Complex division, real part

Percentage Accurate: 61.3% → 82.8%

Time: 3.1s

Alternatives: 8

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -8.7 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{c}, \frac{d}{c}, \frac{a}{c}\right)\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -8.7e+111)
     (fma (/ b c) (/ d c) (/ a c))
     (if (<= c -2.2e-29)
       t_0
       (if (<= c 3.8e-145)
         (/ (fma a (/ c d) b) d)
         (if (<= c 2.5e+89) t_0 (/ (fma b (/ d c) a) c)))))))

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 (c <= -8.7e+111) {
		tmp = fma((b / c), (d / c), (a / c));
	} else if (c <= -2.2e-29) {
		tmp = t_0;
	} else if (c <= 3.8e-145) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 2.5e+89) {
		tmp = t_0;
	} else {
		tmp = fma(b, (d / c), a) / c;
	}
	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 (c <= -8.7e+111)
		tmp = fma(Float64(b / c), Float64(d / c), Float64(a / c));
	elseif (c <= -2.2e-29)
		tmp = t_0;
	elseif (c <= 3.8e-145)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 2.5e+89)
		tmp = t_0;
	else
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	end
	return 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[c, -8.7e+111], N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.2e-29], t$95$0, If[LessEqual[c, 3.8e-145], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+89], t$95$0, N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -8.6999999999999999e111

   1. Initial program 37.8%

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

   2. Taylor expanded in d around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 85.7

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

   4. Applied rewrites 85.7%

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

   if -8.6999999999999999e111 < c < -2.1999999999999999e-29 or 3.8000000000000002e-145 < c < 2.49999999999999992e89

   1. Initial program 75.7%

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

   if -2.1999999999999999e-29 < c < 3.8000000000000002e-145

   1. Initial program 71.7%

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

   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 86.9

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

   4. Applied rewrites 86.9%

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

   if 2.49999999999999992e89 < c

   1. Initial program 39.5%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.6

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

   4. Applied rewrites 83.6%

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

Alternative 2: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (/ (fma b (/ d c) a) c)))
   (if (<= c -4.8e+113)
     t_1
     (if (<= c -2.2e-29)
       t_0
       (if (<= c 3.8e-145)
         (/ (fma a (/ c d) b) d)
         (if (<= c 2.5e+89) 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 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -4.8e+113) {
		tmp = t_1;
	} else if (c <= -2.2e-29) {
		tmp = t_0;
	} else if (c <= 3.8e-145) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 2.5e+89) {
		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(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -4.8e+113)
		tmp = t_1;
	elseif (c <= -2.2e-29)
		tmp = t_0;
	elseif (c <= 3.8e-145)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 2.5e+89)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return 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[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.8e+113], t$95$1, If[LessEqual[c, -2.2e-29], t$95$0, If[LessEqual[c, 3.8e-145], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+89], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.79999999999999966e113 or 2.49999999999999992e89 < c

   1. Initial program 38.6%

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

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 84.4

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

   4. Applied rewrites 84.4%

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

   if -4.79999999999999966e113 < c < -2.1999999999999999e-29 or 3.8000000000000002e-145 < c < 2.49999999999999992e89

   1. Initial program 75.6%

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

   if -2.1999999999999999e-29 < c < 3.8000000000000002e-145

   1. Initial program 71.7%

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

   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 86.9

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

   4. Applied rewrites 86.9%

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

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+64}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+64)
   (/ a c)
   (if (<= c 7.5e-9)
     (/ (fma a (/ c d) b) d)
     (if (<= c 6.2e+72) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+64) {
		tmp = a / c;
	} else if (c <= 7.5e-9) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 6.2e+72) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+64)
		tmp = Float64(a / c);
	elseif (c <= 7.5e-9)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 6.2e+72)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+64], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.5e-9], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.2e+72], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.15e64 or 6.19999999999999977e72 < c

   1. Initial program 42.5%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 70.9

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

   4. Applied rewrites 70.9%

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

   if -1.15e64 < c < 7.49999999999999933e-9

   1. Initial program 74.0%

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

   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 78.0

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

   4. Applied rewrites 78.0%

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

   if 7.49999999999999933e-9 < c < 6.19999999999999977e72

   1. Initial program 74.2%

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

   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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{c \cdot c}} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 50.3

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

      9. pow2 50.3

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

      10. pow2 50.3

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

      11. +-commutative 50.3

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

      12. pow2 50.3

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

      13. pow2 50.3

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

   6. Applied rewrites 50.3%

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

Alternative 4: 64.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.8e+59)
   (/ a c)
   (if (<= c 7.5e-9)
     (/ b d)
     (if (<= c 6.2e+72) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.8e+59) {
		tmp = a / c;
	} else if (c <= 7.5e-9) {
		tmp = b / d;
	} else if (c <= 6.2e+72) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.8e+59)
		tmp = Float64(a / c);
	elseif (c <= 7.5e-9)
		tmp = Float64(b / d);
	elseif (c <= 6.2e+72)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.8e+59], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.5e-9], N[(b / d), $MachinePrecision], If[LessEqual[c, 6.2e+72], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.7999999999999998e59 or 6.19999999999999977e72 < c

   1. Initial program 42.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 70.6

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

   4. Applied rewrites 70.6%

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

   if -2.7999999999999998e59 < c < 7.49999999999999933e-9

   1. Initial program 74.0%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 61.4

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

   4. Applied rewrites 61.4%

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

   if 7.49999999999999933e-9 < c < 6.19999999999999977e72

   1. Initial program 74.2%

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

   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 50.3

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

   4. Applied rewrites 50.3%

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{c \cdot c}} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 50.3

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

      9. pow2 50.3

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

      10. pow2 50.3

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

      11. +-commutative 50.3

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

      12. pow2 50.3

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

      13. pow2 50.3

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

   6. Applied rewrites 50.3%

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

Alternative 5: 64.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+165}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.8e+59)
   (/ a c)
   (if (<= c 2.4e-124)
     (/ b d)
     (if (<= c 1.55e+165) (* a (/ c (fma d d (* c c)))) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.8e+59) {
		tmp = a / c;
	} else if (c <= 2.4e-124) {
		tmp = b / d;
	} else if (c <= 1.55e+165) {
		tmp = a * (c / fma(d, d, (c * c)));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.8e+59)
		tmp = Float64(a / c);
	elseif (c <= 2.4e-124)
		tmp = Float64(b / d);
	elseif (c <= 1.55e+165)
		tmp = Float64(a * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.8e+59], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.4e-124], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.55e+165], N[(a * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.7999999999999998e59 or 1.5500000000000001e165 < c

   1. Initial program 38.7%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 72.9

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

   4. Applied rewrites 72.9%

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

   if -2.7999999999999998e59 < c < 2.39999999999999992e-124

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 63.5

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

   4. Applied rewrites 63.5%

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

   if 2.39999999999999992e-124 < c < 1.5500000000000001e165

   1. Initial program 71.5%

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

   2. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 54.1

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

   4. Applied rewrites 54.1%

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

Alternative 6: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -2.8e+59) t_0 (if (<= c 7.5e-9) (/ (fma a (/ c d) b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -2.8e+59) {
		tmp = t_0;
	} else if (c <= 7.5e-9) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -2.8e+59)
		tmp = t_0;
	elseif (c <= 7.5e-9)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.8e+59], t$95$0, If[LessEqual[c, 7.5e-9], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.7999999999999998e59 or 7.49999999999999933e-9 < c

   1. Initial program 47.0%

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

   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 77.4

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

   4. Applied rewrites 77.4%

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

   if -2.7999999999999998e59 < c < 7.49999999999999933e-9

   1. Initial program 74.0%

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

   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 78.3

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

   4. Applied rewrites 78.3%

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

Alternative 7: 64.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+59}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3700000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.8e+59) (/ a c) (if (<= c 3700000000.0) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.8e+59) {
		tmp = a / c;
	} else if (c <= 3700000000.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.8d+59)) then
        tmp = a / c
    else if (c <= 3700000000.0d0) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.8e+59) {
		tmp = a / c;
	} else if (c <= 3700000000.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.8e+59:
		tmp = a / c
	elif c <= 3700000000.0:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.8e+59)
		tmp = Float64(a / c);
	elseif (c <= 3700000000.0)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.8e+59)
		tmp = a / c;
	elseif (c <= 3700000000.0)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.8e+59], N[(a / c), $MachinePrecision], If[LessEqual[c, 3700000000.0], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.7999999999999998e59 or 3.7e9 < c

   1. Initial program 46.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 68.0

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

   4. Applied rewrites 68.0%

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

   if -2.7999999999999998e59 < c < 3.7e9

   1. Initial program 74.1%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

Alternative 8: 43.7% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.3%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 43.7

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

4. Applied rewrites 43.7%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2025107 
(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))))