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

Percentage Accurate: 97.6% → 98.3%

Time: 3.5s

Alternatives: 14

Speedup: 0.7×

• 33.0% 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: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot b}{4} \leq 2 \cdot 10^{+296}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t}{16}, y \cdot x\right) - \left(\frac{b \cdot a}{4} - c\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (/ (* a b) 4.0) 2e+296)
   (- (fma z (/ t 16.0) (* y x)) (- (/ (* b a) 4.0) c))
   (* -0.25 (* b a))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double tmp;
	if (((a * b) / 4.0) <= 2e+296) {
		tmp = fma(z, (t / 16.0), (y * x)) - (((b * a) / 4.0) - c);
	} else {
		tmp = -0.25 * (b * a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(Float64(a * b) / 4.0) <= 2e+296)
		tmp = Float64(fma(z, Float64(t / 16.0), Float64(y * x)) - Float64(Float64(Float64(b * a) / 4.0) - c));
	else
		tmp = Float64(-0.25 * Float64(b * a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision], 2e+296], N[(N[(z * N[(t / 16.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(b * a), $MachinePrecision] / 4.0), $MachinePrecision] - c), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 1.99999999999999996e296

   1. Initial program 97.6%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-+l- N/A

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

      10. lower--.f64 N/A

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

   3. Applied rewrites 98.0%

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

   if 1.99999999999999996e296 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

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

   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 29.2

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

   4. Applied rewrites 29.2%

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

Alternative 2: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(b \cdot a\right)\right) + c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+114)
   (fma (* 0.0625 t) z (fma x y c))
   (if (<= (* x y) 2e+176)
     (+ (fma (* 0.0625 t) z (* -0.25 (* b a))) c)
     (fma (* 0.0625 t) z (* 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+114) {
		tmp = fma((0.0625 * t), z, fma(x, y, c));
	} else if ((x * y) <= 2e+176) {
		tmp = fma((0.0625 * t), z, (-0.25 * (b * a))) + c;
	} else {
		tmp = fma((0.0625 * t), z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+114)
		tmp = fma(Float64(0.0625 * t), z, fma(x, y, c));
	elseif (Float64(x * y) <= 2e+176)
		tmp = Float64(fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(b * a))) + c);
	else
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+114], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+176], N[(N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e114

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -2e114 < (*.f64 x y) < 2e176

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.3

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

   4. Applied rewrites 74.3%

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

   if 2e176 < (*.f64 x y)

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 53.2

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

   11. Applied rewrites 53.2%

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

Alternative 3: 88.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot y \leq -2 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{elif}\;x \cdot y \leq 2 \cdot 10^{+176}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot z, 0.0625, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (if (<= (* x y) -2e+114)
   (fma (* 0.0625 t) z (fma x y c))
   (if (<= (* x y) 2e+176)
     (- (fma (* t z) 0.0625 c) (* 0.25 (* b a)))
     (fma (* 0.0625 t) z (* 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+114) {
		tmp = fma((0.0625 * t), z, fma(x, y, c));
	} else if ((x * y) <= 2e+176) {
		tmp = fma((t * z), 0.0625, c) - (0.25 * (b * a));
	} else {
		tmp = fma((0.0625 * t), z, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	tmp = 0.0
	if (Float64(x * y) <= -2e+114)
		tmp = fma(Float64(0.0625 * t), z, fma(x, y, c));
	elseif (Float64(x * y) <= 2e+176)
		tmp = Float64(fma(Float64(t * z), 0.0625, c) - Float64(0.25 * Float64(b * a)));
	else
		tmp = fma(Float64(0.0625 * t), z, Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := If[LessEqual[N[(x * y), $MachinePrecision], -2e+114], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * y), $MachinePrecision], 2e+176], N[(N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(y * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < -2e114

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -2e114 < (*.f64 x y) < 2e176

   1. Initial program 97.6%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   if 2e176 < (*.f64 x y)

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 53.2

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

   11. Applied rewrites 53.2%

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

Alternative 4: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, -0.25 \cdot \left(a \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -2e+179)
     (- (fma y x c) (* 0.25 (* b a)))
     (if (<= t_1 2e+208)
       (fma (* 0.0625 t) z (fma x y c))
       (fma (* 0.0625 t) z (* -0.25 (* a b)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -2e+179) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else if (t_1 <= 2e+208) {
		tmp = fma((0.0625 * t), z, fma(x, y, c));
	} else {
		tmp = fma((0.0625 * t), z, (-0.25 * (a * b)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -2e+179)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	elseif (t_1 <= 2e+208)
		tmp = fma(Float64(0.0625 * t), z, fma(x, y, c));
	else
		tmp = fma(Float64(0.0625 * t), z, Float64(-0.25 * Float64(a * b)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+179], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+208], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(-0.25 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999996e179

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   if -1.99999999999999996e179 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e208

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 2e208 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in c around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 53.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

Alternative 5: 87.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+179}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.25 \cdot a, b, \left(t \cdot z\right) \cdot 0.0625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -2e+179)
     (- (fma y x c) (* 0.25 (* b a)))
     (if (<= t_1 2e+208)
       (fma (* 0.0625 t) z (fma x y c))
       (fma (* -0.25 a) b (* (* t z) 0.0625))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -2e+179) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} else if (t_1 <= 2e+208) {
		tmp = fma((0.0625 * t), z, fma(x, y, c));
	} else {
		tmp = fma((-0.25 * a), b, ((t * z) * 0.0625));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -2e+179)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	elseif (t_1 <= 2e+208)
		tmp = fma(Float64(0.0625 * t), z, fma(x, y, c));
	else
		tmp = fma(Float64(-0.25 * a), b, Float64(Float64(t * z) * 0.0625));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+179], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+208], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * a), $MachinePrecision] * b + N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -1.99999999999999996e179

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   if -1.99999999999999996e179 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e208

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 2e208 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

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

   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 73.9

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

   4. Applied rewrites 73.9%

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

   5. Taylor expanded in c around 0

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

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

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

      2. associate-*r* N/A

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

      3. metadata-eval N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 53.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 53.8

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

   9. Applied rewrites 53.8%

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

Alternative 6: 87.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right) - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 x y c))))
   (if (<= t_1 -1e+54)
     t_2
     (if (<= t_1 5e+44) (- (fma y x c) (* 0.25 (* b a))) 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(x, y, c));
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 5e+44) {
		tmp = fma(y, x, c) - (0.25 * (b * a));
	} 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(x, y, c))
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 5e+44)
		tmp = Float64(fma(y, x, c) - Float64(0.25 * Float64(b * a)));
	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[(x * y + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 5e+44], N[(N[(y * x + c), $MachinePrecision] - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 4.9999999999999996e44 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.9999999999999996e44

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

Alternative 7: 86.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+229}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, a \cdot b, x \cdot y\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(0.0625 \cdot t, z, \mathsf{fma}\left(x, y, c\right)\right)\\ \mathbf{else}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* a b) 4.0)))
   (if (<= t_1 -1e+229)
     (fma -0.25 (* a b) (* x y))
     (if (<= t_1 2e+208)
       (fma (* 0.0625 t) z (fma x y c))
       (- c (* 0.25 (* b a)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (a * b) / 4.0;
	double tmp;
	if (t_1 <= -1e+229) {
		tmp = fma(-0.25, (a * b), (x * y));
	} else if (t_1 <= 2e+208) {
		tmp = fma((0.0625 * t), z, fma(x, y, c));
	} else {
		tmp = c - (0.25 * (b * a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(a * b) / 4.0)
	tmp = 0.0
	if (t_1 <= -1e+229)
		tmp = fma(-0.25, Float64(a * b), Float64(x * y));
	elseif (t_1 <= 2e+208)
		tmp = fma(Float64(0.0625 * t), z, fma(x, y, c));
	else
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+229], N[(-0.25 * N[(a * b), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+208], N[(N[(0.0625 * t), $MachinePrecision] * z + N[(x * y + c), $MachinePrecision]), $MachinePrecision], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 a b) #s(literal 4 binary64)) < -9.9999999999999999e228

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto x \cdot y - \color{blue}{\frac{1}{4} \cdot \left(a \cdot b\right)} \]
   6. 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. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   if -9.9999999999999999e228 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e208

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   if 2e208 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.5%

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

Alternative 8: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(0.0625 \cdot t, z, y \cdot x\right)\\ t_2 := x \cdot y + \frac{z \cdot t}{16}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+95}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e+114) t_1 (if (<= t_2 5e+95) (- c (* 0.25 (* b a))) 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 <= -1e+114) {
		tmp = t_1;
	} else if (t_2 <= 5e+95) {
		tmp = c - (0.25 * (b * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(0.0625 * t), z, Float64(y * x))
	t_2 = Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0))
	tmp = 0.0
	if (t_2 <= -1e+114)
		tmp = t_1;
	elseif (t_2 <= 5e+95)
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(0.0625 * t), $MachinePrecision] * z + 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, -1e+114], t$95$1, If[LessEqual[t$95$2, 5e+95], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $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))) < -1e114 or 5.00000000000000025e95 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fma.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lift-*.f64 53.2

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

   11. Applied rewrites 53.2%

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

   if -1e114 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 5.00000000000000025e95

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.5%

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

Alternative 9: 64.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-307}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+71}:\\ \;\;\;\;c - 0.25 \cdot \left(b \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* t z) 0.0625 c)))
   (if (<= t_1 -1e+54)
     t_2
     (if (<= t_1 1e-307)
       (fma y x c)
       (if (<= t_1 5e+71) (- c (* 0.25 (* b a))) 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((t * z), 0.0625, c);
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 1e-307) {
		tmp = fma(y, x, c);
	} else if (t_1 <= 5e+71) {
		tmp = c - (0.25 * (b * a));
	} 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(t * z), 0.0625, c)
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 1e-307)
		tmp = fma(y, x, c);
	elseif (t_1 <= 5e+71)
		tmp = Float64(c - Float64(0.25 * Float64(b * a)));
	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[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 1e-307], N[(y * x + c), $MachinePrecision], If[LessEqual[t$95$1, 5e+71], N[(c - N[(0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 4.99999999999999972e71 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 48.2

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

   7. Applied rewrites 48.2%

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

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999909e-308

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   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 48.4

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

   12. Applied rewrites 48.4%

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

   if 9.99999999999999909e-308 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 4.99999999999999972e71

   1. Initial program 97.6%

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

   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 74.3

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 49.5%

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

Alternative 10: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \mathsf{fma}\left(t \cdot z, 0.0625, c\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (fma (* t z) 0.0625 c)))
   (if (<= t_1 -1e+54) t_2 (if (<= t_1 1e+88) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = fma((t * z), 0.0625, c);
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 1e+88) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = fma(Float64(t * z), 0.0625, c)
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 1e+88)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625 + c), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 1e+88], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 9.99999999999999959e87 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

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

   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 48.2

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

   7. Applied rewrites 48.2%

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

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999959e87

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   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 48.4

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

   12. Applied rewrites 48.4%

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

Alternative 11: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z \cdot t}{16}\\ t_2 := \left(t \cdot z\right) \cdot 0.0625\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (/ (* z t) 16.0)) (t_2 (* (* t z) 0.0625)))
   (if (<= t_1 -1e+54) t_2 (if (<= t_1 1e+88) (fma y x c) t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = (z * t) / 16.0;
	double t_2 = (t * z) * 0.0625;
	double tmp;
	if (t_1 <= -1e+54) {
		tmp = t_2;
	} else if (t_1 <= 1e+88) {
		tmp = fma(y, x, c);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = Float64(Float64(z * t) / 16.0)
	t_2 = Float64(Float64(t * z) * 0.0625)
	tmp = 0.0
	if (t_1 <= -1e+54)
		tmp = t_2;
	elseif (t_1 <= 1e+88)
		tmp = fma(y, x, c);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * z), $MachinePrecision] * 0.0625), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+54], t$95$2, If[LessEqual[t$95$1, 1e+88], N[(y * x + c), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 z t) #s(literal 16 binary64)) < -1.0000000000000001e54 or 9.99999999999999959e87 < (/.f64 (*.f64 z t) #s(literal 16 binary64))

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 28.1

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

   9. Applied rewrites 28.1%

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

   if -1.0000000000000001e54 < (/.f64 (*.f64 z t) #s(literal 16 binary64)) < 9.99999999999999959e87

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   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 48.4

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

   12. Applied rewrites 48.4%

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

Alternative 12: 61.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a \cdot b}{4}\\ t_2 := -0.25 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+82}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+208}:\\ \;\;\;\;\mathsf{fma}\left(y, x, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+82) t_2 (if (<= t_1 2e+208) (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 <= -5e+82) {
		tmp = t_2;
	} else if (t_1 <= 2e+208) {
		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 <= -5e+82)
		tmp = t_2;
	elseif (t_1 <= 2e+208)
		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, -5e+82], t$95$2, If[LessEqual[t$95$1, 2e+208], 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)) < -5.00000000000000015e82 or 2e208 < (/.f64 (*.f64 a b) #s(literal 4 binary64))

   1. Initial program 97.6%

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

   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 29.2

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

   4. Applied rewrites 29.2%

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

   if -5.00000000000000015e82 < (/.f64 (*.f64 a b) #s(literal 4 binary64)) < 2e208

   1. Initial program 97.6%

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

   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 73.1

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

   4. Applied rewrites 73.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
   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. *-commutative N/A

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

      11. lower-fma.f64 73.5

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

   6. Applied rewrites 73.5%

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

   7. Taylor expanded in c around 0

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

      1. associate-*l* N/A

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

      2. associate-+r+ N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 52.8

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

   9. Applied rewrites 52.8%

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

   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 48.4

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

   12. Applied rewrites 48.4%

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

Alternative 13: 48.4% 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 97.6%

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

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 73.1

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

4. Applied rewrites 73.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
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. *-commutative N/A

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

   11. lower-fma.f64 73.5

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

6. Applied rewrites 73.5%

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

7. Taylor expanded in c around 0

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

   1. associate-*l* N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 52.8

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

9. Applied rewrites 52.8%

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

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 48.4

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

12. Applied rewrites 48.4%

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

Alternative 14: 22.2% 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 97.6%

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

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 73.1

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

4. Applied rewrites 73.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot z, 0.0625, y \cdot x\right) + c} \]
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. *-commutative N/A

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

   11. lower-fma.f64 73.5

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

6. Applied rewrites 73.5%

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

7. Taylor expanded in c around 0

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

   1. associate-*l* N/A

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

   2. associate-+r+ N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 52.8

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

9. Applied rewrites 52.8%

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

10. Taylor expanded in c around inf

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

12. Applied rewrites 22.2%

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

Reproduce

herbie shell --seed 2025142 
(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))