Complex division, real part

Percentage Accurate: 60.7% → 82.2%

Time: 3.3s

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: 60.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (fma c c (* d d))))
        (t_1 (/ (fma b (/ d c) a) c)))
   (if (<= c -1.05e+61)
     t_1
     (if (<= c -1.8e-156)
       t_0
       (if (<= c 2.6e-153)
         (/ (fma (/ a d) c b) d)
         (if (<= c 2.2e+83) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / fma(c, c, (d * d));
	double t_1 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -1.05e+61) {
		tmp = t_1;
	} else if (c <= -1.8e-156) {
		tmp = t_0;
	} else if (c <= 2.6e-153) {
		tmp = fma((a / d), c, b) / d;
	} else if (c <= 2.2e+83) {
		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)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -1.05e+61)
		tmp = t_1;
	elseif (c <= -1.8e-156)
		tmp = t_0;
	elseif (c <= 2.6e-153)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (c <= 2.2e+83)
		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[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.05e+61], t$95$1, If[LessEqual[c, -1.8e-156], t$95$0, If[LessEqual[c, 2.6e-153], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.2e+83], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.0500000000000001e61 or 2.19999999999999999e83 < c

   1. Initial program 42.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

   if -1.0500000000000001e61 < c < -1.79999999999999999e-156 or 2.6000000000000001e-153 < c < 2.19999999999999999e83

   1. Initial program 83.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if -1.79999999999999999e-156 < c < 2.6000000000000001e-153

   1. Initial program 70.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 94.6

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

   7. Applied rewrites 94.6%

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

Alternative 2: 63.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, b, c \cdot a\right)\\ \mathbf{if}\;c \leq -2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-102}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d b (* c a))))
   (if (<= c -2.5e+144)
     (/ a c)
     (if (<= c -4.1e-102)
       (* a (/ c (fma d d (* c c))))
       (if (<= c 1.7e-78)
         (/ t_0 (* d d))
         (if (<= c 9.5e+81) (/ t_0 (* c c)) (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a));
	double tmp;
	if (c <= -2.5e+144) {
		tmp = a / c;
	} else if (c <= -4.1e-102) {
		tmp = a * (c / fma(d, d, (c * c)));
	} else if (c <= 1.7e-78) {
		tmp = t_0 / (d * d);
	} else if (c <= 9.5e+81) {
		tmp = t_0 / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, b, Float64(c * a))
	tmp = 0.0
	if (c <= -2.5e+144)
		tmp = Float64(a / c);
	elseif (c <= -4.1e-102)
		tmp = Float64(a * Float64(c / fma(d, d, Float64(c * c))));
	elseif (c <= 1.7e-78)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 9.5e+81)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.5e+144], N[(a / c), $MachinePrecision], If[LessEqual[c, -4.1e-102], N[(a * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e-78], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.5e+81], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.5e144 or 9.50000000000000083e81 < c

   1. Initial program 38.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -2.5e144 < c < -4.1000000000000003e-102

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{a \cdot c}{{c}^{2} + {d}^{2}}} \]
   4. 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 70.2

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

   5. Applied rewrites 70.2%

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

   if -4.1000000000000003e-102 < c < 1.70000000000000006e-78

   1. Initial program 77.2%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   6. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if 1.70000000000000006e-78 < c < 9.50000000000000083e81

   1. Initial program 79.0%

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

   3. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.8

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

      9. pow2 64.8

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

      10. pow2 64.8

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

      11. +-commutative 64.8

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

      12. pow2 64.8

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

      13. pow2 64.8

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

   7. Applied rewrites 64.8%

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

Alternative 3: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;c \leq -2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+164}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* a (/ c (fma d d (* c c))))))
   (if (<= c -2.5e+144)
     (/ a c)
     (if (<= c -5.8e-102)
       t_0
       (if (<= c 5.2e-151) (/ b d) (if (<= c 8.6e+164) t_0 (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a * (c / fma(d, d, (c * c)));
	double tmp;
	if (c <= -2.5e+144) {
		tmp = a / c;
	} else if (c <= -5.8e-102) {
		tmp = t_0;
	} else if (c <= 5.2e-151) {
		tmp = b / d;
	} else if (c <= 8.6e+164) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a * Float64(c / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (c <= -2.5e+144)
		tmp = Float64(a / c);
	elseif (c <= -5.8e-102)
		tmp = t_0;
	elseif (c <= 5.2e-151)
		tmp = Float64(b / d);
	elseif (c <= 8.6e+164)
		tmp = t_0;
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.5e+144], N[(a / c), $MachinePrecision], If[LessEqual[c, -5.8e-102], t$95$0, If[LessEqual[c, 5.2e-151], N[(b / d), $MachinePrecision], If[LessEqual[c, 8.6e+164], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.5e144 or 8.6e164 < c

   1. Initial program 33.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -2.5e144 < c < -5.79999999999999973e-102 or 5.2000000000000001e-151 < c < 8.6e164

   1. Initial program 77.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{a \cdot c}{{c}^{2} + {d}^{2}}} \]
   4. 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 63.5

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

   5. Applied rewrites 63.5%

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

   if -5.79999999999999973e-102 < c < 5.2000000000000001e-151

   1. Initial program 74.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 71.0

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

   5. Applied rewrites 71.0%

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

Alternative 4: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e-12)
   (/ a c)
   (if (<= c 2.4e-78)
     (/ (fma a (/ c d) b) d)
     (if (<= c 9.5e+81) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.8e-12) {
		tmp = a / c;
	} else if (c <= 2.4e-78) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 9.5e+81) {
		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 <= -6.8e-12)
		tmp = Float64(a / c);
	elseif (c <= 2.4e-78)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 9.5e+81)
		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, -6.8e-12], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.4e-78], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.5e+81], 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 < -6.8000000000000001e-12 or 9.50000000000000083e81 < c

   1. Initial program 48.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if -6.8000000000000001e-12 < c < 2.4e-78

   1. Initial program 78.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if 2.4e-78 < c < 9.50000000000000083e81

   1. Initial program 79.0%

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

   3. Taylor expanded in c around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.8

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

      9. pow2 64.8

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

      10. pow2 64.8

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

      11. +-commutative 64.8

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

      12. pow2 64.8

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

      13. pow2 64.8

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

   7. Applied rewrites 64.8%

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

Alternative 5: 77.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6e-12) (not (<= c 2.4e-78)))
   (/ (fma b (/ d c) a) c)
   (/ (fma a (/ c d) b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6e-12) || !(c <= 2.4e-78)) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = fma(a, (c / d), b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6e-12) || !(c <= 2.4e-78))
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6e-12], N[Not[LessEqual[c, 2.4e-78]], $MachinePrecision]], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.0000000000000003e-12 or 2.4e-78 < c

   1. Initial program 54.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   if -6.0000000000000003e-12 < c < 2.4e-78

   1. Initial program 78.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.8

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

   5. Applied rewrites 82.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.2e-12) (not (<= c 2.4e-78)))
   (/ a c)
   (/ (fma d b (* c a)) (* d d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.2e-12) || !(c <= 2.4e-78)) {
		tmp = a / c;
	} else {
		tmp = fma(d, b, (c * a)) / (d * d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.2e-12) || !(c <= 2.4e-78))
		tmp = Float64(a / c);
	else
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(d * d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.2e-12], N[Not[LessEqual[c, 2.4e-78]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.2000000000000001e-12 or 2.4e-78 < c

   1. Initial program 54.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -3.2000000000000001e-12 < c < 2.4e-78

   1. Initial program 78.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in d around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 65.6

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

   8. Applied rewrites 65.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6.5e-12) (not (<= c 2.4e-78))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.5e-12) || !(c <= 2.4e-78)) {
		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 <= (-6.5d-12)) .or. (.not. (c <= 2.4d-78))) 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 <= -6.5e-12) || !(c <= 2.4e-78)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6.5e-12) or not (c <= 2.4e-78):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6.5e-12) || !(c <= 2.4e-78))
		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 <= -6.5e-12) || ~((c <= 2.4e-78)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6.5e-12], N[Not[LessEqual[c, 2.4e-78]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.5000000000000002e-12 or 2.4e-78 < c

   1. Initial program 54.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if -6.5000000000000002e-12 < c < 2.4e-78

   1. Initial program 78.4%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 60.0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{-12} \lor \neg \left(c \leq 2.4 \cdot 10^{-78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 46.9

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

5. Applied rewrites 46.9%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2025037 
(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))))