Complex division, real part

Percentage Accurate: 62.2% → 83.2%

Time: 5.2s

Alternatives: 9

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: 62.2% 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: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+114}:\\ \;\;\;\;\frac{a}{d} \cdot \frac{c}{d} + \frac{b}{d}\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{d \cdot b}{c} + a}{c}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{d} + b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -7.2e+114)
     (+ (* (/ a d) (/ c d)) (/ b d))
     (if (<= d -4.2e-74)
       t_0
       (if (<= d 7.5e-145)
         (/ (+ (/ (* d b) c) a) c)
         (if (<= d 3.7e+102) t_0 (/ (+ (* c (/ a d)) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -7.2e+114) {
		tmp = ((a / d) * (c / d)) + (b / d);
	} else if (d <= -4.2e-74) {
		tmp = t_0;
	} else if (d <= 7.5e-145) {
		tmp = (((d * b) / c) + a) / c;
	} else if (d <= 3.7e+102) {
		tmp = t_0;
	} else {
		tmp = ((c * (a / d)) + 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-7.2d+114)) then
        tmp = ((a / d) * (c / d)) + (b / d)
    else if (d <= (-4.2d-74)) then
        tmp = t_0
    else if (d <= 7.5d-145) then
        tmp = (((d * b) / c) + a) / c
    else if (d <= 3.7d+102) then
        tmp = t_0
    else
        tmp = ((c * (a / d)) + b) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -7.2e+114) {
		tmp = ((a / d) * (c / d)) + (b / d);
	} else if (d <= -4.2e-74) {
		tmp = t_0;
	} else if (d <= 7.5e-145) {
		tmp = (((d * b) / c) + a) / c;
	} else if (d <= 3.7e+102) {
		tmp = t_0;
	} else {
		tmp = ((c * (a / d)) + b) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -7.2e+114:
		tmp = ((a / d) * (c / d)) + (b / d)
	elif d <= -4.2e-74:
		tmp = t_0
	elif d <= 7.5e-145:
		tmp = (((d * b) / c) + a) / c
	elif d <= 3.7e+102:
		tmp = t_0
	else:
		tmp = ((c * (a / d)) + b) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -7.2e+114)
		tmp = Float64(Float64(Float64(a / d) * Float64(c / d)) + Float64(b / d));
	elseif (d <= -4.2e-74)
		tmp = t_0;
	elseif (d <= 7.5e-145)
		tmp = Float64(Float64(Float64(Float64(d * b) / c) + a) / c);
	elseif (d <= 3.7e+102)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(a / d)) + b) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -7.2e+114)
		tmp = ((a / d) * (c / d)) + (b / d);
	elseif (d <= -4.2e-74)
		tmp = t_0;
	elseif (d <= 7.5e-145)
		tmp = (((d * b) / c) + a) / c;
	elseif (d <= 3.7e+102)
		tmp = t_0;
	else
		tmp = ((c * (a / d)) + b) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.2e+114], N[(N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -4.2e-74], t$95$0, If[LessEqual[d, 7.5e-145], N[(N[(N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.7e+102], t$95$0, N[(N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.2000000000000001e114

   1. Initial program 44.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 86.6

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

   5. Applied rewrites 86.6%

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

   if -7.2000000000000001e114 < d < -4.2e-74 or 7.4999999999999996e-145 < d < 3.70000000000000023e102

   1. Initial program 81.7%

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

   if -4.2e-74 < d < 7.4999999999999996e-145

   1. Initial program 68.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

   if 3.70000000000000023e102 < d

   1. Initial program 44.7%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 91.5

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

   7. Applied rewrites 91.5%

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

Alternative 2: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{a}{d} + b}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -4.2 \cdot 10^{-74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{d \cdot b}{c} + a}{c}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (/ (+ (* c (/ a d)) b) d)))
   (if (<= d -7.2e+114)
     t_1
     (if (<= d -4.2e-74)
       t_0
       (if (<= d 7.5e-145)
         (/ (+ (/ (* d b) c) a) c)
         (if (<= d 3.7e+102) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (a / d)) + b) / d;
	double tmp;
	if (d <= -7.2e+114) {
		tmp = t_1;
	} else if (d <= -4.2e-74) {
		tmp = t_0;
	} else if (d <= 7.5e-145) {
		tmp = (((d * b) / c) + a) / c;
	} else if (d <= 3.7e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = ((c * (a / d)) + b) / d
    if (d <= (-7.2d+114)) then
        tmp = t_1
    else if (d <= (-4.2d-74)) then
        tmp = t_0
    else if (d <= 7.5d-145) then
        tmp = (((d * b) / c) + a) / c
    else if (d <= 3.7d+102) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (a / d)) + b) / d;
	double tmp;
	if (d <= -7.2e+114) {
		tmp = t_1;
	} else if (d <= -4.2e-74) {
		tmp = t_0;
	} else if (d <= 7.5e-145) {
		tmp = (((d * b) / c) + a) / c;
	} else if (d <= 3.7e+102) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = ((c * (a / d)) + b) / d
	tmp = 0
	if d <= -7.2e+114:
		tmp = t_1
	elif d <= -4.2e-74:
		tmp = t_0
	elif d <= 7.5e-145:
		tmp = (((d * b) / c) + a) / c
	elif d <= 3.7e+102:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(c * Float64(a / d)) + b) / d)
	tmp = 0.0
	if (d <= -7.2e+114)
		tmp = t_1;
	elseif (d <= -4.2e-74)
		tmp = t_0;
	elseif (d <= 7.5e-145)
		tmp = Float64(Float64(Float64(Float64(d * b) / c) + a) / c);
	elseif (d <= 3.7e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = ((c * (a / d)) + b) / d;
	tmp = 0.0;
	if (d <= -7.2e+114)
		tmp = t_1;
	elseif (d <= -4.2e-74)
		tmp = t_0;
	elseif (d <= 7.5e-145)
		tmp = (((d * b) / c) + a) / c;
	elseif (d <= 3.7e+102)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.2e+114], t$95$1, If[LessEqual[d, -4.2e-74], t$95$0, If[LessEqual[d, 7.5e-145], N[(N[(N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.7e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.2000000000000001e114 or 3.70000000000000023e102 < d

   1. Initial program 44.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.5

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

   5. Applied rewrites 85.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 89.3

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

   7. Applied rewrites 89.3%

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

   if -7.2000000000000001e114 < d < -4.2e-74 or 7.4999999999999996e-145 < d < 3.70000000000000023e102

   1. Initial program 81.7%

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

   if -4.2e-74 < d < 7.4999999999999996e-145

   1. Initial program 68.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.0

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

   5. Applied rewrites 89.0%

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

Alternative 3: 80.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{c} \cdot \frac{d}{c} + \frac{a}{c}\\ \mathbf{if}\;c \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.95 \cdot 10^{-135}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{c \cdot a}{d} + b}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* (/ b c) (/ d c)) (/ a c))))
   (if (<= c -9.2e+82)
     t_0
     (if (<= c -1.95e-135)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 2.05e+22) (/ (+ (/ (* c a) d) b) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b / c) * (d / c)) + (a / c);
	double tmp;
	if (c <= -9.2e+82) {
		tmp = t_0;
	} else if (c <= -1.95e-135) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 2.05e+22) {
		tmp = (((c * a) / d) + b) / d;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((b / c) * (d / c)) + (a / c)
    if (c <= (-9.2d+82)) then
        tmp = t_0
    else if (c <= (-1.95d-135)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (c <= 2.05d+22) then
        tmp = (((c * a) / d) + b) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b / c) * (d / c)) + (a / c);
	double tmp;
	if (c <= -9.2e+82) {
		tmp = t_0;
	} else if (c <= -1.95e-135) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 2.05e+22) {
		tmp = (((c * a) / d) + b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b / c) * (d / c)) + (a / c)
	tmp = 0
	if c <= -9.2e+82:
		tmp = t_0
	elif c <= -1.95e-135:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif c <= 2.05e+22:
		tmp = (((c * a) / d) + b) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b / c) * Float64(d / c)) + Float64(a / c))
	tmp = 0.0
	if (c <= -9.2e+82)
		tmp = t_0;
	elseif (c <= -1.95e-135)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.05e+22)
		tmp = Float64(Float64(Float64(Float64(c * a) / d) + b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b / c) * (d / c)) + (a / c);
	tmp = 0.0;
	if (c <= -9.2e+82)
		tmp = t_0;
	elseif (c <= -1.95e-135)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (c <= 2.05e+22)
		tmp = (((c * a) / d) + b) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.2e+82], t$95$0, If[LessEqual[c, -1.95e-135], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.05e+22], N[(N[(N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.19999999999999953e82 or 2.0499999999999999e22 < c

   1. Initial program 49.7%

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

   3. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -9.19999999999999953e82 < c < -1.95000000000000011e-135

   1. Initial program 83.1%

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

   if -1.95000000000000011e-135 < c < 2.0499999999999999e22

   1. Initial program 72.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

Alternative 4: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-61} \lor \neg \left(d \leq 5.4 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{c \cdot \frac{a}{d} + b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot b}{c} + a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.5e-61) (not (<= d 5.4e+18)))
   (/ (+ (* c (/ a d)) b) d)
   (/ (+ (/ (* d b) c) a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.5e-61) || !(d <= 5.4e+18)) {
		tmp = ((c * (a / d)) + b) / d;
	} else {
		tmp = (((d * b) / c) + a) / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.5d-61)) .or. (.not. (d <= 5.4d+18))) then
        tmp = ((c * (a / d)) + b) / d
    else
        tmp = (((d * b) / c) + a) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.5e-61) || !(d <= 5.4e+18)) {
		tmp = ((c * (a / d)) + b) / d;
	} else {
		tmp = (((d * b) / c) + a) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.5e-61) or not (d <= 5.4e+18):
		tmp = ((c * (a / d)) + b) / d
	else:
		tmp = (((d * b) / c) + a) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.5e-61) || !(d <= 5.4e+18))
		tmp = Float64(Float64(Float64(c * Float64(a / d)) + b) / d);
	else
		tmp = Float64(Float64(Float64(Float64(d * b) / c) + a) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.5e-61) || ~((d <= 5.4e+18)))
		tmp = ((c * (a / d)) + b) / d;
	else
		tmp = (((d * b) / c) + a) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.5e-61], N[Not[LessEqual[d, 5.4e+18]], $MachinePrecision]], N[(N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.4999999999999999e-61 or 5.4e18 < d

   1. Initial program 59.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 79.1

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

   7. Applied rewrites 79.1%

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

   if -2.4999999999999999e-61 < d < 5.4e18

   1. Initial program 73.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{-61} \lor \neg \left(d \leq 5.4 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{c \cdot \frac{a}{d} + b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{d \cdot b}{c} + a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+82} \lor \neg \left(c \leq 2.1 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{d} + b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.2e+82) (not (<= c 2.1e+71)))
   (/ a c)
   (/ (+ (/ (* c a) d) b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+82) || !(c <= 2.1e+71)) {
		tmp = a / c;
	} else {
		tmp = (((c * a) / d) + b) / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.2d+82)) .or. (.not. (c <= 2.1d+71))) then
        tmp = a / c
    else
        tmp = (((c * a) / d) + b) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+82) || !(c <= 2.1e+71)) {
		tmp = a / c;
	} else {
		tmp = (((c * a) / d) + b) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.2e+82) or not (c <= 2.1e+71):
		tmp = a / c
	else:
		tmp = (((c * a) / d) + b) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.2e+82) || !(c <= 2.1e+71))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * a) / d) + b) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.2e+82) || ~((c <= 2.1e+71)))
		tmp = a / c;
	else
		tmp = (((c * a) / d) + b) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.2e+82], N[Not[LessEqual[c, 2.1e+71]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.1999999999999997e82 or 2.09999999999999989e71 < c

   1. Initial program 46.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -5.1999999999999997e82 < c < 2.09999999999999989e71

   1. Initial program 76.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+82} \lor \neg \left(c \leq 2.1 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{d} + b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{+82} \lor \neg \left(c \leq 3.1 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{d} + b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.4e+82) (not (<= c 3.1e+71)))
   (/ a c)
   (/ (+ (* c (/ a d)) b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.4e+82) || !(c <= 3.1e+71)) {
		tmp = a / c;
	} else {
		tmp = ((c * (a / d)) + b) / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.4d+82)) .or. (.not. (c <= 3.1d+71))) then
        tmp = a / c
    else
        tmp = ((c * (a / d)) + b) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.4e+82) || !(c <= 3.1e+71)) {
		tmp = a / c;
	} else {
		tmp = ((c * (a / d)) + b) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.4e+82) or not (c <= 3.1e+71):
		tmp = a / c
	else:
		tmp = ((c * (a / d)) + b) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.4e+82) || !(c <= 3.1e+71))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(a / d)) + b) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.4e+82) || ~((c <= 3.1e+71)))
		tmp = a / c;
	else
		tmp = ((c * (a / d)) + b) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.4e+82], N[Not[LessEqual[c, 3.1e+71]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.3999999999999999e82 or 3.10000000000000018e71 < c

   1. Initial program 46.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.0

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

   5. Applied rewrites 78.0%

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

   if -5.3999999999999999e82 < c < 3.10000000000000018e71

   1. Initial program 76.3%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 75.2

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

   5. Applied rewrites 75.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.7

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

   7. Applied rewrites 72.7%

      \[\leadsto \frac{c \cdot \frac{a}{d} + b}{d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{+82} \lor \neg \left(c \leq 3.1 \cdot 10^{+71}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a}{d} + b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -1.55 \cdot 10^{-117}:\\ \;\;\;\;\frac{c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9.2e+82)
   (/ a c)
   (if (<= c -1.55e-117)
     (/ (* c a) (+ (* c c) (* d d)))
     (if (<= c 2.05e+22) (/ b d) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.2e+82) {
		tmp = a / c;
	} else if (c <= -1.55e-117) {
		tmp = (c * a) / ((c * c) + (d * d));
	} else if (c <= 2.05e+22) {
		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 <= (-9.2d+82)) then
        tmp = a / c
    else if (c <= (-1.55d-117)) then
        tmp = (c * a) / ((c * c) + (d * d))
    else if (c <= 2.05d+22) 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 <= -9.2e+82) {
		tmp = a / c;
	} else if (c <= -1.55e-117) {
		tmp = (c * a) / ((c * c) + (d * d));
	} else if (c <= 2.05e+22) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -9.2e+82:
		tmp = a / c
	elif c <= -1.55e-117:
		tmp = (c * a) / ((c * c) + (d * d))
	elif c <= 2.05e+22:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9.2e+82)
		tmp = Float64(a / c);
	elseif (c <= -1.55e-117)
		tmp = Float64(Float64(c * a) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.05e+22)
		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 <= -9.2e+82)
		tmp = a / c;
	elseif (c <= -1.55e-117)
		tmp = (c * a) / ((c * c) + (d * d));
	elseif (c <= 2.05e+22)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9.2e+82], N[(a / c), $MachinePrecision], If[LessEqual[c, -1.55e-117], N[(N[(c * a), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.05e+22], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.19999999999999953e82 or 2.0499999999999999e22 < c

   1. Initial program 49.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -9.19999999999999953e82 < c < -1.55000000000000005e-117

   1. Initial program 81.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 55.5

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

   5. Applied rewrites 55.5%

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

   if -1.55000000000000005e-117 < c < 2.0499999999999999e22

   1. Initial program 73.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

Alternative 8: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+82} \lor \neg \left(c \leq 2.05 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.2e+82) (not (<= c 2.05e+22))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+82) || !(c <= 2.05e+22)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.2d+82)) .or. (.not. (c <= 2.05d+22))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e+82) || !(c <= 2.05e+22)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.2e+82) or not (c <= 2.05e+22):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.2e+82) || !(c <= 2.05e+22))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.2e+82) || ~((c <= 2.05e+22)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.2e+82], N[Not[LessEqual[c, 2.05e+22]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.1999999999999997e82 or 2.0499999999999999e22 < c

   1. Initial program 49.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -5.1999999999999997e82 < c < 2.0499999999999999e22

   1. Initial program 75.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+82} \lor \neg \left(c \leq 2.05 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.5% 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 66.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.4

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

5. Applied rewrites 42.4%

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

Reproduce

herbie shell --seed 2025058 
(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))))