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

Percentage Accurate: 97.8% → 97.8%

Time: 3.6s

Alternatives: 12

Speedup: 1.0×

• 33.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% 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 97.8%

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

Alternative 2: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot a, -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -4e+118)
     (fma (* 0.0625 t) z (fma y x c))
     (if (<= t_1 1e+133)
       (fma y x (fma (* b a) -0.25 c))
       (- (fma (* t z) 0.0625 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 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -4e+118) {
		tmp = fma((0.0625 * t), z, fma(y, x, c));
	} else if (t_1 <= 1e+133) {
		tmp = fma(y, x, fma((b * a), -0.25, c));
	} else {
		tmp = fma((t * z), 0.0625, c) - (0.25 * (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -4e+118)
		tmp = fma(Float64(0.0625 * t), z, fma(y, x, c));
	elseif (t_1 <= 1e+133)
		tmp = fma(y, x, fma(Float64(b * a), -0.25, c));
	else
		tmp = Float64(fma(Float64(t * z), 0.0625, 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[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+118], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+133], N[(y * x + N[(N[(b * a), $MachinePrecision] * -0.25 + c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -3.99999999999999987e118

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 74.2

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

   6. Applied rewrites 74.2%

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

   if -3.99999999999999987e118 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1e133

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if 1e133 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.8%

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

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.6

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

   4. Applied rewrites 72.6%

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

Alternative 3: 89.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(y, x, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+118}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot a, -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* 0.0625 t) z (fma y x c))))
   (if (<= t_1 -4e+118)
     t_2
     (if (<= t_1 1e+138) (fma y x (fma (* b a) -0.25 c)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((0.0625 * t), z, fma(y, x, c));
	double tmp;
	if (t_1 <= -4e+118) {
		tmp = t_2;
	} else if (t_1 <= 1e+138) {
		tmp = fma(y, x, fma((b * a), -0.25, c));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(0.0625 * t), z, fma(y, x, c))
	tmp = 0.0
	if (t_1 <= -4e+118)
		tmp = t_2;
	elseif (t_1 <= 1e+138)
		tmp = fma(y, x, fma(Float64(b * a), -0.25, c));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+118], t$95$2, If[LessEqual[t$95$1, 1e+138], N[(y * x + N[(N[(b * a), $MachinePrecision] * -0.25 + c), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -3.99999999999999987e118 or 1e138 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-fma.f64 74.2

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

   6. Applied rewrites 74.2%

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

   if -3.99999999999999987e118 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1e138

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   8. Applied rewrites 74.3%

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

Alternative 4: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+118}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, t \cdot z, y \cdot x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \mathsf{fma}\left(b \cdot a, -0.25, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)))
   (if (<= t_1 -4e+118)
     (fma 0.0625 (* t z) (* y x))
     (if (<= t_1 1e+138)
       (fma y x (fma (* b a) -0.25 c))
       (fma (* 0.0625 t) z c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double tmp;
	if (t_1 <= -4e+118) {
		tmp = fma(0.0625, (t * z), (y * x));
	} else if (t_1 <= 1e+138) {
		tmp = fma(y, x, fma((b * a), -0.25, c));
	} else {
		tmp = fma((0.0625 * t), z, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	tmp = 0.0
	if (t_1 <= -4e+118)
		tmp = fma(0.0625, Float64(t * z), Float64(y * x));
	elseif (t_1 <= 1e+138)
		tmp = fma(y, x, fma(Float64(b * a), -0.25, c));
	else
		tmp = fma(Float64(0.0625 * t), z, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+118], N[(0.0625 * N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+138], N[(y * x + N[(N[(b * a), $MachinePrecision] * -0.25 + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -3.99999999999999987e118

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 54.2

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

   10. Applied rewrites 54.2%

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

   if -3.99999999999999987e118 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 1e138

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   8. Applied rewrites 74.3%

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

   if 1e138 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

Alternative 5: 77.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625, t \cdot z, y \cdot x\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma 0.0625 (* t z) (* y x))) (t_2 (+ (* x y) (/ (* z t) 16.0))))
   (if (<= t_2 -4e+118) t_1 (if (<= t_2 5e+151) (fma -0.25 (* b a) c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(0.0625, (t * z), (y * x));
	double t_2 = (x * y) + ((z * t) / 16.0);
	double tmp;
	if (t_2 <= -4e+118) {
		tmp = t_1;
	} else if (t_2 <= 5e+151) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(0.0625, Float64(t * z), Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -4e+118)
		tmp = t_1;
	elseif (t_2 <= 5e+151)
		tmp = fma(-0.25, Float64(b * a), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(0.0625 * N[(t * z), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+118], t$95$1, If[LessEqual[t$95$2, 5e+151], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -3.99999999999999987e118 or 5.0000000000000002e151 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in c around 0

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

      1. lower-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 54.2

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

   10. Applied rewrites 54.2%

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

   if -3.99999999999999987e118 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.0000000000000002e151

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 47.7

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

   9. Applied rewrites 47.7%

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

Alternative 6: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, y \cdot x\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+41)
   (fma -0.25 (* b a) (* y x))
   (if (<= (* x y) 2e-161)
     (fma -0.25 (* b a) c)
     (if (<= (* x y) 1e+108) (fma (* 0.0625 t) z c) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -2e+41) {
		tmp = fma(-0.25, (b * a), (y * x));
	} else if ((x * y) <= 2e-161) {
		tmp = fma(-0.25, (b * a), c);
	} else if ((x * y) <= 1e+108) {
		tmp = fma((0.0625 * t), z, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+41)
		tmp = fma(-0.25, Float64(b * a), Float64(y * x));
	elseif (Float64(x * y) <= 2e-161)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (Float64(x * y) <= 1e+108)
		tmp = fma(Float64(0.0625 * t), z, c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+41], N[(-0.25 * N[(b * a), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-161], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+108], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 x y) < -2.00000000000000001e41

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in c around 0

      \[\leadsto x \cdot y - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   8. 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. lower-*.f64 54.4

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

   9. Applied rewrites 54.4%

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

   if -2.00000000000000001e41 < (*.f64 x y) < 2.00000000000000006e-161

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 47.7

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

   9. Applied rewrites 47.7%

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

   if 2.00000000000000006e-161 < (*.f64 x y) < 1e108

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   if 1e108 < (*.f64 x y)

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 49.0

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

   12. Applied rewrites 49.0%

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

Alternative 7: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;x \cdot y \leq 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1e+144)
   (fma y x c)
   (if (<= (* x y) 2e-161)
     (fma -0.25 (* b a) c)
     (if (<= (* x y) 1e+108) (fma (* 0.0625 t) z c) (fma y x c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1e+144) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 2e-161) {
		tmp = fma(-0.25, (b * a), c);
	} else if ((x * y) <= 1e+108) {
		tmp = fma((0.0625 * t), z, c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1e+144)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 2e-161)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (Float64(x * y) <= 1e+108)
		tmp = fma(Float64(0.0625 * t), z, c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+144], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e-161], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 1e+108], N[(N[(0.0625 * t), $MachinePrecision] * z + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -1.00000000000000002e144 or 1e108 < (*.f64 x y)

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 49.0

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

   12. Applied rewrites 49.0%

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

   if -1.00000000000000002e144 < (*.f64 x y) < 2.00000000000000006e-161

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 47.7

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

   9. Applied rewrites 47.7%

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

   if 2.00000000000000006e-161 < (*.f64 x y) < 1e108

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

Alternative 8: 64.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -1 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -1e+144)
   (fma y x c)
   (if (<= (* x y) 5e+71) (fma -0.25 (* b a) c) (fma y x c))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if ((x * y) <= -1e+144) {
		tmp = fma(y, x, c);
	} else if ((x * y) <= 5e+71) {
		tmp = fma(-0.25, (b * a), c);
	} else {
		tmp = fma(y, x, c);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -1e+144)
		tmp = fma(y, x, c);
	elseif (Float64(x * y) <= 5e+71)
		tmp = fma(-0.25, Float64(b * a), c);
	else
		tmp = fma(y, x, c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -1e+144], N[(y * x + c), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+71], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], N[(y * x + c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -1.00000000000000002e144 or 4.99999999999999972e71 < (*.f64 x y)

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 49.0

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

   12. Applied rewrites 49.0%

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

   if -1.00000000000000002e144 < (*.f64 x y) < 4.99999999999999972e71

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in x around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 47.7

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

   9. Applied rewrites 47.7%

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

Alternative 9: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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+153) t_2 (if (<= t_1 4e+103) (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+153) {
		tmp = t_2;
	} else if (t_1 <= 4e+103) {
		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+153)
		tmp = t_2;
	elseif (t_1 <= 4e+103)
		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+153], t$95$2, If[LessEqual[t$95$1, 4e+103], 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)) < -2e153 or 4e103 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.8%

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

   2. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 28.4

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

   4. Applied rewrites 28.4%

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

   if -2e153 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 4e103

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 49.0

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

   12. Applied rewrites 49.0%

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

Alternative 10: 49.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 73.7

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

4. Applied rewrites 73.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 47.5

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

7. Applied rewrites 47.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 47.6

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

9. Applied rewrites 47.6%

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

10. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 49.0

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

12. Applied rewrites 49.0%

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

Alternative 11: 41.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+41}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{+151}:\\ \;\;\;\;c\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+41) (* y x) (if (<= (* x y) 5e+151) 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) <= -2e+41) {
		tmp = y * x;
	} else if ((x * y) <= 5e+151) {
		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) <= (-2d+41)) then
        tmp = y * x
    else if ((x * y) <= 5d+151) 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) <= -2e+41) {
		tmp = y * x;
	} else if ((x * y) <= 5e+151) {
		tmp = c;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c):
	tmp = 0
	if (x * y) <= -2e+41:
		tmp = y * x
	elif (x * y) <= 5e+151:
		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) <= -2e+41)
		tmp = Float64(y * x);
	elseif (Float64(x * y) <= 5e+151)
		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) <= -2e+41)
		tmp = y * x;
	elseif ((x * y) <= 5e+151)
		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], -2e+41], N[(y * x), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 5e+151], c, N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -2.00000000000000001e41 or 5.0000000000000002e151 < (*.f64 x y)

   1. Initial program 97.8%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   3. 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 74.0

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

   4. Applied rewrites 74.0%

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

      1. lift--.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 74.3

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

   6. Applied rewrites 74.3%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 29.7

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

   9. Applied rewrites 29.7%

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

   if -2.00000000000000001e41 < (*.f64 x y) < 5.0000000000000002e151

   1. Initial program 97.8%

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

   2. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 73.7

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 47.5

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

   7. Applied rewrites 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 47.6

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

   9. Applied rewrites 47.6%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 21.3%

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

Alternative 12: 21.3% accurate, 24.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.8%

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

2. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 73.7

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

4. Applied rewrites 73.7%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-*.f64 47.5

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

7. Applied rewrites 47.5%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lift-*.f64 47.6

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

9. Applied rewrites 47.6%

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

10. Taylor expanded in z around 0

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

12. Applied rewrites 21.3%

    \[\leadsto c \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025127 
(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))