Complex division, real part

Percentage Accurate: 61.3% → 83.6%

Time: 5.6s

Alternatives: 9

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, d)
    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: 83.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+75}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.12 \cdot 10^{-110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-78}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (fma c c (* d d)))))
   (if (<= d -1.35e+75)
     (/ (fma (/ a d) c b) d)
     (if (<= d -1.12e-110)
       t_0
       (if (<= d 1.65e-78)
         (/ (fma (/ d c) b a) c)
         (if (<= d 2.3e+86) t_0 (/ (fma (/ c d) a b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / fma(c, c, (d * d));
	double tmp;
	if (d <= -1.35e+75) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -1.12e-110) {
		tmp = t_0;
	} else if (d <= 1.65e-78) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 2.3e+86) {
		tmp = t_0;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (d <= -1.35e+75)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -1.12e-110)
		tmp = t_0;
	elseif (d <= 1.65e-78)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 2.3e+86)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.35e+75], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.12e-110], t$95$0, If[LessEqual[d, 1.65e-78], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.3e+86], t$95$0, N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.34999999999999999e75

   1. Initial program 48.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -1.34999999999999999e75 < d < -1.11999999999999998e-110 or 1.64999999999999991e-78 < d < 2.2999999999999999e86

   1. Initial program 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 81.0

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

   4. Applied rewrites 81.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 81.0

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

   6. Applied rewrites 81.0%

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

   if -1.11999999999999998e-110 < d < 1.64999999999999991e-78

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 66.1

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 2.2999999999999999e86 < d

   1. Initial program 38.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

Alternative 2: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-108}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+84}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.2e+74)
   (/ (fma (/ a d) c b) d)
   (if (<= d -2.3e-6)
     (/ (fma (/ b c) d a) c)
     (if (<= d -2.8e-108)
       (* (/ d (fma d d (* c c))) b)
       (if (<= d 4.1e+84) (/ (fma (/ d c) b a) c) (/ (fma (/ c d) a b) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.2e+74) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -2.3e-6) {
		tmp = fma((b / c), d, a) / c;
	} else if (d <= -2.8e-108) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 4.1e+84) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.2e+74)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -2.3e-6)
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	elseif (d <= -2.8e-108)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 4.1e+84)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.2e+74], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.3e-6], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, -2.8e-108], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 4.1e+84], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.2000000000000001e74

   1. Initial program 48.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -2.2000000000000001e74 < d < -2.3e-6

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if -2.3e-6 < d < -2.8e-108

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 85.5

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

   7. Applied rewrites 85.5%

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

   if -2.8e-108 < d < 4.1000000000000003e84

   1. Initial program 69.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 69.2

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 79.2

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

   7. Applied rewrites 79.2%

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

   if 4.1000000000000003e84 < d

   1. Initial program 38.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

Alternative 3: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= d -2.2e+74)
     (/ (fma (/ a d) c b) d)
     (if (<= d -2.3e-6)
       t_0
       (if (<= d -6.4e-105)
         (* (/ d (fma d d (* c c))) b)
         (if (<= d 4.1e+84) t_0 (/ (fma (/ c d) a b) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (d <= -2.2e+74) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= -2.3e-6) {
		tmp = t_0;
	} else if (d <= -6.4e-105) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 4.1e+84) {
		tmp = t_0;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (d <= -2.2e+74)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= -2.3e-6)
		tmp = t_0;
	elseif (d <= -6.4e-105)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 4.1e+84)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[d, -2.2e+74], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.3e-6], t$95$0, If[LessEqual[d, -6.4e-105], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 4.1e+84], t$95$0, N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.2000000000000001e74

   1. Initial program 48.4%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 94.8

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

   5. Applied rewrites 94.8%

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

   if -2.2000000000000001e74 < d < -2.3e-6 or -6.39999999999999962e-105 < d < 4.1000000000000003e84

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -2.3e-6 < d < -6.39999999999999962e-105

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

   if 4.1000000000000003e84 < d

   1. Initial program 38.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 86.0

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

   7. Applied rewrites 86.0%

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

Alternative 4: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)) (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -2.2e+74)
     t_1
     (if (<= d -2.3e-6)
       t_0
       (if (<= d -6.4e-105)
         (* (/ d (fma d d (* c c))) b)
         (if (<= d 4.1e+84) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -2.2e+74) {
		tmp = t_1;
	} else if (d <= -2.3e-6) {
		tmp = t_0;
	} else if (d <= -6.4e-105) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 4.1e+84) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -2.2e+74)
		tmp = t_1;
	elseif (d <= -2.3e-6)
		tmp = t_0;
	elseif (d <= -6.4e-105)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 4.1e+84)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.2e+74], t$95$1, If[LessEqual[d, -2.3e-6], t$95$0, If[LessEqual[d, -6.4e-105], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 4.1e+84], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.2000000000000001e74 or 4.1000000000000003e84 < d

   1. Initial program 43.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if -2.2000000000000001e74 < d < -2.3e-6 or -6.39999999999999962e-105 < d < 4.1000000000000003e84

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 79.2

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

   5. Applied rewrites 79.2%

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

   if -2.3e-6 < d < -6.39999999999999962e-105

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 84.8

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

   7. Applied rewrites 84.8%

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

Alternative 5: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ a d) c b) d)))
   (if (<= d -2.2e+74)
     t_0
     (if (<= d -7e-5)
       (/ a c)
       (if (<= d -4.8e-114)
         (* (/ d (fma d d (* c c))) b)
         (if (<= d 1.25e-26) (/ 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 <= -2.2e+74) {
		tmp = t_0;
	} else if (d <= -7e-5) {
		tmp = a / c;
	} else if (d <= -4.8e-114) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 1.25e-26) {
		tmp = 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 <= -2.2e+74)
		tmp = t_0;
	elseif (d <= -7e-5)
		tmp = Float64(a / c);
	elseif (d <= -4.8e-114)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 1.25e-26)
		tmp = Float64(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, -2.2e+74], t$95$0, If[LessEqual[d, -7e-5], N[(a / c), $MachinePrecision], If[LessEqual[d, -4.8e-114], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 1.25e-26], N[(a / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.2000000000000001e74 or 1.25000000000000005e-26 < d

   1. Initial program 48.5%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -2.2000000000000001e74 < d < -6.9999999999999994e-5 or -4.8000000000000002e-114 < d < 1.25000000000000005e-26

   1. Initial program 65.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -6.9999999999999994e-5 < d < -4.8000000000000002e-114

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

Alternative 6: 63.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.2e+74)
   (/ b d)
   (if (<= d -7e-5)
     (/ a c)
     (if (<= d -4.8e-114)
       (* (/ d (fma d d (* c c))) b)
       (if (<= d 1.25e-26) (/ a c) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.2e+74) {
		tmp = b / d;
	} else if (d <= -7e-5) {
		tmp = a / c;
	} else if (d <= -4.8e-114) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 1.25e-26) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.2e+74)
		tmp = Float64(b / d);
	elseif (d <= -7e-5)
		tmp = Float64(a / c);
	elseif (d <= -4.8e-114)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 1.25e-26)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.2e+74], N[(b / d), $MachinePrecision], If[LessEqual[d, -7e-5], N[(a / c), $MachinePrecision], If[LessEqual[d, -4.8e-114], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 1.25e-26], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.2000000000000001e74 or 1.25000000000000005e-26 < d

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -2.2000000000000001e74 < d < -6.9999999999999994e-5 or -4.8000000000000002e-114 < d < 1.25000000000000005e-26

   1. Initial program 65.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -6.9999999999999994e-5 < d < -4.8000000000000002e-114

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

Alternative 7: 63.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -7 \cdot 10^{-5}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot d\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.2e+74)
   (/ b d)
   (if (<= d -7e-5)
     (/ a c)
     (if (<= d -4.8e-114)
       (* (/ b (fma d d (* c c))) d)
       (if (<= d 1.25e-26) (/ a c) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.2e+74) {
		tmp = b / d;
	} else if (d <= -7e-5) {
		tmp = a / c;
	} else if (d <= -4.8e-114) {
		tmp = (b / fma(d, d, (c * c))) * d;
	} else if (d <= 1.25e-26) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.2e+74)
		tmp = Float64(b / d);
	elseif (d <= -7e-5)
		tmp = Float64(a / c);
	elseif (d <= -4.8e-114)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * d);
	elseif (d <= 1.25e-26)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.2e+74], N[(b / d), $MachinePrecision], If[LessEqual[d, -7e-5], N[(a / c), $MachinePrecision], If[LessEqual[d, -4.8e-114], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision], If[LessEqual[d, 1.25e-26], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.2000000000000001e74 or 1.25000000000000005e-26 < d

   1. Initial program 48.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -2.2000000000000001e74 < d < -6.9999999999999994e-5 or -4.8000000000000002e-114 < d < 1.25000000000000005e-26

   1. Initial program 65.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -6.9999999999999994e-5 < d < -4.8000000000000002e-114

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

Alternative 8: 62.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-108} \lor \neg \left(d \leq 1.25 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.8e-108) (not (<= d 1.25e-26))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.8e-108) || !(d <= 1.25e-26)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    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 <= (-2.8d-108)) .or. (.not. (d <= 1.25d-26))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.8e-108) || !(d <= 1.25e-26)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.8e-108) or not (d <= 1.25e-26):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.8e-108) || !(d <= 1.25e-26))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.8e-108) || ~((d <= 1.25e-26)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.8e-108], N[Not[LessEqual[d, 1.25e-26]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.8e-108 or 1.25000000000000005e-26 < d

   1. Initial program 55.6%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 65.5

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

   5. Applied rewrites 65.5%

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

   if -2.8e-108 < d < 1.25000000000000005e-26

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{-108} \lor \neg \left(d \leq 1.25 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, d)
    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 60.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 40.7

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

5. Applied rewrites 40.7%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c, d)
    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 2024309 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))