Complex division, real part

Percentage Accurate: 62.9% → 82.4%

Time: 2.8s

Alternatives: 8

Speedup: 0.9×

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: 62.9% 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: 82.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, -b \cdot {\left(\frac{c}{d}\right)}^{2}\right) + b}{d}\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-79}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(d, b, -\frac{\left(d \cdot d\right) \cdot a}{c}\right)}{c}, -1, -a\right)}{c}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma d d (* c c)))))
   (if (<= d -8.6e+76)
     (/ (+ (fma (/ c d) a (- (* b (pow (/ c d) 2.0)))) b) d)
     (if (<= d -9.6e-160)
       t_0
       (if (<= d 4.7e-79)
         (- (/ (fma (/ (fma d b (- (/ (* (* d d) a) c))) c) -1.0 (- a)) c))
         (if (<= d 8.6e+102) t_0 (/ (fma a (/ c 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 <= -8.6e+76) {
		tmp = (fma((c / d), a, -(b * pow((c / d), 2.0))) + b) / d;
	} else if (d <= -9.6e-160) {
		tmp = t_0;
	} else if (d <= 4.7e-79) {
		tmp = -(fma((fma(d, b, -(((d * d) * a) / c)) / c), -1.0, -a) / c);
	} else if (d <= 8.6e+102) {
		tmp = t_0;
	} else {
		tmp = fma(a, (c / 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 <= -8.6e+76)
		tmp = Float64(Float64(fma(Float64(c / d), a, Float64(-Float64(b * (Float64(c / d) ^ 2.0)))) + b) / d);
	elseif (d <= -9.6e-160)
		tmp = t_0;
	elseif (d <= 4.7e-79)
		tmp = Float64(-Float64(fma(Float64(fma(d, b, Float64(-Float64(Float64(Float64(d * d) * a) / c))) / c), -1.0, Float64(-a)) / c));
	elseif (d <= 8.6e+102)
		tmp = t_0;
	else
		tmp = Float64(fma(a, Float64(c / d), 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, -8.6e+76], N[(N[(N[(N[(c / d), $MachinePrecision] * a + (-N[(b * N[Power[N[(c / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -9.6e-160], t$95$0, If[LessEqual[d, 4.7e-79], (-N[(N[(N[(N[(d * b + (-N[(N[(N[(d * d), $MachinePrecision] * a), $MachinePrecision] / c), $MachinePrecision])), $MachinePrecision] / c), $MachinePrecision] * -1.0 + (-a)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[d, 8.6e+102], t$95$0, N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.59999999999999957e76

   1. Initial program 42.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 42.6

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

      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 42.6

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

   3. Applied rewrites 42.6%

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

   4. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

   6. Applied rewrites 80.8%

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

   if -8.59999999999999957e76 < d < -9.59999999999999964e-160 or 4.7000000000000002e-79 < d < 8.6000000000000002e102

   1. Initial program 76.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 76.6

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

      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 76.6

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

   3. Applied rewrites 76.6%

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

   if -9.59999999999999964e-160 < d < 4.7000000000000002e-79

   1. Initial program 72.1%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 88.8%

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

   if 8.6000000000000002e102 < d

   1. Initial program 40.0%

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

   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 83.5

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

   4. Applied rewrites 83.5%

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

Alternative 2: 82.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-79}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(d, b, -\frac{\left(d \cdot d\right) \cdot a}{c}\right)}{c}, -1, -a\right)}{c}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -8.6e+76)
     t_1
     (if (<= d -9.6e-160)
       t_0
       (if (<= d 4.7e-79)
         (- (/ (fma (/ (fma d b (- (/ (* (* d d) a) c))) c) -1.0 (- a)) c))
         (if (<= d 8.6e+102) 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 <= -8.6e+76) {
		tmp = t_1;
	} else if (d <= -9.6e-160) {
		tmp = t_0;
	} else if (d <= 4.7e-79) {
		tmp = -(fma((fma(d, b, -(((d * d) * a) / c)) / c), -1.0, -a) / c);
	} else if (d <= 8.6e+102) {
		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 <= -8.6e+76)
		tmp = t_1;
	elseif (d <= -9.6e-160)
		tmp = t_0;
	elseif (d <= 4.7e-79)
		tmp = Float64(-Float64(fma(Float64(fma(d, b, Float64(-Float64(Float64(Float64(d * d) * a) / c))) / c), -1.0, Float64(-a)) / c));
	elseif (d <= 8.6e+102)
		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, -8.6e+76], t$95$1, If[LessEqual[d, -9.6e-160], t$95$0, If[LessEqual[d, 4.7e-79], (-N[(N[(N[(N[(d * b + (-N[(N[(N[(d * d), $MachinePrecision] * a), $MachinePrecision] / c), $MachinePrecision])), $MachinePrecision] / c), $MachinePrecision] * -1.0 + (-a)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[d, 8.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.59999999999999957e76 or 8.6000000000000002e102 < d

   1. Initial program 41.4%

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

   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 82.3

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

   4. Applied rewrites 82.3%

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

   if -8.59999999999999957e76 < d < -9.59999999999999964e-160 or 4.7000000000000002e-79 < d < 8.6000000000000002e102

   1. Initial program 76.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 76.6

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

      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 76.6

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

   3. Applied rewrites 76.6%

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

   if -9.59999999999999964e-160 < d < 4.7000000000000002e-79

   1. Initial program 72.1%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 88.8%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(d, b, -\frac{\left(d \cdot d\right) \cdot a}{c}\right)}{c}, -1, -a\right)}{c}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -8.6 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-157}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 8.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -8.6e+76)
     t_1
     (if (<= d -6.6e-157)
       t_0
       (if (<= d 4.7e-79)
         (/ (fma b (/ d c) a) c)
         (if (<= d 8.6e+102) 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 <= -8.6e+76) {
		tmp = t_1;
	} else if (d <= -6.6e-157) {
		tmp = t_0;
	} else if (d <= 4.7e-79) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 8.6e+102) {
		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 <= -8.6e+76)
		tmp = t_1;
	elseif (d <= -6.6e-157)
		tmp = t_0;
	elseif (d <= 4.7e-79)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 8.6e+102)
		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, -8.6e+76], t$95$1, If[LessEqual[d, -6.6e-157], t$95$0, If[LessEqual[d, 4.7e-79], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 8.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.59999999999999957e76 or 8.6000000000000002e102 < d

   1. Initial program 41.4%

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

   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 82.3

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

   4. Applied rewrites 82.3%

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

   if -8.59999999999999957e76 < d < -6.59999999999999998e-157 or 4.7000000000000002e-79 < d < 8.6000000000000002e102

   1. Initial program 76.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 76.6

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

      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 76.6

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

   3. Applied rewrites 76.6%

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

   if -6.59999999999999998e-157 < d < 4.7000000000000002e-79

   1. Initial program 72.1%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 89.4

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

   4. Applied rewrites 89.4%

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

Alternative 4: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -5.4 \cdot 10^{+101}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-101}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* b (/ d (fma d d (* c c))))))
   (if (<= d -5.4e+101)
     (/ b d)
     (if (<= d -8.5e-136)
       t_0
       (if (<= d 3e-101) (/ a c) (if (<= d 2.2e+139) t_0 (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b * (d / fma(d, d, (c * c)));
	double tmp;
	if (d <= -5.4e+101) {
		tmp = b / d;
	} else if (d <= -8.5e-136) {
		tmp = t_0;
	} else if (d <= 3e-101) {
		tmp = a / c;
	} else if (d <= 2.2e+139) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(b * Float64(d / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (d <= -5.4e+101)
		tmp = Float64(b / d);
	elseif (d <= -8.5e-136)
		tmp = t_0;
	elseif (d <= 3e-101)
		tmp = Float64(a / c);
	elseif (d <= 2.2e+139)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.4e+101], N[(b / d), $MachinePrecision], If[LessEqual[d, -8.5e-136], t$95$0, If[LessEqual[d, 3e-101], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.2e+139], t$95$0, N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.40000000000000012e101 or 2.1999999999999999e139 < d

   1. Initial program 36.6%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 73.5

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

   4. Applied rewrites 73.5%

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

   if -5.40000000000000012e101 < d < -8.49999999999999973e-136 or 3.0000000000000003e-101 < d < 2.1999999999999999e139

   1. Initial program 75.8%

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

   2. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{b \cdot d}{{c}^{2} + {d}^{2}}} \]
   3. 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 55.1

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

   4. Applied rewrites 55.1%

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

   if -8.49999999999999973e-136 < d < 3.0000000000000003e-101

   1. Initial program 71.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 71.4

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

   4. Applied rewrites 71.4%

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

Alternative 5: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -2e-34)
     t_0
     (if (<= d -2.8e-133)
       (/ (fma a c (* b d)) (* c c))
       (if (<= d 2.75e-67) (/ a c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -2e-34) {
		tmp = t_0;
	} else if (d <= -2.8e-133) {
		tmp = fma(a, c, (b * d)) / (c * c);
	} else if (d <= 2.75e-67) {
		tmp = a / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -2e-34)
		tmp = t_0;
	elseif (d <= -2.8e-133)
		tmp = Float64(fma(a, c, Float64(b * d)) / Float64(c * c));
	elseif (d <= 2.75e-67)
		tmp = Float64(a / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2e-34], t$95$0, If[LessEqual[d, -2.8e-133], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.75e-67], N[(a / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.99999999999999986e-34 or 2.7500000000000001e-67 < d

   1. Initial program 54.7%

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

   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 71.0

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

   4. Applied rewrites 71.0%

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

   if -1.99999999999999986e-34 < d < -2.7999999999999999e-133

   1. Initial program 80.0%

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

   2. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 52.5

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

   4. Applied rewrites 52.5%

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{c \cdot c}} \]
   5. 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. lower-fma.f64 N/A

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

      5. lift-*.f64 52.5

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

   6. Applied rewrites 52.5%

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

   if -2.7999999999999999e-133 < d < 2.7500000000000001e-67

   1. Initial program 72.6%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 69.4

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

   4. Applied rewrites 69.4%

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

Alternative 6: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -1.65e-32) t_0 (if (<= d 9.5e-67) (/ (fma b (/ d c) a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -1.65e-32) {
		tmp = t_0;
	} else if (d <= 9.5e-67) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -1.65e-32)
		tmp = t_0;
	elseif (d <= 9.5e-67)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.65e-32], t$95$0, If[LessEqual[d, 9.5e-67], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.65000000000000013e-32 or 9.4999999999999994e-67 < d

   1. Initial program 54.7%

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

   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 71.2

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

   4. Applied rewrites 71.2%

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

   if -1.65000000000000013e-32 < d < 9.4999999999999994e-67

   1. Initial program 74.0%

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

   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 85.0

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

   4. Applied rewrites 85.0%

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

Alternative 7: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.8e+40) (/ b d) (if (<= d 9.5e-67) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.8e+40) {
		tmp = b / d;
	} else if (d <= 9.5e-67) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-6.8d+40)) then
        tmp = b / d
    else if (d <= 9.5d-67) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.8e+40) {
		tmp = b / d;
	} else if (d <= 9.5e-67) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6.8e+40:
		tmp = b / d
	elif d <= 9.5e-67:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6.8e+40)
		tmp = Float64(b / d);
	elseif (d <= 9.5e-67)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6.8e+40)
		tmp = b / d;
	elseif (d <= 9.5e-67)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6.8e+40], N[(b / d), $MachinePrecision], If[LessEqual[d, 9.5e-67], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.79999999999999977e40 or 9.4999999999999994e-67 < d

   1. Initial program 51.8%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 61.5

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

   4. Applied rewrites 61.5%

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

   if -6.79999999999999977e40 < d < 9.4999999999999994e-67

   1. Initial program 74.7%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

Alternative 8: 42.5% 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 62.9%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 42.5

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

4. Applied rewrites 42.5%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
5. 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 2025105 
(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))))