Complex division, real part

Percentage Accurate: 61.5% → 84.1%

Time: 5.4s

Alternatives: 11

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((a * c) + (b * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a * c) + (b * d)) / ((c * c) + (d * d));
}

Python

def code(a, b, c, d):
	return ((a * c) + (b * d)) / ((c * c) + (d * d))

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;c \leq -3.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{d}{c}, -a, b\right)}{c}, d, a\right)}{c}\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{t\_0}, c \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{t\_0}, d \cdot \frac{b}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= c -3.6e+151)
     (/ (fma (/ (fma (/ d c) (- a) b) c) d a) c)
     (if (<= c -1.35e-125)
       (fma b (/ d t_0) (* c (/ a t_0)))
       (if (<= c 1.65e-52)
         (/ (fma (/ c d) a b) d)
         (if (<= c 1e+142)
           (fma a (/ c t_0) (* d (/ b t_0)))
           (/ (fma (/ d c) b a) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (c <= -3.6e+151) {
		tmp = fma((fma((d / c), -a, b) / c), d, a) / c;
	} else if (c <= -1.35e-125) {
		tmp = fma(b, (d / t_0), (c * (a / t_0)));
	} else if (c <= 1.65e-52) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 1e+142) {
		tmp = fma(a, (c / t_0), (d * (b / t_0)));
	} else {
		tmp = fma((d / c), b, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (c <= -3.6e+151)
		tmp = Float64(fma(Float64(fma(Float64(d / c), Float64(-a), b) / c), d, a) / c);
	elseif (c <= -1.35e-125)
		tmp = fma(b, Float64(d / t_0), Float64(c * Float64(a / t_0)));
	elseif (c <= 1.65e-52)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 1e+142)
		tmp = fma(a, Float64(c / t_0), Float64(d * Float64(b / t_0)));
	else
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.6e+151], N[(N[(N[(N[(N[(d / c), $MachinePrecision] * (-a) + b), $MachinePrecision] / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.35e-125], N[(b * N[(d / t$95$0), $MachinePrecision] + N[(c * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.65e-52], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1e+142], N[(a * N[(c / t$95$0), $MachinePrecision] + N[(d * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.6e151

   1. Initial program 28.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \left(-1 \cdot \frac{a \cdot {d}^{2}}{{c}^{2}} + \frac{b \cdot d}{c}\right)}{c}} \]

   5. Applied rewrites 80.0%

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

   7. Applied rewrites 91.7%

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

   if -3.6e151 < c < -1.3499999999999999e-125

   1. Initial program 72.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 78.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 78.3

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

   4. Applied rewrites 78.3%

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

   if -1.3499999999999999e-125 < c < 1.64999999999999998e-52

   1. Initial program 68.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 66.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 66.8

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

   4. Applied rewrites 66.8%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{d} + b}}{d} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \frac{c}{d}} + b}{d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{c}{d} \cdot a} + b}{d} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{d} \]

      9. lower-/.f64 88.9

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{c}{d}}, a, b\right)}{d} \]

   7. Applied rewrites 88.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}} \]

   if 1.64999999999999998e-52 < c < 1.00000000000000005e142

   1. Initial program 77.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{a \cdot c}{c \cdot c + d \cdot d} + \frac{b \cdot d}{c \cdot c + d \cdot d}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c}}{c \cdot c + d \cdot d} + \frac{b \cdot d}{c \cdot c + d \cdot d} \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{c}{c \cdot c + d \cdot d}} + \frac{b \cdot d}{c \cdot c + d \cdot d} \]

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 84.8

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 84.8

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

   4. Applied rewrites 84.8%

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

   if 1.00000000000000005e142 < c

   1. Initial program 45.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 46.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 46.5

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

   4. Applied rewrites 46.5%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{b \cdot d}{c} + a}}{c} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{d}{c}} + a}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{d}{c} \cdot b} + a}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}}{c} \]

      6. lower-/.f64 87.8

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{c}}, b, a\right)}{c} \]

   7. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{d}{c}, -a, b\right)}{c}, d, a\right)}{c}\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, c \cdot \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, d \cdot \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -6.6 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c))))
        (t_1 (/ (fma (/ b c) d a) c)))
   (if (<= c -6.6e+88)
     t_1
     (if (<= c -1.35e-125)
       t_0
       (if (<= c 6.3e-105)
         (/ (fma (/ c d) a b) d)
         (if (<= c 9.5e+122) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / fma(d, d, (c * c));
	double t_1 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -6.6e+88) {
		tmp = t_1;
	} else if (c <= -1.35e-125) {
		tmp = t_0;
	} else if (c <= 6.3e-105) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 9.5e+122) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -6.6e+88)
		tmp = t_1;
	elseif (c <= -1.35e-125)
		tmp = t_0;
	elseif (c <= 6.3e-105)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 9.5e+122)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.6e+88], t$95$1, If[LessEqual[c, -1.35e-125], t$95$0, If[LessEqual[c, 6.3e-105], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.5e+122], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.6000000000000006e88 or 9.49999999999999986e122 < c

   1. Initial program 40.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{b}{c} \cdot d + \color{blue}{a}}{c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}}{c} \]

      20. lower-/.f64 86.7

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{b}{c}}, d, a\right)}{c} \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}} \]

   if -6.6000000000000006e88 < c < -1.3499999999999999e-125 or 6.3e-105 < c < 9.49999999999999986e122

   1. Initial program 77.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d} + a \cdot c}{c \cdot c + d \cdot d} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{d \cdot b} + a \cdot c}{c \cdot c + d \cdot d} \]

      5. lower-fma.f64 77.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 77.7

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if -1.3499999999999999e-125 < c < 6.3e-105

   1. Initial program 66.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 68.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 68.7

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

   4. Applied rewrites 68.7%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{d} + b}}{d} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \frac{c}{d}} + b}{d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{c}{d} \cdot a} + b}{d} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{d} \]

      9. lower-/.f64 93.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{c}{d}}, a, b\right)}{d} \]

   7. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, d, c\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot a\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ a d) (/ c d) (/ b d))))
   (if (<= d -6.8e+76)
     t_0
     (if (<= d -2.2e-161)
       (* (/ (fma (/ b a) d c) (fma c c (* d d))) a)
       (if (<= d 7e+31) (/ (fma (/ d c) b a) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), (c / d), (b / d));
	double tmp;
	if (d <= -6.8e+76) {
		tmp = t_0;
	} else if (d <= -2.2e-161) {
		tmp = (fma((b / a), d, c) / fma(c, c, (d * d))) * a;
	} else if (d <= 7e+31) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(a / d), Float64(c / d), Float64(b / d))
	tmp = 0.0
	if (d <= -6.8e+76)
		tmp = t_0;
	elseif (d <= -2.2e-161)
		tmp = Float64(Float64(fma(Float64(b / a), d, c) / fma(c, c, Float64(d * d))) * a);
	elseif (d <= 7e+31)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.8e+76], t$95$0, If[LessEqual[d, -2.2e-161], N[(N[(N[(N[(b / a), $MachinePrecision] * d + c), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[d, 7e+31], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.7999999999999994e76 or 7e31 < d

   1. Initial program 38.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d} + a \cdot c}{c \cdot c + d \cdot d} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{d \cdot b} + a \cdot c}{c \cdot c + d \cdot d} \]

      5. lower-fma.f64 38.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 38.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 38.2

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. unpow2 N/A

         \[\leadsto \frac{a \cdot c}{\color{blue}{d \cdot d}} + \frac{b}{d} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{a}{d} \cdot \frac{c}{d}} + \frac{b}{d} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{a}{d}}, \frac{c}{d}, \frac{b}{d}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{a}{d}, \color{blue}{\frac{c}{d}}, \frac{b}{d}\right) \]

      7. lower-/.f64 80.6

         \[\leadsto \mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \color{blue}{\frac{b}{d}}\right) \]

   7. Applied rewrites 80.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)} \]

   if -6.7999999999999994e76 < d < -2.20000000000000002e-161

   1. Initial program 80.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 78.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 78.5

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(\frac{c}{{c}^{2} + {d}^{2}} + \frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{c}{{c}^{2} + {d}^{2}} + \frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right) \cdot a} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{c}{{c}^{2} + {d}^{2}} + \frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right) \cdot a} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{b \cdot d}{a \cdot \left({c}^{2} + {d}^{2}\right)} + \frac{c}{{c}^{2} + {d}^{2}}\right)} \cdot a \]

      4. times-frac N/A

         \[\leadsto \left(\color{blue}{\frac{b}{a} \cdot \frac{d}{{c}^{2} + {d}^{2}}} + \frac{c}{{c}^{2} + {d}^{2}}\right) \cdot a \]

      5. associate-*r/ N/A

         \[\leadsto \left(\color{blue}{\frac{\frac{b}{a} \cdot d}{{c}^{2} + {d}^{2}}} + \frac{c}{{c}^{2} + {d}^{2}}\right) \cdot a \]

      6. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{a} \cdot d + c}{{c}^{2} + {d}^{2}}} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{b}{a} \cdot d + c}{{c}^{2} + {d}^{2}}} \cdot a \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{b}{a}, d, c\right)}}{{c}^{2} + {d}^{2}} \cdot a \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{b}{a}}, d, c\right)}{{c}^{2} + {d}^{2}} \cdot a \]

      10. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{b}{a}, d, c\right)}{\color{blue}{c \cdot c} + {d}^{2}} \cdot a \]

      11. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{b}{a}, d, c\right)}{\color{blue}{\mathsf{fma}\left(c, c, {d}^{2}\right)}} \cdot a \]

      12. unpow2 N/A

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

      13. lower-*.f64 81.8

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

   7. Applied rewrites 81.8%

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

   if -2.20000000000000002e-161 < d < 7e31

   1. Initial program 70.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 65.4

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 65.4

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{b \cdot d}{c} + a}}{c} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{d}{c}} + a}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{d}{c} \cdot b} + a}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}}{c} \]

      6. lower-/.f64 88.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{c}}, b, a\right)}{c} \]

   7. Applied rewrites 88.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, d, c\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot a\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+125}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-15)
   (/ a c)
   (if (<= c 1.65e-52)
     (/ b d)
     (if (<= c 1.4e+125) (* a (/ c (fma c c (* d d)))) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-15) {
		tmp = a / c;
	} else if (c <= 1.65e-52) {
		tmp = b / d;
	} else if (c <= 1.4e+125) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-15)
		tmp = Float64(a / c);
	elseif (c <= 1.65e-52)
		tmp = Float64(b / d);
	elseif (c <= 1.4e+125)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-15], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.65e-52], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.4e+125], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3e-15 or 1.4e125 < c

   1. Initial program 47.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.0

         \[\leadsto \color{blue}{\frac{a}{c}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -3e-15 < c < 1.64999999999999998e-52

   1. Initial program 69.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.2

         \[\leadsto \color{blue}{\frac{b}{d}} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if 1.64999999999999998e-52 < c < 1.4e125

   1. Initial program 78.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d} + a \cdot c}{c \cdot c + d \cdot d} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{d \cdot b} + a \cdot c}{c \cdot c + d \cdot d} \]

      5. lower-fma.f64 78.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{a \cdot c}{{c}^{2} + {d}^{2}}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{a \cdot \frac{c}{{c}^{2} + {d}^{2}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot \frac{c}{{c}^{2} + {d}^{2}}} \]

      3. lower-/.f64 N/A

         \[\leadsto a \cdot \color{blue}{\frac{c}{{c}^{2} + {d}^{2}}} \]

      4. unpow2 N/A

         \[\leadsto a \cdot \frac{c}{\color{blue}{c \cdot c} + {d}^{2}} \]

      5. lower-fma.f64 N/A

         \[\leadsto a \cdot \frac{c}{\color{blue}{\mathsf{fma}\left(c, c, {d}^{2}\right)}} \]

      6. unpow2 N/A

         \[\leadsto a \cdot \frac{c}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

      7. lower-*.f64 62.2

         \[\leadsto a \cdot \frac{c}{\mathsf{fma}\left(c, c, \color{blue}{d \cdot d}\right)} \]

   7. Applied rewrites 62.2%

      \[\leadsto \color{blue}{a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+125}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 7.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-15)
   (/ a c)
   (if (<= c 2.8e-51)
     (/ b d)
     (if (<= c 7.4e+122) (* (/ a (fma d d (* c c))) c) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-15) {
		tmp = a / c;
	} else if (c <= 2.8e-51) {
		tmp = b / d;
	} else if (c <= 7.4e+122) {
		tmp = (a / fma(d, d, (c * c))) * c;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-15)
		tmp = Float64(a / c);
	elseif (c <= 2.8e-51)
		tmp = Float64(b / d);
	elseif (c <= 7.4e+122)
		tmp = Float64(Float64(a / fma(d, d, Float64(c * c))) * c);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-15], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.8e-51], N[(b / d), $MachinePrecision], If[LessEqual[c, 7.4e+122], N[(N[(a / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3e-15 or 7.3999999999999993e122 < c

   1. Initial program 47.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.0

         \[\leadsto \color{blue}{\frac{a}{c}} \]

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -3e-15 < c < 2.8e-51

   1. Initial program 69.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.2

         \[\leadsto \color{blue}{\frac{b}{d}} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if 2.8e-51 < c < 7.3999999999999993e122

   1. Initial program 78.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{a \cdot c}{{c}^{2} + {d}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{c}^{2} + {d}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{c \cdot \frac{a}{{c}^{2} + {d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{a}{{c}^{2} + {d}^{2}} \cdot c} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{{c}^{2} + {d}^{2}} \cdot c} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{{c}^{2} + {d}^{2}}} \cdot c \]

      6. +-commutative N/A

         \[\leadsto \frac{a}{\color{blue}{{d}^{2} + {c}^{2}}} \cdot c \]

      7. unpow2 N/A

         \[\leadsto \frac{a}{{d}^{2} + \color{blue}{c \cdot c}} \cdot c \]

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

         \[\leadsto \frac{a}{\color{blue}{{d}^{2} - \left(\mathsf{neg}\left(c\right)\right) \cdot c}} \cdot c \]

      9. mul-1-neg N/A

         \[\leadsto \frac{a}{{d}^{2} - \color{blue}{\left(-1 \cdot c\right)} \cdot c} \cdot c \]

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

          \[\leadsto \frac{a}{\color{blue}{{d}^{2} + \left(\mathsf{neg}\left(-1 \cdot c\right)\right) \cdot c}} \cdot c \]

      11. unpow2 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

          \[\leadsto \frac{a}{d \cdot d + \color{blue}{c} \cdot c} \cdot c \]

      15. unpow2 N/A

          \[\leadsto \frac{a}{d \cdot d + \color{blue}{{c}^{2}}} \cdot c \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{a}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \cdot c \]

      17. unpow2 N/A

          \[\leadsto \frac{a}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot c \]

      18. lower-*.f64 59.1

          \[\leadsto \frac{a}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot c \]

   5. Applied rewrites 59.1%

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

Alternative 6: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.42 \cdot 10^{+24} \lor \neg \left(c \leq 4.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.42e+24) (not (<= c 4.5e-13)))
   (/ (fma (/ b c) d a) c)
   (/ (fma (/ c d) a b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.42e+24) || !(c <= 4.5e-13)) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.42e+24) || !(c <= 4.5e-13))
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.42e+24], N[Not[LessEqual[c, 4.5e-13]], $MachinePrecision]], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.42e24 or 4.5e-13 < c

   1. Initial program 51.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

          \[\leadsto \frac{\frac{b}{c} \cdot d + \color{blue}{a}}{c} \]

      19. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}}{c} \]

      20. lower-/.f64 80.1

          \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{b}{c}}, d, a\right)}{c} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}} \]

   if -1.42e24 < c < 4.5e-13

   1. Initial program 71.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 70.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 70.1

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

   4. Applied rewrites 70.1%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{d} + b}}{d} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \frac{c}{d}} + b}{d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{c}{d} \cdot a} + b}{d} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{d} \]

      9. lower-/.f64 80.2

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{c}{d}}, a, b\right)}{d} \]

   7. Applied rewrites 80.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.42 \cdot 10^{+24} \lor \neg \left(c \leq 4.5 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+24} \lor \neg \left(c \leq 2.6 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.5e+24) (not (<= c 2.6e+45)))
   (/ (fma (/ d c) b a) c)
   (/ (fma (/ c d) a b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.5e+24) || !(c <= 2.6e+45)) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.5e+24) || !(c <= 2.6e+45))
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.5e+24], N[Not[LessEqual[c, 2.6e+45]], $MachinePrecision]], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.49999999999999997e24 or 2.60000000000000007e45 < c

   1. Initial program 48.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 52.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 52.9

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{b \cdot d}{c} + a}}{c} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{b \cdot \frac{d}{c}} + a}{c} \]

      4. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{d}{c} \cdot b} + a}{c} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}}{c} \]

      6. lower-/.f64 81.7

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{d}{c}}, b, a\right)}{c} \]

   7. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}} \]

   if -1.49999999999999997e24 < c < 2.60000000000000007e45

   1. Initial program 72.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 71.0

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 71.0

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{d} + b}}{d} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \frac{c}{d}} + b}{d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{c}{d} \cdot a} + b}{d} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{d} \]

      9. lower-/.f64 78.0

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{c}{d}}, a, b\right)}{d} \]

   7. Applied rewrites 78.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{+24} \lor \neg \left(c \leq 2.6 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{-14} \lor \neg \left(c \leq 1.24 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.8e-14) (not (<= c 1.24e+82)))
   (/ a c)
   (/ (fma (/ c d) a b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.8e-14) || !(c <= 1.24e+82)) {
		tmp = a / c;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.8e-14) || !(c <= 1.24e+82))
		tmp = Float64(a / c);
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.8e-14], N[Not[LessEqual[c, 1.24e+82]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.7999999999999996e-14 or 1.24e82 < c

   1. Initial program 50.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.6

         \[\leadsto \color{blue}{\frac{a}{c}} \]

   5. Applied rewrites 69.6%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -7.7999999999999996e-14 < c < 1.24e82

   1. Initial program 71.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{a \cdot c + b \cdot d}}{c \cdot c + d \cdot d} \]

      3. +-commutative N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + a \cdot c}}{c \cdot c + d \cdot d} \]

      4. div-add N/A

         \[\leadsto \color{blue}{\frac{b \cdot d}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot d}}{c \cdot c + d \cdot d} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{b \cdot \frac{d}{c \cdot c + d \cdot d}} + \frac{a \cdot c}{c \cdot c + d \cdot d} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 70.3

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-*.f64 N/A

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

      21. lower-fma.f64 70.3

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

   4. Applied rewrites 70.3%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{b}{d} + \frac{a \cdot c}{\color{blue}{d \cdot d}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]

      3. div-add N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]

      5. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{d} + b}}{d} \]

      6. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{a \cdot \frac{c}{d}} + b}{d} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{c}{d} \cdot a} + b}{d} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}}{d} \]

      9. lower-/.f64 78.3

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{c}{d}}, a, b\right)}{d} \]

   7. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{-14} \lor \neg \left(c \leq 1.24 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.25e+154)
   (/ b d)
   (if (<= d -2.4e-129)
     (* (/ d (fma d d (* c c))) b)
     (if (<= d 7.5e+54) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e+154) {
		tmp = b / d;
	} else if (d <= -2.4e-129) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 7.5e+54) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.25e+154)
		tmp = Float64(b / d);
	elseif (d <= -2.4e-129)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 7.5e+54)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.25e+154], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.4e-129], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 7.5e+54], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.25000000000000001e154 or 7.50000000000000042e54 < d

   1. Initial program 31.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d}} \]
   4. Step-by-step derivation

      1. lower-/.f64 71.4

         \[\leadsto \color{blue}{\frac{b}{d}} \]

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{b}{d}} \]

   if -1.25000000000000001e154 < d < -2.39999999999999989e-129

   1. Initial program 74.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{b \cdot d}{{c}^{2} + {d}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{d \cdot b}}{{c}^{2} + {d}^{2}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{d \cdot \frac{b}{{c}^{2} + {d}^{2}}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{b}{{c}^{2} + {d}^{2}} \cdot d} \]

      4. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{{c}^{2} + {d}^{2}} \cdot d} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{b}{{c}^{2} + {d}^{2}}} \cdot d \]

      6. +-commutative N/A

         \[\leadsto \frac{b}{\color{blue}{{d}^{2} + {c}^{2}}} \cdot d \]

      7. unpow2 N/A

         \[\leadsto \frac{b}{{d}^{2} + \color{blue}{c \cdot c}} \cdot d \]

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

         \[\leadsto \frac{b}{\color{blue}{{d}^{2} - \left(\mathsf{neg}\left(c\right)\right) \cdot c}} \cdot d \]

      9. mul-1-neg N/A

         \[\leadsto \frac{b}{{d}^{2} - \color{blue}{\left(-1 \cdot c\right)} \cdot c} \cdot d \]

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

          \[\leadsto \frac{b}{\color{blue}{{d}^{2} + \left(\mathsf{neg}\left(-1 \cdot c\right)\right) \cdot c}} \cdot d \]

      11. unpow2 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

          \[\leadsto \frac{b}{d \cdot d + \color{blue}{c} \cdot c} \cdot d \]

      15. unpow2 N/A

          \[\leadsto \frac{b}{d \cdot d + \color{blue}{{c}^{2}}} \cdot d \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \cdot d \]

      17. unpow2 N/A

          \[\leadsto \frac{b}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot d \]

      18. lower-*.f64 56.1

          \[\leadsto \frac{b}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \cdot d \]

   5. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d} \]
   6. Step-by-step derivation

   7. Applied rewrites 67.4%

      \[\leadsto \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \color{blue}{b} \]

   if -2.39999999999999989e-129 < d < 7.50000000000000042e54

   1. Initial program 72.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.3

         \[\leadsto \color{blue}{\frac{a}{c}} \]

   5. Applied rewrites 69.3%

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

Alternative 10: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15} \lor \neg \left(c \leq 2.8 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3e-15) (not (<= c 2.8e-51))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3e-15) || !(c <= 2.8e-51)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3d-15)) .or. (.not. (c <= 2.8d-51))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3e-15) || !(c <= 2.8e-51)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3e-15) or not (c <= 2.8e-51):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3e-15) || !(c <= 2.8e-51))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3e-15) || ~((c <= 2.8e-51)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3e-15], N[Not[LessEqual[c, 2.8e-51]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3e-15 or 2.8e-51 < c

   1. Initial program 54.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.3

         \[\leadsto \color{blue}{\frac{a}{c}} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{a}{c}} \]

   if -3e-15 < c < 2.8e-51

   1. Initial program 69.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d}} \]
   4. Step-by-step derivation

      1. lower-/.f64 69.2

         \[\leadsto \color{blue}{\frac{b}{d}} \]

   5. Applied rewrites 69.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15} \lor \neg \left(c \leq 2.8 \cdot 10^{-51}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.0% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (a b c d) :precision binary64 (/ a c))

C

double code(double a, double b, double c, double d) {
	return a / 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(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = a / c
end function

Java

public static double code(double a, double b, double c, double d) {
	return a / c;
}

Python

def code(a, b, c, d):
	return a / c

Julia

function code(a, b, c, d)
	return Float64(a / c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = a / c;
end

Wolfram

code[a_, b_, c_, d_] := N[(a / c), $MachinePrecision]

Derivation

1. Initial program 60.7%

   \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
2. Add Preprocessing

3. Taylor expanded in c around inf

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

   1. lower-/.f64 44.9

      \[\leadsto \color{blue}{\frac{a}{c}} \]

5. Applied rewrites 44.9%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Less[N[Abs[d], $MachinePrecision], N[Abs[c], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2025007 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))