Complex division, real part

Percentage Accurate: 61.8% → 83.1%

Time: 10.1s

Alternatives: 9

Speedup: 1.1×

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: 61.8% 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: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-156}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ b (* a (/ c d)))))
   (if (<= d -1.65e+49)
     (/ t_1 d)
     (if (<= d -2.2e-118)
       t_0
       (if (<= d 1.25e-156)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 7e+81) t_0 (/ t_1 (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -1.65e+49) {
		tmp = t_1 / d;
	} else if (d <= -2.2e-118) {
		tmp = t_0;
	} else if (d <= 1.25e-156) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 7e+81) {
		tmp = t_0;
	} else {
		tmp = t_1 / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -1.65e+49) {
		tmp = t_1 / d;
	} else if (d <= -2.2e-118) {
		tmp = t_0;
	} else if (d <= 1.25e-156) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 7e+81) {
		tmp = t_0;
	} else {
		tmp = t_1 / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = b + (a * (c / d))
	tmp = 0
	if d <= -1.65e+49:
		tmp = t_1 / d
	elif d <= -2.2e-118:
		tmp = t_0
	elif d <= 1.25e-156:
		tmp = (a + (b * (d / c))) / c
	elif d <= 7e+81:
		tmp = t_0
	else:
		tmp = t_1 / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -1.65e+49)
		tmp = Float64(t_1 / d);
	elseif (d <= -2.2e-118)
		tmp = t_0;
	elseif (d <= 1.25e-156)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 7e+81)
		tmp = t_0;
	else
		tmp = Float64(t_1 / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = b + (a * (c / d));
	tmp = 0.0;
	if (d <= -1.65e+49)
		tmp = t_1 / d;
	elseif (d <= -2.2e-118)
		tmp = t_0;
	elseif (d <= 1.25e-156)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 7e+81)
		tmp = t_0;
	else
		tmp = t_1 / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = 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$1 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.65e+49], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -2.2e-118], t$95$0, If[LessEqual[d, 1.25e-156], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7e+81], t$95$0, N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6499999999999999e49

   1. Initial program 48.2%

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

   3. Taylor expanded in d around inf 88.0%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -1.6499999999999999e49 < d < -2.19999999999999984e-118 or 1.25000000000000002e-156 < d < 7.0000000000000001e81

   1. Initial program 81.1%

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

   if -2.19999999999999984e-118 < d < 1.25000000000000002e-156

   1. Initial program 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. add-sqr-sqrt 65.9%

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

      3. times-frac 65.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 65.9%

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

      5. fma-define 65.8%

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

      6. hypot-define 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in c around inf 92.3%

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

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if 7.0000000000000001e81 < d

   1. Initial program 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. add-sqr-sqrt 40.3%

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

      3. times-frac 40.4%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 40.4%

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

      5. fma-define 40.4%

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

      6. hypot-define 59.1%

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

   4. Applied egg-rr 59.1%

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

      1. *-commutative 59.1%

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

      2. associate-*l/ 59.0%

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

      3. div-inv 59.1%

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

      4. fma-undefine 59.0%

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

      5. *-commutative 59.0%

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

      6. fma-define 59.1%

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

      7. *-commutative 59.1%

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

   6. Applied egg-rr 59.1%

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

   7. Taylor expanded in c around 0 79.5%

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

      1. associate-/l* 82.8%

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

   9. Simplified 82.8%

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

Alternative 2: 86.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+306)
   (/ (/ (fma c a (* b d)) (hypot c d)) (hypot c d))
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1e+306) {
		tmp = (fma(c, a, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+306)
		tmp = Float64(Float64(fma(c, a, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+306], N[(N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.00000000000000002e306

   1. Initial program 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.5%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 79.5%

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

      5. fma-define 79.5%

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

      6. hypot-define 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l/ 96.3%

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

      3. div-inv 96.4%

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

      4. fma-undefine 96.5%

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

      5. *-commutative 96.5%

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

      6. fma-define 96.4%

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

      7. *-commutative 96.4%

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

   6. Applied egg-rr 96.4%

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

   if 1.00000000000000002e306 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.4%

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

      1. *-un-lft-identity 12.4%

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

      2. add-sqr-sqrt 12.4%

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

      3. times-frac 12.4%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 12.4%

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

      5. fma-define 12.4%

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

      6. hypot-define 18.3%

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

   4. Applied egg-rr 18.3%

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

   5. Taylor expanded in c around inf 51.8%

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

      1. associate-/l* 57.7%

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

   7. Simplified 57.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+306}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.05 \cdot 10^{+48}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-122}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \left(c \cdot \frac{1}{d}\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -2.05e+48)
     (/ (+ b (* a (/ c d))) d)
     (if (<= d -8.5e-122)
       t_0
       (if (<= d 2.9e-162)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 1.7e+79) t_0 (/ (+ b (* a (* c (/ 1.0 d)))) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.05e+48) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -8.5e-122) {
		tmp = t_0;
	} else if (d <= 2.9e-162) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.7e+79) {
		tmp = t_0;
	} else {
		tmp = (b + (a * (c * (1.0 / d)))) / 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-2.05d+48)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-8.5d-122)) then
        tmp = t_0
    else if (d <= 2.9d-162) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 1.7d+79) then
        tmp = t_0
    else
        tmp = (b + (a * (c * (1.0d0 / d)))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.05e+48) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -8.5e-122) {
		tmp = t_0;
	} else if (d <= 2.9e-162) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.7e+79) {
		tmp = t_0;
	} else {
		tmp = (b + (a * (c * (1.0 / d)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -2.05e+48:
		tmp = (b + (a * (c / d))) / d
	elif d <= -8.5e-122:
		tmp = t_0
	elif d <= 2.9e-162:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1.7e+79:
		tmp = t_0
	else:
		tmp = (b + (a * (c * (1.0 / d)))) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.05e+48)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -8.5e-122)
		tmp = t_0;
	elseif (d <= 2.9e-162)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1.7e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c * Float64(1.0 / d)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -2.05e+48)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -8.5e-122)
		tmp = t_0;
	elseif (d <= 2.9e-162)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1.7e+79)
		tmp = t_0;
	else
		tmp = (b + (a * (c * (1.0 / d)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.05e+48], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -8.5e-122], t$95$0, If[LessEqual[d, 2.9e-162], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.7e+79], t$95$0, N[(N[(b + N[(a * N[(c * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.0500000000000001e48

   1. Initial program 48.2%

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

   3. Taylor expanded in d around inf 88.0%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -2.0500000000000001e48 < d < -8.50000000000000003e-122 or 2.9000000000000001e-162 < d < 1.70000000000000016e79

   1. Initial program 81.1%

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

   if -8.50000000000000003e-122 < d < 2.9000000000000001e-162

   1. Initial program 65.9%

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

      1. *-un-lft-identity 65.9%

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

      2. add-sqr-sqrt 65.9%

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

      3. times-frac 65.9%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 65.9%

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

      5. fma-define 65.8%

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

      6. hypot-define 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in c around inf 92.3%

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

      1. associate-/l* 93.2%

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

   7. Simplified 93.2%

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

   if 1.70000000000000016e79 < d

   1. Initial program 40.3%

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

   3. Taylor expanded in d around inf 79.3%

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

      1. associate-/l* 82.5%

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

   5. Simplified 82.5%

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

      1. div-inv 82.6%

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

   7. Applied egg-rr 82.6%

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

Alternative 4: 78.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+89} \lor \neg \left(c \leq 3900000000000\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \left(c \cdot \frac{1}{d}\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.5e+89) (not (<= c 3900000000000.0)))
   (/ (+ a (* d (/ b c))) c)
   (/ (+ b (* a (* c (/ 1.0 d)))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e+89) || !(c <= 3900000000000.0)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c * (1.0 / d)))) / 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 ((c <= (-7.5d+89)) .or. (.not. (c <= 3900000000000.0d0))) then
        tmp = (a + (d * (b / c))) / c
    else
        tmp = (b + (a * (c * (1.0d0 / d)))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e+89) || !(c <= 3900000000000.0)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c * (1.0 / d)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.5e+89) or not (c <= 3900000000000.0):
		tmp = (a + (d * (b / c))) / c
	else:
		tmp = (b + (a * (c * (1.0 / d)))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.5e+89) || !(c <= 3900000000000.0))
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c * Float64(1.0 / d)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -7.5e+89) || ~((c <= 3900000000000.0)))
		tmp = (a + (d * (b / c))) / c;
	else
		tmp = (b + (a * (c * (1.0 / d)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.5e+89], N[Not[LessEqual[c, 3900000000000.0]], $MachinePrecision]], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.49999999999999947e89 or 3.9e12 < c

   1. Initial program 52.5%

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

   3. Taylor expanded in c around inf 77.2%

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

      1. *-commutative 77.2%

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

   5. Simplified 77.2%

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

      1. associate-/l* 80.4%

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

   7. Applied egg-rr 80.4%

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

   if -7.49999999999999947e89 < c < 3.9e12

   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 d around inf 82.8%

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

      1. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. div-inv 82.9%

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

   7. Applied egg-rr 82.9%

      \[\leadsto \frac{b + a \cdot \color{blue}{\left(c \cdot \frac{1}{d}\right)}}{d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+89} \lor \neg \left(c \leq 3900000000000\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \left(c \cdot \frac{1}{d}\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+91} \lor \neg \left(c \leq 4000000000000\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4e+91) (not (<= c 4000000000000.0)))
   (/ (+ a (* d (/ b c))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e+91) || !(c <= 4000000000000.0)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / 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 ((c <= (-4d+91)) .or. (.not. (c <= 4000000000000.0d0))) then
        tmp = (a + (d * (b / c))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e+91) || !(c <= 4000000000000.0)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4e+91) or not (c <= 4000000000000.0):
		tmp = (a + (d * (b / c))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4e+91) || !(c <= 4000000000000.0))
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4e+91) || ~((c <= 4000000000000.0)))
		tmp = (a + (d * (b / c))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4e+91], N[Not[LessEqual[c, 4000000000000.0]], $MachinePrecision]], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.00000000000000032e91 or 4e12 < c

   1. Initial program 52.5%

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

   3. Taylor expanded in c around inf 77.2%

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

      1. *-commutative 77.2%

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

   5. Simplified 77.2%

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

      1. associate-/l* 80.4%

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

   7. Applied egg-rr 80.4%

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

   if -4.00000000000000032e91 < c < 4e12

   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 d around inf 82.8%

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

      1. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+91} \lor \neg \left(c \leq 4000000000000\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+49} \lor \neg \left(d \leq 1.6 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.05e+49) (not (<= d 1.6e+29)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e+49) || !(d <= 1.6e+29)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / 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 <= (-1.05d+49)) .or. (.not. (d <= 1.6d+29))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e+49) || !(d <= 1.6e+29)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.05e+49) or not (d <= 1.6e+29):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.05e+49) || !(d <= 1.6e+29))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.05e+49) || ~((d <= 1.6e+29)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.05e+49], N[Not[LessEqual[d, 1.6e+29]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.05000000000000005e49 or 1.59999999999999993e29 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0 73.7%

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

   if -1.05000000000000005e49 < d < 1.59999999999999993e29

   1. Initial program 72.1%

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

      1. *-un-lft-identity 72.1%

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

      2. add-sqr-sqrt 72.1%

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

      3. times-frac 72.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 72.2%

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

      5. fma-define 72.2%

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

      6. hypot-define 81.6%

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

   4. Applied egg-rr 81.6%

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

   5. Taylor expanded in c around inf 75.7%

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

      1. associate-/l* 76.2%

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

   7. Simplified 76.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+49} \lor \neg \left(d \leq 1.6 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+30} \lor \neg \left(d \leq 8.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.2e+30) (not (<= d 8.4e+24))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e+30) || !(d <= 8.4e+24)) {
		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 <= (-9.2d+30)) .or. (.not. (d <= 8.4d+24))) 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 <= -9.2e+30) || !(d <= 8.4e+24)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9.2e+30) or not (d <= 8.4e+24):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9.2e+30) || !(d <= 8.4e+24))
		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 <= -9.2e+30) || ~((d <= 8.4e+24)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9.2e+30], N[Not[LessEqual[d, 8.4e+24]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.2e30 or 8.4000000000000005e24 < d

   1. Initial program 50.0%

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

   3. Taylor expanded in c around 0 73.4%

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

   if -9.2e30 < d < 8.4000000000000005e24

   1. Initial program 71.7%

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

   3. Taylor expanded in c around inf 62.5%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+30} \lor \neg \left(d \leq 8.4 \cdot 10^{+24}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -2e+156) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2e+156) {
		tmp = a / 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 <= (-2d+156)) then
        tmp = a / 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 <= -2e+156) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2e+156:
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2e+156)
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2e+156)
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2e+156], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2e156

   1. Initial program 36.2%

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

      1. *-un-lft-identity 36.2%

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

      2. add-sqr-sqrt 36.2%

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

      3. times-frac 36.2%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{a \cdot c + b \cdot d}{\sqrt{c \cdot c + d \cdot d}}} \]

      4. hypot-define 36.2%

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

      5. fma-define 36.2%

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

      6. hypot-define 65.7%

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

   4. Applied egg-rr 65.7%

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

   5. Taylor expanded in c around inf 5.7%

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

   6. Taylor expanded in d around inf 6.2%

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

   7. Taylor expanded in a around inf 19.9%

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

   if -2e156 < d

   1. Initial program 65.9%

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

   3. Taylor expanded in c around inf 46.7%

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

Alternative 9: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 61.7%

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

3. Taylor expanded in c around inf 41.2%

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

Developer Target 1: 99.3% accurate, 0.1× 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 2024146 
(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))))