Complex division, real part

Percentage Accurate: 61.1% → 83.0%

Time: 3.0s

Alternatives: 8

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: 61.1% 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: 83.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -3.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot a}{d} + \frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c)))))
   (if (<= d -3.05e+100)
     (/ (fma a (/ c d) b) d)
     (if (<= d -1.25e-158)
       t_0
       (if (<= d 4.5e-102)
         (/ (fma b (/ d c) a) c)
         (if (<= d 5e+84) t_0 (+ (/ (* (/ c d) a) d) (/ b d))))))))

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 tmp;
	if (d <= -3.05e+100) {
		tmp = fma(a, (c / d), b) / d;
	} else if (d <= -1.25e-158) {
		tmp = t_0;
	} else if (d <= 4.5e-102) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 5e+84) {
		tmp = t_0;
	} else {
		tmp = (((c / d) * a) / d) + (b / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)))
	tmp = 0.0
	if (d <= -3.05e+100)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (d <= -1.25e-158)
		tmp = t_0;
	elseif (d <= 4.5e-102)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 5e+84)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(Float64(c / d) * a) / d) + Float64(b / d));
	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]}, If[LessEqual[d, -3.05e+100], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.25e-158], t$95$0, If[LessEqual[d, 4.5e-102], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+84], t$95$0, N[(N[(N[(N[(c / d), $MachinePrecision] * a), $MachinePrecision] / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.05e100

   1. Initial program 37.6%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   3. 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 83.6

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

   4. Applied rewrites 83.6%

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

   if -3.05e100 < d < -1.24999999999999993e-158 or 4.49999999999999999e-102 < d < 5.0000000000000001e84

   1. Initial program 75.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      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 75.8

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

      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. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 75.8

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

   3. Applied rewrites 75.8%

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

   if -1.24999999999999993e-158 < d < 4.49999999999999999e-102

   1. Initial program 69.5%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   3. 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 92.3

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

   4. Applied rewrites 92.3%

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

   if 5.0000000000000001e84 < d

   1. Initial program 41.3%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   3. 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 81.9

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

   4. Applied rewrites 81.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. div-add N/A

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

      5. associate-*r/ N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 81.9

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

   6. Applied rewrites 81.9%

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

Alternative 2: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3.05 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c))))
        (t_1 (/ (fma a (/ c d) b) d)))
   (if (<= d -3.05e+100)
     t_1
     (if (<= d -1.25e-158)
       t_0
       (if (<= d 4.5e-102)
         (/ (fma b (/ d c) a) c)
         (if (<= d 5e+84) 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(a, (c / d), b) / d;
	double tmp;
	if (d <= -3.05e+100) {
		tmp = t_1;
	} else if (d <= -1.25e-158) {
		tmp = t_0;
	} else if (d <= 4.5e-102) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 5e+84) {
		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(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -3.05e+100)
		tmp = t_1;
	elseif (d <= -1.25e-158)
		tmp = t_0;
	elseif (d <= 4.5e-102)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 5e+84)
		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[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.05e+100], t$95$1, If[LessEqual[d, -1.25e-158], t$95$0, If[LessEqual[d, 4.5e-102], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+84], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.05e100 or 5.0000000000000001e84 < d

   1. Initial program 39.5%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   3. 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 82.7

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

   4. Applied rewrites 82.7%

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

   if -3.05e100 < d < -1.24999999999999993e-158 or 4.49999999999999999e-102 < d < 5.0000000000000001e84

   1. Initial program 75.8%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      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 75.8

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

      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. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 75.8

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

   3. Applied rewrites 75.8%

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

   if -1.24999999999999993e-158 < d < 4.49999999999999999e-102

   1. Initial program 69.5%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   3. 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 92.3

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

   4. Applied rewrites 92.3%

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

Alternative 3: 66.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+154)
   (/ a c)
   (if (<= c -7.5e-35)
     (* a (/ c (fma d d (* c c))))
     (if (<= c 3.5e-18)
       (/ b d)
       (if (<= c 1.2e+154) (/ (fma d b (* c a)) (* c c)) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+154) {
		tmp = a / c;
	} else if (c <= -7.5e-35) {
		tmp = a * (c / fma(d, d, (c * c)));
	} else if (c <= 3.5e-18) {
		tmp = b / d;
	} else if (c <= 1.2e+154) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+154)
		tmp = Float64(a / c);
	elseif (c <= -7.5e-35)
		tmp = Float64(a * Float64(c / fma(d, d, Float64(c * c))));
	elseif (c <= 3.5e-18)
		tmp = Float64(b / d);
	elseif (c <= 1.2e+154)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+154], N[(a / c), $MachinePrecision], If[LessEqual[c, -7.5e-35], N[(a * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.5e-18], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.2e+154], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.1000000000000001e154 or 1.20000000000000007e154 < c

   1. Initial program 28.3%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if -1.1000000000000001e154 < c < -7.5e-35

   1. Initial program 72.0%

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

   2. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.1

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

   4. Applied rewrites 60.1%

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

   if -7.5e-35 < c < 3.4999999999999999e-18

   1. Initial program 74.2%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 66.1

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

   4. Applied rewrites 66.1%

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

   if 3.4999999999999999e-18 < c < 1.20000000000000007e154

   1. Initial program 68.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      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 68.5

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

      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. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.6

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

   6. Applied rewrites 54.6%

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

Alternative 4: 67.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* a (/ c (fma d d (* c c))))))
   (if (<= c -1.1e+154)
     (/ a c)
     (if (<= c -7.5e-35)
       t_0
       (if (<= c 5e-31) (/ b d) (if (<= c 3.1e+156) t_0 (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a * (c / fma(d, d, (c * c)));
	double tmp;
	if (c <= -1.1e+154) {
		tmp = a / c;
	} else if (c <= -7.5e-35) {
		tmp = t_0;
	} else if (c <= 5e-31) {
		tmp = b / d;
	} else if (c <= 3.1e+156) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a * Float64(c / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (c <= -1.1e+154)
		tmp = Float64(a / c);
	elseif (c <= -7.5e-35)
		tmp = t_0;
	elseif (c <= 5e-31)
		tmp = Float64(b / d);
	elseif (c <= 3.1e+156)
		tmp = t_0;
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.1e+154], N[(a / c), $MachinePrecision], If[LessEqual[c, -7.5e-35], t$95$0, If[LessEqual[c, 5e-31], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.1e+156], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1000000000000001e154 or 3.1000000000000002e156 < c

   1. Initial program 28.4%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 77.8

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

   4. Applied rewrites 77.8%

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

   if -1.1000000000000001e154 < c < -7.5e-35 or 5e-31 < c < 3.1000000000000002e156

   1. Initial program 70.0%

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

   2. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 58.7

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

   4. Applied rewrites 58.7%

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

   if -7.5e-35 < c < 5e-31

   1. Initial program 74.1%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 66.4

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

   4. Applied rewrites 66.4%

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

Alternative 5: 73.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.6e+40)
   (/ a c)
   (if (<= c 3.5e-18)
     (/ (fma a (/ c d) b) d)
     (if (<= c 1.2e+154) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.6e+40) {
		tmp = a / c;
	} else if (c <= 3.5e-18) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.2e+154) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.6e+40)
		tmp = Float64(a / c);
	elseif (c <= 3.5e-18)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.2e+154)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.6e+40], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.5e-18], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.2e+154], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.59999999999999996e40 or 1.20000000000000007e154 < c

   1. Initial program 38.6%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 71.9

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

   4. Applied rewrites 71.9%

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

   if -3.59999999999999996e40 < c < 3.4999999999999999e-18

   1. Initial program 74.6%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   3. 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 80.2

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

   4. Applied rewrites 80.2%

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

   if 3.4999999999999999e-18 < c < 1.20000000000000007e154

   1. Initial program 68.5%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      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 68.5

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

      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. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 68.5

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

   3. Applied rewrites 68.5%

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

   4. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 54.6

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

   6. Applied rewrites 54.6%

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

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -7.4 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -7.4e-29) t_0 (if (<= c 3.5e-18) (/ (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 <= -7.4e-29) {
		tmp = t_0;
	} else if (c <= 3.5e-18) {
		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 <= -7.4e-29)
		tmp = t_0;
	elseif (c <= 3.5e-18)
		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, -7.4e-29], t$95$0, If[LessEqual[c, 3.5e-18], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.3999999999999995e-29 or 3.4999999999999999e-18 < c

   1. Initial program 50.3%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   3. 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 73.9

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

   4. Applied rewrites 73.9%

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

   if -7.3999999999999995e-29 < c < 3.4999999999999999e-18

   1. Initial program 74.2%

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

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{b + \frac{a \cdot c}{d}}{d}} \]
   3. 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 83.3

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

   4. Applied rewrites 83.3%

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

Alternative 7: 64.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+40) (/ a c) (if (<= c 9e+17) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+40) {
		tmp = a / c;
	} else if (c <= 9e+17) {
		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 (c <= (-1.1d+40)) then
        tmp = a / c
    else if (c <= 9d+17) 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 (c <= -1.1e+40) {
		tmp = a / c;
	} else if (c <= 9e+17) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.1e+40:
		tmp = a / c
	elif c <= 9e+17:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+40)
		tmp = Float64(a / c);
	elseif (c <= 9e+17)
		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 (c <= -1.1e+40)
		tmp = a / c;
	elseif (c <= 9e+17)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+40], N[(a / c), $MachinePrecision], If[LessEqual[c, 9e+17], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.0999999999999999e40 or 9e17 < c

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 67.4

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

   4. Applied rewrites 67.4%

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

   if -1.0999999999999999e40 < c < 9e17

   1. Initial program 74.9%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 61.8

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

   4. Applied rewrites 61.8%

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

Alternative 8: 43.1% 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 61.1%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 43.1

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

4. Applied rewrites 43.1%

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

Developer Target 1: 99.4% 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 2025102 
(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))))