Complex division, real part

Percentage Accurate: 62.0% → 87.0%

Time: 4.4s

Alternatives: 8

Speedup: 1.8×

Specification

Math

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

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: 62.0% accurate, 1.0× speedup

Math

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

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: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, a, \frac{b}{t\_0} \cdot d\right)\\ \mathbf{if}\;c \leq -1.15 \cdot 10^{+135}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 3.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{+175}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (/ c t_0) a (* (/ b t_0) d))))
   (if (<= c -1.15e+135)
     (/ (fma d (/ b c) a) c)
     (if (<= c -2.7e-17)
       t_1
       (if (<= c 3.25e-24)
         (/ (fma (/ c d) a b) d)
         (if (<= c 2.9e+175) t_1 (fma (/ d c) (/ b c) (/ a c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / t_0), a, ((b / t_0) * d));
	double tmp;
	if (c <= -1.15e+135) {
		tmp = fma(d, (b / c), a) / c;
	} else if (c <= -2.7e-17) {
		tmp = t_1;
	} else if (c <= 3.25e-24) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 2.9e+175) {
		tmp = t_1;
	} else {
		tmp = fma((d / c), (b / c), (a / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), a, Float64(Float64(b / t_0) * d))
	tmp = 0.0
	if (c <= -1.15e+135)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	elseif (c <= -2.7e-17)
		tmp = t_1;
	elseif (c <= 3.25e-24)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 2.9e+175)
		tmp = t_1;
	else
		tmp = fma(Float64(d / c), Float64(b / c), Float64(a / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / t$95$0), $MachinePrecision] * a + N[(N[(b / t$95$0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.15e+135], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -2.7e-17], t$95$1, If[LessEqual[c, 3.25e-24], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.9e+175], t$95$1, N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.1500000000000001e135

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.7%

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

   8. Applied rewrites 53.7%

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

   if -1.1500000000000001e135 < c < -2.7000000000000001e-17 or 3.25e-24 < c < 2.9e175

   1. Initial program 62.0%

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

   3. Applied rewrites 61.7%

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

   if -2.7000000000000001e-17 < c < 3.25e-24

   1. Initial program 62.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   4. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   if 2.9e175 < c

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 53.0%

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

   8. Applied rewrites 53.0%

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

Alternative 2: 82.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{d}{c}, b, a\right)\\ t_1 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_2 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_3 := \mathsf{fma}\left(d, b, c \cdot a\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+301}:\\ \;\;\;\;\frac{t\_0}{c}\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\frac{t\_2}{t\_3}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(a, c, b \cdot d\right)}, c, \frac{d}{b}\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\frac{t\_3}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\frac{1}{t\_0}, c, \frac{d}{b}\right)}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ d c) b a))
        (t_1 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_2 (fma d d (* c c)))
        (t_3 (fma d b (* c a))))
   (if (<= t_1 -5e+301)
     (/ t_0 c)
     (if (<= t_1 -5e-130)
       (/ 1.0 (/ t_2 t_3))
       (if (<= t_1 0.0)
         (/ 1.0 (fma (/ c (fma a c (* b d))) c (/ d b)))
         (if (<= t_1 2e+302)
           (/ t_3 t_2)
           (/ 1.0 (fma (/ 1.0 t_0) c (/ d b)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((d / c), b, a);
	double t_1 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_2 = fma(d, d, (c * c));
	double t_3 = fma(d, b, (c * a));
	double tmp;
	if (t_1 <= -5e+301) {
		tmp = t_0 / c;
	} else if (t_1 <= -5e-130) {
		tmp = 1.0 / (t_2 / t_3);
	} else if (t_1 <= 0.0) {
		tmp = 1.0 / fma((c / fma(a, c, (b * d))), c, (d / b));
	} else if (t_1 <= 2e+302) {
		tmp = t_3 / t_2;
	} else {
		tmp = 1.0 / fma((1.0 / t_0), c, (d / b));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(d / c), b, a)
	t_1 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_2 = fma(d, d, Float64(c * c))
	t_3 = fma(d, b, Float64(c * a))
	tmp = 0.0
	if (t_1 <= -5e+301)
		tmp = Float64(t_0 / c);
	elseif (t_1 <= -5e-130)
		tmp = Float64(1.0 / Float64(t_2 / t_3));
	elseif (t_1 <= 0.0)
		tmp = Float64(1.0 / fma(Float64(c / fma(a, c, Float64(b * d))), c, Float64(d / b)));
	elseif (t_1 <= 2e+302)
		tmp = Float64(t_3 / t_2);
	else
		tmp = Float64(1.0 / fma(Float64(1.0 / t_0), c, Float64(d / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+301], N[(t$95$0 / c), $MachinePrecision], If[LessEqual[t$95$1, -5e-130], N[(1.0 / N[(t$95$2 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(1.0 / N[(N[(c / N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c + N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[(t$95$3 / t$95$2), $MachinePrecision], N[(1.0 / N[(N[(1.0 / t$95$0), $MachinePrecision] * c + N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -5.0000000000000004e301

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.5%

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

   8. Applied rewrites 54.5%

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

   if -5.0000000000000004e301 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -4.9999999999999996e-130

   1. Initial program 62.0%

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

   3. Applied rewrites 61.8%

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

   if -4.9999999999999996e-130 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 0.0

   1. Initial program 62.0%

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

   3. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(d, b, c \cdot a\right)}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 67.9%

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

      21. lift-fma.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(a, c, b \cdot d\right)}, c, \color{blue}{\frac{d}{b}}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 68.3%

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

   8. Applied rewrites 68.3%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000002e302

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   if 2.0000000000000002e302 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 62.0%

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

   3. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{\mathsf{fma}\left(d, b, c \cdot a\right)}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 67.9%

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

      21. lift-fma.f64 N/A

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

      22. lift-*.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(a, c, b \cdot d\right)}, c, \color{blue}{\frac{d}{b}}\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 68.3%

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

   8. Applied rewrites 68.3%

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(a, c, b \cdot d\right)}, c, \color{blue}{\frac{d}{b}}\right)} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. lift-fma.f64 N/A

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

      6. add-to-fraction N/A

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

      7. lift-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-unsound-/.f64 74.1%

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 76.3%

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

   10. Applied rewrites 76.3%

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}}, c, \frac{d}{b}\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-139}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.85 \cdot 10^{+175}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{d}{c}, \frac{b}{c}, \frac{a}{c}\right)\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.7e-17)
   (/ (fma d (/ b c) a) c)
   (if (<= c 1.9e-139)
     (/ (fma (/ c d) a b) d)
     (if (<= c 2.85e+175)
       (/ (fma d b (* c a)) (fma d d (* c c)))
       (fma (/ d c) (/ b c) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e-17) {
		tmp = fma(d, (b / c), a) / c;
	} else if (c <= 1.9e-139) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 2.85e+175) {
		tmp = fma(d, b, (c * a)) / fma(d, d, (c * c));
	} else {
		tmp = fma((d / c), (b / c), (a / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.7e-17)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	elseif (c <= 1.9e-139)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 2.85e+175)
		tmp = Float64(fma(d, b, Float64(c * a)) / fma(d, d, Float64(c * c)));
	else
		tmp = fma(Float64(d / c), Float64(b / c), Float64(a / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.7e-17], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.9e-139], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.85e+175], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision] + N[(a / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.7000000000000001e-17

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.7%

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

   8. Applied rewrites 53.7%

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

   if -2.7000000000000001e-17 < c < 1.90000000000000004e-139

   1. Initial program 62.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   4. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.2%

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

   6. Applied rewrites 54.2%

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

   if 1.90000000000000004e-139 < c < 2.85000000000000012e175

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   if 2.85000000000000012e175 < c

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 53.0%

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

   8. Applied rewrites 53.0%

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

Alternative 4: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d (/ b c) a) c)))
   (if (<= c -2.7e-17) t_0 (if (<= c 5.2e+62) (/ (fma (/ c d) a b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, (b / c), a) / c;
	double tmp;
	if (c <= -2.7e-17) {
		tmp = t_0;
	} else if (c <= 5.2e+62) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, Float64(b / c), a) / c)
	tmp = 0.0
	if (c <= -2.7e-17)
		tmp = t_0;
	elseif (c <= 5.2e+62)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.7e-17], t$95$0, If[LessEqual[c, 5.2e+62], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.7000000000000001e-17 or 5.19999999999999968e62 < c

   1. Initial program 62.0%

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   6. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 53.7%

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

   8. Applied rewrites 53.7%

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

   if -2.7000000000000001e-17 < c < 5.19999999999999968e62

   1. Initial program 62.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   4. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.2%

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

   6. Applied rewrites 54.2%

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

Alternative 5: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.7e-17)
   (/ a c)
   (if (<= c 2.6e+45) (/ (fma (/ c d) a b) d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e-17) {
		tmp = a / c;
	} else if (c <= 2.6e+45) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.7e-17)
		tmp = Float64(a / c);
	elseif (c <= 2.6e+45)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.7e-17], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.6e+45], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.7000000000000001e-17 or 2.60000000000000007e45 < c

   1. Initial program 62.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -2.7000000000000001e-17 < c < 2.60000000000000007e45

   1. Initial program 62.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   4. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 54.2%

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

   6. Applied rewrites 54.2%

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

Alternative 6: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.7e-17)
   (/ a c)
   (if (<= c 2.6e+45) (/ (fma (/ a d) c b) d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e-17) {
		tmp = a / c;
	} else if (c <= 2.6e+45) {
		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 <= -2.7e-17)
		tmp = Float64(a / c);
	elseif (c <= 2.6e+45)
		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, -2.7e-17], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.6e+45], 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 < -2.7000000000000001e-17 or 2.60000000000000007e45 < c

   1. Initial program 62.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 42.9%

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

   4. Applied rewrites 42.9%

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

   if -2.7000000000000001e-17 < c < 2.60000000000000007e45

   1. Initial program 62.0%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.3%

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

   4. Applied rewrites 52.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 53.5%

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

   6. Applied rewrites 53.5%

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

Alternative 7: 63.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.5e+78) (/ b d) (if (<= d 9.8e+17) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.5e+78) {
		tmp = b / d;
	} else if (d <= 9.8e+17) {
		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 <= (-5.5d+78)) then
        tmp = b / d
    else if (d <= 9.8d+17) 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 <= -5.5e+78) {
		tmp = b / d;
	} else if (d <= 9.8e+17) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.5e+78:
		tmp = b / d
	elif d <= 9.8e+17:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.5e+78)
		tmp = Float64(b / d);
	elseif (d <= 9.8e+17)
		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 <= -5.5e+78)
		tmp = b / d;
	elseif (d <= 9.8e+17)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.5e+78], N[(b / d), $MachinePrecision], If[LessEqual[d, 9.8e+17], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.4999999999999997e78 or 9.8e17 < d

   1. Initial program 62.0%

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

   2. Taylor expanded in c around 0

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

      1. lower-/.f64 42.5%

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

   4. Applied rewrites 42.5%

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

   if -5.4999999999999997e78 < d < 9.8e17

   1. Initial program 62.0%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 42.9%

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

   4. Applied rewrites 42.9%

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

Alternative 8: 42.9% accurate, 5.0× speedup

Math

\[\frac{a}{c} \]

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 62.0%

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

2. Taylor expanded in c around inf

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

   1. lower-/.f64 42.9%

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

4. Applied rewrites 42.9%

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

Developer Target 1: 99.4% accurate, 0.8× speedup

Math

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

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 2025183 
(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))))