Complex division, real part

Percentage Accurate: 61.3% → 82.2%

Time: 4.0s

Alternatives: 10

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: 61.3% 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: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{d}{c}, b, \left(-b\right) \cdot {\left(\frac{d}{c}\right)}^{3}\right) + a\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(c, a, \frac{\left(c \cdot c\right) \cdot b}{-d}\right)}{d}, -1, -b\right)}{-d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e+132)
   (/
    (-
     (+ (fma (/ d c) b (* (- b) (pow (/ d c) 3.0))) a)
     (* a (pow (/ d c) 2.0)))
    c)
   (if (<= c -1.5e-113)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= c 1.3e-36)
       (/ (fma (/ (fma c a (/ (* (* c c) b) (- d))) d) -1.0 (- b)) (- d))
       (if (<= c 1.5e+99)
         (* (/ (fma b (/ d a) c) (fma d d (* c c))) a)
         (/ (fma b (/ d c) a) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e+132) {
		tmp = ((fma((d / c), b, (-b * pow((d / c), 3.0))) + a) - (a * pow((d / c), 2.0))) / c;
	} else if (c <= -1.5e-113) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.3e-36) {
		tmp = fma((fma(c, a, (((c * c) * b) / -d)) / d), -1.0, -b) / -d;
	} else if (c <= 1.5e+99) {
		tmp = (fma(b, (d / a), c) / fma(d, d, (c * c))) * a;
	} else {
		tmp = fma(b, (d / c), a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e+132)
		tmp = Float64(Float64(Float64(fma(Float64(d / c), b, Float64(Float64(-b) * (Float64(d / c) ^ 3.0))) + a) - Float64(a * (Float64(d / c) ^ 2.0))) / c);
	elseif (c <= -1.5e-113)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.3e-36)
		tmp = Float64(fma(Float64(fma(c, a, Float64(Float64(Float64(c * c) * b) / Float64(-d))) / d), -1.0, Float64(-b)) / Float64(-d));
	elseif (c <= 1.5e+99)
		tmp = Float64(Float64(fma(b, Float64(d / a), c) / fma(d, d, Float64(c * c))) * a);
	else
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e+132], N[(N[(N[(N[(N[(d / c), $MachinePrecision] * b + N[((-b) * N[Power[N[(d / c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] - N[(a * N[Power[N[(d / c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.5e-113], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-36], N[(N[(N[(N[(c * a + N[(N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -1.0 + (-b)), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.5e+99], N[(N[(N[(b * N[(d / a), $MachinePrecision] + c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -5.5999999999999998e132

   1. Initial program 34.1%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.3

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{\left(a + \left(-1 \cdot \frac{b \cdot {d}^{3}}{{c}^{3}} + \frac{b \cdot d}{c}\right)\right) - \frac{a \cdot {d}^{2}}{{c}^{2}}}{c}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(a + \left(-1 \cdot \frac{b \cdot {d}^{3}}{{c}^{3}} + \frac{b \cdot d}{c}\right)\right) - \frac{a \cdot {d}^{2}}{{c}^{2}}}{\color{blue}{c}} \]

   8. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(\frac{d}{c}, b, -b \cdot {\left(\frac{d}{c}\right)}^{3}\right) + a\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}}{c}} \]

   if -5.5999999999999998e132 < c < -1.5e-113

   1. Initial program 78.8%

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

   if -1.5e-113 < c < 1.3e-36

   1. Initial program 70.5%

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

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d} \]

   5. Applied rewrites 86.9%

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

   if 1.3e-36 < c < 1.50000000000000007e99

   1. Initial program 75.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1.50000000000000007e99 < c

   1. Initial program 33.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(\frac{d}{c}, b, \left(-b\right) \cdot {\left(\frac{d}{c}\right)}^{3}\right) + a\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(c, a, \frac{\left(c \cdot c\right) \cdot b}{-d}\right)}{d}, -1, -b\right)}{-d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(c, a, \frac{\left(c \cdot c\right) \cdot b}{-d}\right)}{d}, -1, -b\right)}{-d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -2.2e+132)
     t_0
     (if (<= c -1.5e-113)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 1.3e-36)
         (/ (fma (/ (fma c a (/ (* (* c c) b) (- d))) d) -1.0 (- b)) (- d))
         (if (<= c 1.5e+99)
           (* (/ (fma b (/ d a) c) (fma d d (* c c))) a)
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -2.2e+132) {
		tmp = t_0;
	} else if (c <= -1.5e-113) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.3e-36) {
		tmp = fma((fma(c, a, (((c * c) * b) / -d)) / d), -1.0, -b) / -d;
	} else if (c <= 1.5e+99) {
		tmp = (fma(b, (d / a), c) / fma(d, d, (c * c))) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -2.2e+132)
		tmp = t_0;
	elseif (c <= -1.5e-113)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.3e-36)
		tmp = Float64(fma(Float64(fma(c, a, Float64(Float64(Float64(c * c) * b) / Float64(-d))) / d), -1.0, Float64(-b)) / Float64(-d));
	elseif (c <= 1.5e+99)
		tmp = Float64(Float64(fma(b, Float64(d / a), c) / fma(d, d, Float64(c * c))) * a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+132], t$95$0, If[LessEqual[c, -1.5e-113], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-36], N[(N[(N[(N[(c * a + N[(N[(N[(c * c), $MachinePrecision] * b), $MachinePrecision] / (-d)), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision] * -1.0 + (-b)), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.5e+99], N[(N[(N[(b * N[(d / a), $MachinePrecision] + c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.19999999999999989e132 or 1.50000000000000007e99 < c

   1. Initial program 33.9%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if -2.19999999999999989e132 < c < -1.5e-113

   1. Initial program 78.8%

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

   if -1.5e-113 < c < 1.3e-36

   1. Initial program 70.5%

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

   3. Taylor expanded in d around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot b + -1 \cdot \frac{-1 \cdot \frac{b \cdot {c}^{2}}{d} + a \cdot c}{d}}{d} \]

   5. Applied rewrites 86.9%

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

   if 1.3e-36 < c < 1.50000000000000007e99

   1. Initial program 75.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(c, a, \frac{\left(c \cdot c\right) \cdot b}{-d}\right)}{d}, -1, -b\right)}{-d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{a}, c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -2.2e+132)
     t_0
     (if (<= c -1.85e-113)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 1.3e-36)
         (/ (fma a (/ c d) b) d)
         (if (<= c 1.5e+99)
           (* (/ (fma b (/ d a) c) (fma d d (* c c))) a)
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -2.2e+132) {
		tmp = t_0;
	} else if (c <= -1.85e-113) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.3e-36) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.5e+99) {
		tmp = (fma(b, (d / a), c) / fma(d, d, (c * c))) * a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -2.2e+132)
		tmp = t_0;
	elseif (c <= -1.85e-113)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.3e-36)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.5e+99)
		tmp = Float64(Float64(fma(b, Float64(d / a), c) / fma(d, d, Float64(c * c))) * a);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+132], t$95$0, If[LessEqual[c, -1.85e-113], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-36], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.5e+99], N[(N[(N[(b * N[(d / a), $MachinePrecision] + c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.19999999999999989e132 or 1.50000000000000007e99 < c

   1. Initial program 33.9%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

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

   if -2.19999999999999989e132 < c < -1.8499999999999999e-113

   1. Initial program 78.8%

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

   if -1.8499999999999999e-113 < c < 1.3e-36

   1. Initial program 70.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

   if 1.3e-36 < c < 1.50000000000000007e99

   1. Initial program 75.2%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. lower-fma.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

Alternative 4: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -8.8 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+143}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (* c c))))
   (if (<= c -1.1e+127)
     (/ a c)
     (if (<= c -8.8e-73)
       t_0
       (if (<= c 8.8e-31)
         (/ (fma a (/ c d) b) d)
         (if (<= c 1.15e+143) t_0 (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / (c * c);
	double tmp;
	if (c <= -1.1e+127) {
		tmp = a / c;
	} else if (c <= -8.8e-73) {
		tmp = t_0;
	} else if (c <= 8.8e-31) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.15e+143) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / Float64(c * c))
	tmp = 0.0
	if (c <= -1.1e+127)
		tmp = Float64(a / c);
	elseif (c <= -8.8e-73)
		tmp = t_0;
	elseif (c <= 8.8e-31)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.15e+143)
		tmp = t_0;
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.1e+127], N[(a / c), $MachinePrecision], If[LessEqual[c, -8.8e-73], t$95$0, If[LessEqual[c, 8.8e-31], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.15e+143], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1000000000000001e127 or 1.15e143 < c

   1. Initial program 29.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -1.1000000000000001e127 < c < -8.8000000000000001e-73 or 8.80000000000000038e-31 < c < 1.15e143

   1. Initial program 78.5%

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

   3. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 64.5

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

   5. Applied rewrites 64.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.6

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

      9. pow2 64.6

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

      10. pow2 64.6

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

      11. +-commutative 64.6

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

      12. pow2 64.6

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

      13. pow2 64.6

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

   7. Applied rewrites 64.6%

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

   if -8.8000000000000001e-73 < c < 8.80000000000000038e-31

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

Alternative 5: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.18 \cdot 10^{-297}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+102}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.75e-47)
   (/ b d)
   (if (<= d -1.18e-297)
     (/ a c)
     (if (<= d 2.4e-53)
       (/ (fma d b (* c a)) (* c c))
       (if (<= d 2.2e+102) (* b (/ d (fma c c (* d d)))) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.75e-47) {
		tmp = b / d;
	} else if (d <= -1.18e-297) {
		tmp = a / c;
	} else if (d <= 2.4e-53) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else if (d <= 2.2e+102) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.75e-47)
		tmp = Float64(b / d);
	elseif (d <= -1.18e-297)
		tmp = Float64(a / c);
	elseif (d <= 2.4e-53)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	elseif (d <= 2.2e+102)
		tmp = Float64(b * Float64(d / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.75e-47], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.18e-297], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.4e-53], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e+102], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.7499999999999999e-47 or 2.20000000000000007e102 < d

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.7499999999999999e-47 < d < -1.17999999999999993e-297

   1. Initial program 71.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -1.17999999999999993e-297 < d < 2.40000000000000007e-53

   1. Initial program 77.1%

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

   3. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.7

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

      9. pow2 70.7

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

      10. pow2 70.7

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

      11. +-commutative 70.7

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

      12. pow2 70.7

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

      13. pow2 70.7

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

   7. Applied rewrites 70.7%

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

   if 2.40000000000000007e-53 < d < 2.20000000000000007e102

   1. Initial program 77.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

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

Alternative 6: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -2.2e+132)
     t_0
     (if (<= c -1.85e-113)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 8.8e-31) (/ (fma a (/ c d) b) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -2.2e+132) {
		tmp = t_0;
	} else if (c <= -1.85e-113) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 8.8e-31) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -2.2e+132)
		tmp = t_0;
	elseif (c <= -1.85e-113)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 8.8e-31)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.2e+132], t$95$0, If[LessEqual[c, -1.85e-113], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.8e-31], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.19999999999999989e132 or 8.80000000000000038e-31 < c

   1. Initial program 45.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -2.19999999999999989e132 < c < -1.8499999999999999e-113

   1. Initial program 78.8%

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

   if -1.8499999999999999e-113 < c < 8.80000000000000038e-31

   1. Initial program 70.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.9

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

   5. Applied rewrites 86.9%

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

Alternative 7: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+102}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.75e-47)
   (/ b d)
   (if (<= d 2.4e-53)
     (/ a c)
     (if (<= d 2.2e+102) (* b (/ d (fma c c (* d d)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.75e-47) {
		tmp = b / d;
	} else if (d <= 2.4e-53) {
		tmp = a / c;
	} else if (d <= 2.2e+102) {
		tmp = b * (d / fma(c, c, (d * d)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.75e-47)
		tmp = Float64(b / d);
	elseif (d <= 2.4e-53)
		tmp = Float64(a / c);
	elseif (d <= 2.2e+102)
		tmp = Float64(b * Float64(d / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.75e-47], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.4e-53], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.2e+102], N[(b * N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.7499999999999999e-47 or 2.20000000000000007e102 < d

   1. Initial program 44.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -1.7499999999999999e-47 < d < 2.40000000000000007e-53

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 2.40000000000000007e-53 < d < 2.20000000000000007e102

   1. Initial program 77.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 62.2

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

   5. Applied rewrites 62.2%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lower-*.f64 62.3

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

   7. Applied rewrites 62.3%

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

Alternative 8: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.8 \cdot 10^{-73} \lor \neg \left(c \leq 8.8 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8.8e-73) (not (<= c 8.8e-31)))
   (/ (fma b (/ d c) a) c)
   (/ (fma a (/ c d) b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.8e-73) || !(c <= 8.8e-31)) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8.8e-73) || !(c <= 8.8e-31))
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8.8e-73], N[Not[LessEqual[c, 8.8e-31]], $MachinePrecision]], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.8000000000000001e-73 or 8.80000000000000038e-31 < c

   1. Initial program 53.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -8.8000000000000001e-73 < c < 8.80000000000000038e-31

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.8 \cdot 10^{-73} \lor \neg \left(c \leq 8.8 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-47} \lor \neg \left(d \leq 45000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.75e-47) (not (<= d 45000.0))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.75e-47) || !(d <= 45000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(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 ((d <= (-1.75d-47)) .or. (.not. (d <= 45000.0d0))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.75e-47) || !(d <= 45000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.75e-47) or not (d <= 45000.0):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.75e-47) || !(d <= 45000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.75e-47) || ~((d <= 45000.0)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.75e-47], N[Not[LessEqual[d, 45000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.7499999999999999e-47 or 45000 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -1.7499999999999999e-47 < d < 45000

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.6

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

   5. Applied rewrites 65.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{-47} \lor \neg \left(d \leq 45000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.8% 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 61.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 41.8

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

5. Applied rewrites 41.8%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2025084 
(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))))