Complex division, real part

Percentage Accurate: 62.3% → 82.0%

Time: 3.0s

Alternatives: 8

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% 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: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;d \leq -6.9 \cdot 10^{+103}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -3.95 \cdot 10^{-144}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-162}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(d, b, -\frac{\mathsf{fma}\left(b, \frac{{d}^{3}}{c}, 1 \cdot \left(\left(d \cdot d\right) \cdot a\right)\right)}{c}\right)}{c}, -1, -a\right)}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+179}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, -b \cdot {\left(\frac{c}{d}\right)}^{2}\right) + b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (fma c c (* d d)))))
   (if (<= d -6.9e+103)
     (fma (/ a d) (/ c d) (/ b d))
     (if (<= d -3.95e-144)
       t_0
       (if (<= d 2e-162)
         (-
          (/
           (fma
            (/
             (fma
              d
              b
              (- (/ (fma b (/ (pow d 3.0) c) (* 1.0 (* (* d d) a))) c)))
             c)
            -1.0
            (- a))
           c))
         (if (<= d 2.2e+179)
           t_0
           (/ (+ (fma (/ c d) a (- (* b (pow (/ c d) 2.0)))) b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / fma(c, c, (d * d));
	double tmp;
	if (d <= -6.9e+103) {
		tmp = fma((a / d), (c / d), (b / d));
	} else if (d <= -3.95e-144) {
		tmp = t_0;
	} else if (d <= 2e-162) {
		tmp = -(fma((fma(d, b, -(fma(b, (pow(d, 3.0) / c), (1.0 * ((d * d) * a))) / c)) / c), -1.0, -a) / c);
	} else if (d <= 2.2e+179) {
		tmp = t_0;
	} else {
		tmp = (fma((c / d), a, -(b * pow((c / d), 2.0))) + b) / d;
	}
	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)))
	tmp = 0.0
	if (d <= -6.9e+103)
		tmp = fma(Float64(a / d), Float64(c / d), Float64(b / d));
	elseif (d <= -3.95e-144)
		tmp = t_0;
	elseif (d <= 2e-162)
		tmp = Float64(-Float64(fma(Float64(fma(d, b, Float64(-Float64(fma(b, Float64((d ^ 3.0) / c), Float64(1.0 * Float64(Float64(d * d) * a))) / c))) / c), -1.0, Float64(-a)) / c));
	elseif (d <= 2.2e+179)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(Float64(c / d), a, Float64(-Float64(b * (Float64(c / d) ^ 2.0)))) + b) / d);
	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]}, If[LessEqual[d, -6.9e+103], N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.95e-144], t$95$0, If[LessEqual[d, 2e-162], (-N[(N[(N[(N[(d * b + (-N[(N[(b * N[(N[Power[d, 3.0], $MachinePrecision] / c), $MachinePrecision] + N[(1.0 * N[(N[(d * d), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision])), $MachinePrecision] / c), $MachinePrecision] * -1.0 + (-a)), $MachinePrecision] / c), $MachinePrecision]), If[LessEqual[d, 2.2e+179], t$95$0, N[(N[(N[(N[(c / d), $MachinePrecision] * a + (-N[(b * N[Power[N[(c / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.8999999999999999e103

   1. Initial program 36.8%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. 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 84.1

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

   4. Applied rewrites 84.1%

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

   if -6.8999999999999999e103 < d < -3.95000000000000026e-144 or 1.99999999999999991e-162 < d < 2.2e179

   1. Initial program 72.7%

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

      1. lift-+.f64 N/A

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

      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 72.7

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

   3. Applied rewrites 72.7%

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

   if -3.95000000000000026e-144 < d < 1.99999999999999991e-162

   1. Initial program 72.1%

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

   2. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{-1 \cdot a + -1 \cdot \frac{-1 \cdot \frac{\frac{b \cdot {d}^{3}}{c} - -1 \cdot \left(a \cdot {d}^{2}\right)}{c} + b \cdot d}{c}}{c}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{-1 \cdot a + -1 \cdot \frac{-1 \cdot \frac{\frac{b \cdot {d}^{3}}{c} - -1 \cdot \left(a \cdot {d}^{2}\right)}{c} + b \cdot d}{c}}{c} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{-1 \cdot a + -1 \cdot \frac{-1 \cdot \frac{\frac{b \cdot {d}^{3}}{c} - -1 \cdot \left(a \cdot {d}^{2}\right)}{c} + b \cdot d}{c}}{c} \]

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(d, b, -\frac{\mathsf{fma}\left(b, \frac{{d}^{3}}{c}, 1 \cdot \left(\left(d \cdot d\right) \cdot a\right)\right)}{c}\right)}{c}, -1, -a\right)}{c}} \]

   if 2.2e179 < d

   1. Initial program 31.8%

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

   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 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 90.8%

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

Alternative 2: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+179}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, -b \cdot {\left(\frac{c}{d}\right)}^{2}\right) + b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.2e-10)
   (fma (/ a d) (/ c d) (/ b d))
   (if (<= d 2.15e-162)
     (/ (fma b (/ d c) a) c)
     (if (<= d 2.2e+179)
       (/ (+ (* a c) (* b d)) (fma c c (* d d)))
       (/ (+ (fma (/ c d) a (- (* b (pow (/ c d) 2.0)))) b) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.2e-10) {
		tmp = fma((a / d), (c / d), (b / d));
	} else if (d <= 2.15e-162) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 2.2e+179) {
		tmp = ((a * c) + (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (fma((c / d), a, -(b * pow((c / d), 2.0))) + b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.2e-10)
		tmp = fma(Float64(a / d), Float64(c / d), Float64(b / d));
	elseif (d <= 2.15e-162)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 2.2e+179)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(fma(Float64(c / d), a, Float64(-Float64(b * (Float64(c / d) ^ 2.0)))) + b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.2e-10], N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e-162], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.2e+179], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c / d), $MachinePrecision] * a + (-N[(b * N[Power[N[(c / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.19999999999999981e-10

   1. Initial program 49.8%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. 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 75.6

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

   4. Applied rewrites 75.6%

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

   if -3.19999999999999981e-10 < d < 2.14999999999999998e-162

   1. Initial program 73.8%

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

   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 86.2

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

   4. Applied rewrites 86.2%

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

   if 2.14999999999999998e-162 < d < 2.2e179

   1. Initial program 70.3%

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

      1. lift-+.f64 N/A

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

      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 70.3

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

   3. Applied rewrites 70.3%

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

   if 2.2e179 < d

   1. Initial program 31.8%

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

   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 91.2

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

   4. Applied rewrites 91.2%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 90.8%

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

Alternative 3: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ a d) (/ c d) (/ b d))))
   (if (<= d -3.2e-10)
     t_0
     (if (<= d 2.15e-162)
       (/ (fma b (/ d c) a) c)
       (if (<= d 3e+153) (/ (+ (* a c) (* b d)) (fma c c (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), (c / d), (b / d));
	double tmp;
	if (d <= -3.2e-10) {
		tmp = t_0;
	} else if (d <= 2.15e-162) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 3e+153) {
		tmp = ((a * c) + (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(a / d), Float64(c / d), Float64(b / d))
	tmp = 0.0
	if (d <= -3.2e-10)
		tmp = t_0;
	elseif (d <= 2.15e-162)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 3e+153)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.2e-10], t$95$0, If[LessEqual[d, 2.15e-162], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+153], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.19999999999999981e-10 or 3.00000000000000019e153 < d

   1. Initial program 43.2%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. 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 80.4

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

   4. Applied rewrites 80.4%

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

   if -3.19999999999999981e-10 < d < 2.14999999999999998e-162

   1. Initial program 73.8%

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

   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 86.2

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

   4. Applied rewrites 86.2%

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

   if 2.14999999999999998e-162 < d < 3.00000000000000019e153

   1. Initial program 74.4%

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

      1. lift-+.f64 N/A

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

      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 74.4

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

   3. Applied rewrites 74.4%

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

Alternative 4: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{a}{d}, \frac{c}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ a d) (/ c d) (/ b d))))
   (if (<= d -3.2e-10) t_0 (if (<= d 1.2e-20) (/ (fma b (/ d c) a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), (c / d), (b / d));
	double tmp;
	if (d <= -3.2e-10) {
		tmp = t_0;
	} else if (d <= 1.2e-20) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(a / d), Float64(c / d), Float64(b / d))
	tmp = 0.0
	if (d <= -3.2e-10)
		tmp = t_0;
	elseif (d <= 1.2e-20)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.2e-10], t$95$0, If[LessEqual[d, 1.2e-20], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.19999999999999981e-10 or 1.19999999999999996e-20 < d

   1. Initial program 50.3%

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

   2. Taylor expanded in c around 0

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. 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 75.2

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

   4. Applied rewrites 75.2%

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

   if -3.19999999999999981e-10 < d < 1.19999999999999996e-20

   1. Initial program 75.0%

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

   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 82.7

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

   4. Applied rewrites 82.7%

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

Alternative 5: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.05 \cdot 10^{+129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 130000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -2.05e+129)
     t_0
     (if (<= c 130000000.0) (/ (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 <= -2.05e+129) {
		tmp = t_0;
	} else if (c <= 130000000.0) {
		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 <= -2.05e+129)
		tmp = t_0;
	elseif (c <= 130000000.0)
		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, -2.05e+129], t$95$0, If[LessEqual[c, 130000000.0], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.0500000000000001e129 or 1.3e8 < c

   1. Initial program 43.8%

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

   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 80.7

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

   4. Applied rewrites 80.7%

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

   if -2.0500000000000001e129 < c < 1.3e8

   1. Initial program 73.9%

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

   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 73.6

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

   4. Applied rewrites 73.6%

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

Alternative 6: 70.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{+188}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e+188) (/ a c) (if (<= c 4e+99) (/ (fma a (/ c d) b) d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e+188) {
		tmp = a / c;
	} else if (c <= 4e+99) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e+188)
		tmp = Float64(a / c);
	elseif (c <= 4e+99)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e+188], N[(a / c), $MachinePrecision], If[LessEqual[c, 4e+99], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.0000000000000001e188 or 3.9999999999999999e99 < c

   1. Initial program 36.6%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if -3.0000000000000001e188 < c < 3.9999999999999999e99

   1. Initial program 71.6%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 67.5

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

   4. Applied rewrites 67.5%

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

Alternative 7: 64.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.5e-11) (/ b d) (if (<= d 1.26e-20) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.5e-11) {
		tmp = b / d;
	} else if (d <= 1.26e-20) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-6.5d-11)) then
        tmp = b / d
    else if (d <= 1.26d-20) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.5e-11) {
		tmp = b / d;
	} else if (d <= 1.26e-20) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6.5e-11:
		tmp = b / d
	elif d <= 1.26e-20:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6.5e-11)
		tmp = Float64(b / d);
	elseif (d <= 1.26e-20)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6.5e-11)
		tmp = b / d;
	elseif (d <= 1.26e-20)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6.5e-11], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.26e-20], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.49999999999999953e-11 or 1.26e-20 < d

   1. Initial program 50.3%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 63.3

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

   4. Applied rewrites 63.3%

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

   if -6.49999999999999953e-11 < d < 1.26e-20

   1. Initial program 75.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 65.4

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

   4. Applied rewrites 65.4%

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

Alternative 8: 43.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.3%

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

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

Reproduce

herbie shell --seed 2025101 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64
  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))