp = t_2;
	elseif (t_1 <= -600000000.0)
		tmp = Float64(Float64(t * z) * 0.0625);
	elseif (t_1 <= 2e+140)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+223], t$95$2, If[LessEqual[t$95$1, -600000000.0], N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision], If[LessEqual[t$95$1, 2e+140], N[(y * x + c), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2.00000000000000009e223 or 2.00000000000000012e140 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 94.4%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. 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 77.0

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

   5. Applied rewrites 77.0%

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

   if -2.00000000000000009e223 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -6e8

   1. Initial program 99.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 26.6

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

   5. Applied rewrites 26.6%

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

   if -6e8 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.00000000000000012e140

   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 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 61.3

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

   8. Applied rewrites 61.3%

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

Alternative 7: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ t_2 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_2 \leq -2000000000:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{elif}\;t\_2 \leq 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot b, a, t\_1\right) + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* t z) 0.0625)) (t_2 (/ (* a b) 4.0)))
   (if (<= t_2 -2000000000.0)
     (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)
     (if (<= t_2 1e+72) (+ (fma y x t_1) c) (+ (fma (* -0.25 b) a t_1) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (t * z) * 0.0625;
	double t_2 = (a * b) / 4.0;
	double tmp;
	if (t_2 <= -2000000000.0) {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	} else if (t_2 <= 1e+72) {
		tmp = fma(y, x, t_1) + c;
	} else {
		tmp = fma((-0.25 * b), a, t_1) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(t * z) * 0.0625)
	t_2 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_2 <= -2000000000.0)
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	elseif (t_2 <= 1e+72)
		tmp = Float64(fma(y, x, t_1) + c);
	else
		tmp = Float64(fma(Float64(-0.25 * b), a, t_1) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$2, -2000000000.0], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$2, 1e+72], N[(N[(y * x + t$95$1), $MachinePrecision] + c), $MachinePrecision], N[(N[(N[(-0.25 * b), $MachinePrecision] * a + t$95$1), $MachinePrecision] + c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2e9

   1. Initial program 96.6%

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

   3. Taylor expanded in 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 81.4

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

   5. Applied rewrites 81.4%

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

   if -2e9 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999944e71

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 95.5

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

   7. Applied rewrites 95.5%

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

   if 9.99999999999999944e71 < (/.f64 (*.f64 a b) #s(literal 4 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 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 84.0

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

   5. Applied rewrites 84.0%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 83.9

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

   7. Applied rewrites 83.9%

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

Alternative 8: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t \cdot z\right) \cdot 0.0625\\ t_2 := \frac{a \cdot b}{4}\\ t_3 := \mathsf{fma}\left(-0.25 \cdot b, a, t\_1\right) + c\\ \mathbf{if}\;t\_2 \leq -2000000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(y, x, t\_1\right) + c\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (* (* t z) 0.0625))
        (t_2 (/ (* a b) 4.0))
        (t_3 (+ (fma (* -0.25 b) a t_1) c)))
   (if (<= t_2 -2000000000.0)
     t_3
     (if (<= t_2 1e+72) (+ (fma y x t_1) c) t_3))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (t * z) * 0.0625;
	double t_2 = (a * b) / 4.0;
	double t_3 = fma((-0.25 * b), a, t_1) + c;
	double tmp;
	if (t_2 <= -2000000000.0) {
		tmp = t_3;
	} else if (t_2 <= 1e+72) {
		tmp = fma(y, x, t_1) + c;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(t * z) * 0.0625)
	t_2 = Float64(Float64(a * b) / 4.0)
	t_3 = Float64(fma(Float64(-0.25 * b), a, t_1) + c)
	tmp = 0.0
	if (t_2 <= -2000000000.0)
		tmp = t_3;
	elseif (t_2 <= 1e+72)
		tmp = Float64(fma(y, x, t_1) + c);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(-0.25 * b), $MachinePrecision] * a + t$95$1), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[t$95$2, -2000000000.0], t$95$3, If[LessEqual[t$95$2, 1e+72], N[(N[(y * x + t$95$1), $MachinePrecision] + c), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -2e9 or 9.99999999999999944e71 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.5%

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

   3. Taylor expanded in x around 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 82.6

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

   5. Applied rewrites 82.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*r* N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 82.8

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

   7. Applied rewrites 82.8%

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

   if -2e9 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 9.99999999999999944e71

   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 a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 95.5

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

   7. Applied rewrites 95.5%

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

Alternative 9: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(t \cdot z\right) \cdot 0.0625\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -5e+126)
     (+ (fma y x (* -0.25 (* b a))) c)
     (if (<= t_1 2e+64)
       (+ (fma y x (* (* t z) 0.0625)) c)
       (- (fma y x c) (* 0.25 (* b a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -5e+126) {
		tmp = fma(y, x, (-0.25 * (b * a))) + c;
	} else if (t_1 <= 2e+64) {
		tmp = fma(y, x, ((t * z) * 0.0625)) + c;
	} else {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -5e+126)
		tmp = Float64(fma(y, x, Float64(-0.25 * Float64(b * a))) + c);
	elseif (t_1 <= 2e+64)
		tmp = Float64(fma(y, x, Float64(Float64(t * z) * 0.0625)) + c);
	else
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+126], N[(N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$1, 2e+64], N[(N[(y * x + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -4.99999999999999977e126

   1. Initial program 95.4%

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

   3. Taylor expanded in 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 86.6

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

   5. Applied rewrites 86.6%

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

   if -4.99999999999999977e126 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.00000000000000004e64

   1. Initial program 99.3%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 93.3

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

   5. Applied rewrites 93.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 93.6

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

   7. Applied rewrites 93.6%

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

   if 2.00000000000000004e64 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.5%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 83.3

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

   5. Applied rewrites 83.3%

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

Alternative 10: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+281}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* t z) 0.0625)))
   (if (<= t_1 -2e+281)
     t_2
     (if (<= t_1 1.5e+144) (- (fma y x c) (* 0.25 (* b a))) (+ t_2 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 t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -2e+281) {
		tmp = t_2;
	} else if (t_1 <= 1.5e+144) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = t_2 + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(t * z) * 0.0625)
	tmp = 0.0
	if (t_1 <= -2e+281)
		tmp = t_2;
	elseif (t_1 <= 1.5e+144)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = Float64(t_2 + 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]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+281], t$95$2, If[LessEqual[t$95$1, 1.5e+144], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -2.0000000000000001e281

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 92.1

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

   5. Applied rewrites 92.1%

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

   if -2.0000000000000001e281 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1.49999999999999995e144

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.49999999999999995e144 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 95.2%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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 77.8

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

   5. Applied rewrites 77.8%

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

Alternative 11: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -50000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)) (t_2 (* -0.25 (* b a))))
   (if (<= t_1 -50000000000000.0) t_2 (if (<= t_1 2e+140) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double t_2 = -0.25 * (b * a);
	double tmp;
	if (t_1 <= -50000000000000.0) {
		tmp = t_2;
	} else if (t_1 <= 2e+140) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	t_2 = Float64(-0.25 * Float64(b * a))
	tmp = 0.0
	if (t_1 <= -50000000000000.0)
		tmp = t_2;
	elseif (t_1 <= 2e+140)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, Block[{t$95$2 = N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000000.0], t$95$2, If[LessEqual[t$95$1, 2e+140], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -5e13 or 2.00000000000000012e140 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 96.1%

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

   3. Taylor expanded in a around 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 60.7

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

   5. Applied rewrites 60.7%

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

   if -5e13 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2.00000000000000012e140

   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 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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift-fma.f64 61.3

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

   8. Applied rewrites 61.3%

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

Alternative 12: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{if}\;t \leq 2.7 \cdot 10^{-287}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t, \frac{t\_1}{z}\right) \cdot z + c\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, z, \frac{t\_1}{t}\right) \cdot t + c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* -0.25 (* b a)))))
   (if (<= t 2.7e-287)
     (+ (* (fma 0.0625 t (/ t_1 z)) z) c)
     (if (<= t 4.5e-51)
       (- (fma y x c) (* 0.25 (* b a)))
       (+ (* (fma 0.0625 z (/ t_1 t)) t) c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (-0.25 * (b * a)));
	double tmp;
	if (t <= 2.7e-287) {
		tmp = (fma(0.0625, t, (t_1 / z)) * z) + c;
	} else if (t <= 4.5e-51) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else {
		tmp = (fma(0.0625, z, (t_1 / t)) * t) + c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(-0.25 * Float64(b * a)))
	tmp = 0.0
	if (t <= 2.7e-287)
		tmp = Float64(Float64(fma(0.0625, t, Float64(t_1 / z)) * z) + c);
	elseif (t <= 4.5e-51)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = Float64(Float64(fma(0.0625, z, Float64(t_1 / t)) * t) + c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.7e-287], N[(N[(N[(0.0625 * t + N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t, 4.5e-51], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.0625 * z + N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.7000000000000001e-287

   1. Initial program 98.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.6%

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

   if 2.7000000000000001e-287 < t < 4.49999999999999974e-51

   1. Initial program 99.3%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 4.49999999999999974e-51 < t

   1. Initial program 97.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.7%

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

Alternative 13: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ t_2 := \mathsf{fma}\left(0.0625, t, \frac{t\_1}{z}\right) \cdot z + c\\ \mathbf{if}\;z \leq -1.26 \cdot 10^{-152}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-140}:\\ \;\;\;\;t\_1 + c\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* -0.25 (* b a))))
        (t_2 (+ (* (fma 0.0625 t (/ t_1 z)) z) c)))
   (if (<= z -1.26e-152) t_2 (if (<= z 3.2e-140) (+ t_1 c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (-0.25 * (b * a)));
	double t_2 = (fma(0.0625, t, (t_1 / z)) * z) + c;
	double tmp;
	if (z <= -1.26e-152) {
		tmp = t_2;
	} else if (z <= 3.2e-140) {
		tmp = t_1 + c;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(-0.25 * Float64(b * a)))
	t_2 = Float64(Float64(fma(0.0625, t, Float64(t_1 / z)) * z) + c)
	tmp = 0.0
	if (z <= -1.26e-152)
		tmp = t_2;
	elseif (z <= 3.2e-140)
		tmp = Float64(t_1 + c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.0625 * t + N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] + c), $MachinePrecision]}, If[LessEqual[z, -1.26e-152], t$95$2, If[LessEqual[z, 3.2e-140], N[(t$95$1 + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2600000000000001e-152 or 3.2000000000000001e-140 < z

   1. Initial program 97.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

   if -1.2600000000000001e-152 < z < 3.2000000000000001e-140

   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 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 95.0

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

   5. Applied rewrites 95.0%

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

Alternative 14: 41.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -50000000000:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+25}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -50000000000.0) (* y x) (if (<= (* x y) 5e+25) c (* y x))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -50000000000.0) {
		tmp = y * x;
	} else if ((x * y) <= 5e+25) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if ((x * y) <= (-50000000000.0d0)) then
        tmp = y * x
    else if ((x * y) <= 5d+25) then
        tmp = c
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -50000000000.0) {
		tmp = y * x;
	} else if ((x * y) <= 5e+25) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -50000000000.0:
		tmp = y * x
	elif (x * y) <= 5e+25:
		tmp = c
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -50000000000.0)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 5e+25)
		tmp = c;
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c)
	tmp = 0.0;
	if ((x * y) <= -50000000000.0)
		tmp = y * x;
	elseif ((x * y) <= 5e+25)
		tmp = c;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -50000000000.0], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+25], c, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5e10 or 5.00000000000000024e25 < (*.f64 x y)

   1. Initial program 97.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 54.2

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

   5. Applied rewrites 54.2%

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

   if -5e10 < (*.f64 x y) < 5.00000000000000024e25

   1. Initial program 99.1%

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

   3. Taylor expanded in c around inf

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

   5. Applied rewrites 30.5%

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

Alternative 15: 48.5% 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 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. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 73.4

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

5. Applied rewrites 73.4%

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

6. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. lift-fma.f64 48.5

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

8. Applied rewrites 48.5%

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

Alternative 16: 22.4% 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 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 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 2025089 
(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))