Complex division, real part

Percentage Accurate: 60.9% → 79.0%

Time: 6.6s

Alternatives: 8

Speedup: 1.8×

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: 60.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;d \leq 1.16 \cdot 10^{-108}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{b}, a, d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.02e-84)
   (/ (fma (/ c d) a b) d)
   (if (<= d 1.16e-108)
     (/ (fma (/ d c) b a) c)
     (if (<= d 1.26e+149)
       (* (/ (fma (/ c b) a d) (fma d d (* c c))) b)
       (/ (fma (/ a d) c b) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.02e-84) {
		tmp = fma((c / d), a, b) / d;
	} else if (d <= 1.16e-108) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 1.26e+149) {
		tmp = (fma((c / b), a, d) / fma(d, d, (c * c))) * b;
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.02e-84)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (d <= 1.16e-108)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 1.26e+149)
		tmp = Float64(Float64(fma(Float64(c / b), a, d) / fma(d, d, Float64(c * c))) * b);
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.02e-84], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.16e-108], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.26e+149], N[(N[(N[(N[(c / b), $MachinePrecision] * a + d), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.02000000000000004e-84

   1. Initial program 55.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   if -1.02000000000000004e-84 < d < 1.16e-108

   1. Initial program 70.9%

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

   2. Taylor expanded in c around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if 1.16e-108 < d < 1.26000000000000005e149

   1. Initial program 72.6%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 69.9%

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

   if 1.26000000000000005e149 < d

   1. Initial program 30.0%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 88.6

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

   4. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 89.2

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

   6. Applied rewrites 89.2%

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

Alternative 2: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.02e-84)
   (/ (fma (/ c d) a b) d)
   (if (<= d 4.3e-111)
     (/ (fma (/ d c) b a) c)
     (if (<= d 1.8e+126)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (/ (fma (/ a d) c b) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.02e-84) {
		tmp = fma((c / d), a, b) / d;
	} else if (d <= 4.3e-111) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 1.8e+126) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.02e-84)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (d <= 4.3e-111)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 1.8e+126)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.02e-84], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 4.3e-111], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.8e+126], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.02000000000000004e-84

   1. Initial program 55.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   if -1.02000000000000004e-84 < d < 4.29999999999999981e-111

   1. Initial program 70.9%

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

   2. Taylor expanded in c around inf

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

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 88.3

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

   4. Applied rewrites 88.3%

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

   if 4.29999999999999981e-111 < d < 1.8e126

   1. Initial program 73.8%

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

   if 1.8e126 < d

   1. Initial program 32.8%

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

   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. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 86.4

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

   4. Applied rewrites 86.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 87.1

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

   6. Applied rewrites 87.1%

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

Alternative 3: 77.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.02e-84)
   (/ (fma (/ c d) a b) d)
   (if (<= d 2.65e-12) (/ (fma (/ d c) b a) c) (/ (fma (/ a d) c b) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.02e-84) {
		tmp = fma((c / d), a, b) / d;
	} else if (d <= 2.65e-12) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.02e-84)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (d <= 2.65e-12)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.02e-84], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.65e-12], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.02000000000000004e-84

   1. Initial program 55.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   if -1.02000000000000004e-84 < d < 2.64999999999999982e-12

   1. Initial program 72.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 84.0

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

   4. Applied rewrites 84.0%

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

   if 2.64999999999999982e-12 < d

   1. Initial program 49.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 77.3

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

   6. Applied rewrites 77.3%

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

Alternative 4: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.02e-84)
   (/ (fma (/ c d) a b) d)
   (if (<= d 2.65e-12) (/ (fma (/ b c) d a) c) (/ (fma (/ a d) c b) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.02e-84) {
		tmp = fma((c / d), a, b) / d;
	} else if (d <= 2.65e-12) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.02e-84)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (d <= 2.65e-12)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.02e-84], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.65e-12], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.02000000000000004e-84

   1. Initial program 55.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 68.5

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

   4. Applied rewrites 68.5%

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

   if -1.02000000000000004e-84 < d < 2.64999999999999982e-12

   1. Initial program 72.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.1

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

   6. Applied rewrites 82.1%

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

   if 2.64999999999999982e-12 < d

   1. Initial program 49.6%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 76.5

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

   4. Applied rewrites 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 77.3

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

   6. Applied rewrites 77.3%

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

Alternative 5: 76.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -1.02 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ a d) c b) d)))
   (if (<= d -1.02e-84) t_0 (if (<= d 2.65e-12) (/ (fma (/ b c) d a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -1.02e-84) {
		tmp = t_0;
	} else if (d <= 2.65e-12) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -1.02e-84)
		tmp = t_0;
	elseif (d <= 2.65e-12)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.02e-84], t$95$0, If[LessEqual[d, 2.65e-12], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.02000000000000004e-84 or 2.64999999999999982e-12 < d

   1. Initial program 52.9%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 72.2

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

   4. Applied rewrites 72.2%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 72.5

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

   6. Applied rewrites 72.5%

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

   if -1.02000000000000004e-84 < d < 2.64999999999999982e-12

   1. Initial program 72.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.1

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

   6. Applied rewrites 82.1%

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

Alternative 6: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -0.0008:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -0.0008)
   (/ a c)
   (if (<= c 7.5e+49) (/ (fma (/ a d) c b) d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -0.0008) {
		tmp = a / c;
	} else if (c <= 7.5e+49) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -0.0008)
		tmp = Float64(a / c);
	elseif (c <= 7.5e+49)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -0.0008], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.5e+49], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.00000000000000038e-4 or 7.4999999999999995e49 < c

   1. Initial program 47.5%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 66.8

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

   4. Applied rewrites 66.8%

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

   if -8.00000000000000038e-4 < c < 7.4999999999999995e49

   1. Initial program 72.1%

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

   2. Taylor expanded in c around 0

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

      1. pow2 N/A

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

      2. associate-/l/ N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 79.3

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

   4. Applied rewrites 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l/ N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 77.9

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

   6. Applied rewrites 77.9%

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

Alternative 7: 63.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-86}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.9e-86) (/ b d) (if (<= d 2.15e-12) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.9e-86) {
		tmp = b / d;
	} else if (d <= 2.15e-12) {
		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 <= (-2.9d-86)) then
        tmp = b / d
    else if (d <= 2.15d-12) 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 <= -2.9e-86) {
		tmp = b / d;
	} else if (d <= 2.15e-12) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.9e-86:
		tmp = b / d
	elif d <= 2.15e-12:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.9e-86)
		tmp = Float64(b / d);
	elseif (d <= 2.15e-12)
		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 <= -2.9e-86)
		tmp = b / d;
	elseif (d <= 2.15e-12)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.9e-86], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.15e-12], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.8999999999999999e-86 or 2.14999999999999993e-12 < d

   1. Initial program 53.0%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 60.2

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

   4. Applied rewrites 60.2%

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

   if -2.8999999999999999e-86 < d < 2.14999999999999993e-12

   1. Initial program 72.1%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 66.8

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

   4. Applied rewrites 66.8%

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

Alternative 8: 42.1% 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 60.9%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 42.1

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

4. Applied rewrites 42.1%

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

Developer Target 1: 99.3% accurate, 0.8× 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 2025130 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform c (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))))