Complex division, real part

Percentage Accurate: 61.8% → 77.7%

Time: 11.2s

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: 77.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -7.6 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.7e+93)
     (/ (- (/ (- b) (/ c d)) a) (hypot c d))
     (if (<= c -7.6e-142)
       t_0
       (if (<= c 5.6e-160)
         (+ (/ b d) (/ a (* d (/ d c))))
         (if (<= c 7.2e-43)
           t_0
           (if (<= c 2.7e+147)
             (+ (/ b d) (* (/ c d) (/ a d)))
             (if (<= c 2e+166)
               (/ (+ a (/ b (/ c d))) (hypot c d))
               (* (/ c (hypot c d)) (/ a (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 tmp;
	if (c <= -1.7e+93) {
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	} else if (c <= -7.6e-142) {
		tmp = t_0;
	} else if (c <= 5.6e-160) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 7.2e-43) {
		tmp = t_0;
	} else if (c <= 2.7e+147) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= 2e+166) {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	} else {
		tmp = (c / hypot(c, d)) * (a / 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 tmp;
	if (c <= -1.7e+93) {
		tmp = ((-b / (c / d)) - a) / Math.hypot(c, d);
	} else if (c <= -7.6e-142) {
		tmp = t_0;
	} else if (c <= 5.6e-160) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 7.2e-43) {
		tmp = t_0;
	} else if (c <= 2.7e+147) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= 2e+166) {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	} else {
		tmp = (c / Math.hypot(c, d)) * (a / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.7e+93:
		tmp = ((-b / (c / d)) - a) / math.hypot(c, d)
	elif c <= -7.6e-142:
		tmp = t_0
	elif c <= 5.6e-160:
		tmp = (b / d) + (a / (d * (d / c)))
	elif c <= 7.2e-43:
		tmp = t_0
	elif c <= 2.7e+147:
		tmp = (b / d) + ((c / d) * (a / d))
	elif c <= 2e+166:
		tmp = (a + (b / (c / d))) / math.hypot(c, d)
	else:
		tmp = (c / math.hypot(c, d)) * (a / 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)))
	tmp = 0.0
	if (c <= -1.7e+93)
		tmp = Float64(Float64(Float64(Float64(-b) / Float64(c / d)) - a) / hypot(c, d));
	elseif (c <= -7.6e-142)
		tmp = t_0;
	elseif (c <= 5.6e-160)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	elseif (c <= 7.2e-43)
		tmp = t_0;
	elseif (c <= 2.7e+147)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (c <= 2e+166)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	else
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / 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));
	tmp = 0.0;
	if (c <= -1.7e+93)
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	elseif (c <= -7.6e-142)
		tmp = t_0;
	elseif (c <= 5.6e-160)
		tmp = (b / d) + (a / (d * (d / c)));
	elseif (c <= 7.2e-43)
		tmp = t_0;
	elseif (c <= 2.7e+147)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (c <= 2e+166)
		tmp = (a + (b / (c / d))) / hypot(c, d);
	else
		tmp = (c / hypot(c, d)) * (a / 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]}, If[LessEqual[c, -1.7e+93], N[(N[(N[((-b) / N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -7.6e-142], t$95$0, If[LessEqual[c, 5.6e-160], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e-43], t$95$0, If[LessEqual[c, 2.7e+147], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2e+166], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -1.7e93

   1. Initial program 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. add-sqr-sqrt 29.6%

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

      3. times-frac 29.6%

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

      4. hypot-def 29.6%

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

      5. fma-def 29.6%

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

      6. hypot-def 50.4%

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

   4. Applied egg-rr 50.4%

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

      1. associate-*l/ 50.5%

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

      2. *-un-lft-identity 50.5%

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

   6. Applied egg-rr 50.5%

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

   7. Taylor expanded in c around -inf 80.8%

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

      1. neg-mul-1 80.8%

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

      2. +-commutative 80.8%

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

      3. unsub-neg 80.8%

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

      4. mul-1-neg 80.8%

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

      5. associate-/l* 86.2%

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

      6. distribute-neg-frac 86.2%

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

   9. Simplified 86.2%

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

   if -1.7e93 < c < -7.59999999999999944e-142 or 5.60000000000000032e-160 < c < 7.1999999999999998e-43

   1. Initial program 85.5%

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

   if -7.59999999999999944e-142 < c < 5.60000000000000032e-160

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0 85.5%

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

      1. associate-/l* 81.5%

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

   5. Simplified 81.5%

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

      1. pow2 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 93.1%

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

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 93.1%

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

      2. clear-num 93.1%

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

      3. un-div-inv 93.1%

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

   9. Applied egg-rr 93.1%

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

       1. associate-/r/ 93.1%

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

   11. Simplified 93.1%

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

   if 7.1999999999999998e-43 < c < 2.69999999999999998e147

   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 0 64.0%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

      1. pow2 64.0%

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

      2. *-un-lft-identity 64.0%

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

      3. times-frac 67.6%

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

   7. Applied egg-rr 67.6%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 67.6%

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

      2. *-un-lft-identity 67.6%

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

      3. *-commutative 67.6%

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

      4. times-frac 67.7%

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

      5. clear-num 67.7%

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

   9. Applied egg-rr 67.7%

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

   if 2.69999999999999998e147 < c < 1.99999999999999988e166

   1. Initial program 26.1%

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

      1. *-un-lft-identity 26.1%

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

      2. add-sqr-sqrt 26.1%

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

      3. times-frac 26.1%

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

      4. hypot-def 26.1%

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

      5. fma-def 26.1%

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

      6. hypot-def 51.7%

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

   4. Applied egg-rr 51.7%

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

      1. associate-*l/ 51.7%

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

      2. *-un-lft-identity 51.7%

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

   6. Applied egg-rr 51.7%

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

   7. Taylor expanded in c around inf 75.8%

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

      1. associate-/l* 100.0%

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

   9. Simplified 100.0%

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

   if 1.99999999999999988e166 < c

   1. Initial program 27.6%

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

   3. Taylor expanded in a around inf 27.9%

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

      1. *-commutative 27.9%

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

   5. Simplified 27.9%

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

      1. add-sqr-sqrt 27.9%

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

      2. hypot-udef 27.9%

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

      3. hypot-udef 27.9%

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

      4. times-frac 93.5%

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

   7. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -7.6 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+166}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	}
	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))) <= Inf)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	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], Infinity], N[(N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $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))) < +inf.0

   1. Initial program 77.5%

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

      1. *-un-lft-identity 77.5%

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

      2. add-sqr-sqrt 77.5%

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

      3. times-frac 77.6%

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

      4. hypot-def 77.6%

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

      5. fma-def 77.6%

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

      6. hypot-def 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. associate-*l/ 95.2%

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

      2. *-un-lft-identity 95.2%

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

   6. Applied egg-rr 95.2%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf 1.7%

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

      1. *-commutative 1.7%

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

   5. Simplified 1.7%

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

      1. add-sqr-sqrt 1.7%

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

      2. hypot-udef 1.7%

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

      3. hypot-udef 1.7%

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

      4. times-frac 55.5%

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

   7. Applied egg-rr 55.5%

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

4. Final simplification 84.8%

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

Alternative 3: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.55 \cdot 10^{+94}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.52 \cdot 10^{-158}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.55e+94)
     (+ (/ a c) (/ b (* c (/ c d))))
     (if (<= c -1.25e-142)
       t_0
       (if (<= c 1.52e-158)
         (+ (/ b d) (/ a (* d (/ d c))))
         (if (<= c 8.6e+23) t_0 (/ (+ a (/ b (/ c d))) (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 tmp;
	if (c <= -2.55e+94) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= -1.25e-142) {
		tmp = t_0;
	} else if (c <= 1.52e-158) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 8.6e+23) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / 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 tmp;
	if (c <= -2.55e+94) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= -1.25e-142) {
		tmp = t_0;
	} else if (c <= 1.52e-158) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 8.6e+23) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.55e+94:
		tmp = (a / c) + (b / (c * (c / d)))
	elif c <= -1.25e-142:
		tmp = t_0
	elif c <= 1.52e-158:
		tmp = (b / d) + (a / (d * (d / c)))
	elif c <= 8.6e+23:
		tmp = t_0
	else:
		tmp = (a + (b / (c / d))) / 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)))
	tmp = 0.0
	if (c <= -2.55e+94)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (c <= -1.25e-142)
		tmp = t_0;
	elseif (c <= 1.52e-158)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	elseif (c <= 8.6e+23)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / 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));
	tmp = 0.0;
	if (c <= -2.55e+94)
		tmp = (a / c) + (b / (c * (c / d)));
	elseif (c <= -1.25e-142)
		tmp = t_0;
	elseif (c <= 1.52e-158)
		tmp = (b / d) + (a / (d * (d / c)));
	elseif (c <= 8.6e+23)
		tmp = t_0;
	else
		tmp = (a + (b / (c / d))) / 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]}, If[LessEqual[c, -2.55e+94], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.25e-142], t$95$0, If[LessEqual[c, 1.52e-158], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.6e+23], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.5500000000000002e94

   1. Initial program 29.6%

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

   3. Taylor expanded in c around inf 78.4%

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

      1. associate-/l* 80.6%

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

   5. Simplified 80.6%

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

      1. div-inv 80.6%

         \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{{c}^{2} \cdot \frac{1}{d}}} \]

      2. unpow2 80.6%

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

      3. associate-*l* 82.6%

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

      4. div-inv 82.6%

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

   7. Applied egg-rr 82.6%

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

   if -2.5500000000000002e94 < c < -1.2500000000000001e-142 or 1.52e-158 < c < 8.5999999999999997e23

   1. Initial program 82.2%

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

   if -1.2500000000000001e-142 < c < 1.52e-158

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0 85.5%

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

      1. associate-/l* 81.5%

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

   5. Simplified 81.5%

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

      1. pow2 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 93.1%

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

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 93.1%

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

      2. clear-num 93.1%

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

      3. un-div-inv 93.1%

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

   9. Applied egg-rr 93.1%

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

       1. associate-/r/ 93.1%

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

   11. Simplified 93.1%

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

   if 8.5999999999999997e23 < c

   1. Initial program 35.5%

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

      1. *-un-lft-identity 35.5%

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

      2. add-sqr-sqrt 35.5%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. fma-def 35.5%

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

      6. hypot-def 51.1%

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

   4. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.3%

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

      2. *-un-lft-identity 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in c around inf 72.5%

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

      1. associate-/l* 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.55 \cdot 10^{+94}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.52 \cdot 10^{-158}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.9e+93)
     (/ (- (/ (- b) (/ c d)) a) (hypot c d))
     (if (<= c -7.5e-140)
       t_0
       (if (<= c 3.5e-158)
         (+ (/ b d) (/ a (* d (/ d c))))
         (if (<= c 8.8e+23) t_0 (/ (+ a (/ b (/ c d))) (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 tmp;
	if (c <= -1.9e+93) {
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	} else if (c <= -7.5e-140) {
		tmp = t_0;
	} else if (c <= 3.5e-158) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 8.8e+23) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / 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 tmp;
	if (c <= -1.9e+93) {
		tmp = ((-b / (c / d)) - a) / Math.hypot(c, d);
	} else if (c <= -7.5e-140) {
		tmp = t_0;
	} else if (c <= 3.5e-158) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 8.8e+23) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.9e+93:
		tmp = ((-b / (c / d)) - a) / math.hypot(c, d)
	elif c <= -7.5e-140:
		tmp = t_0
	elif c <= 3.5e-158:
		tmp = (b / d) + (a / (d * (d / c)))
	elif c <= 8.8e+23:
		tmp = t_0
	else:
		tmp = (a + (b / (c / d))) / 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)))
	tmp = 0.0
	if (c <= -1.9e+93)
		tmp = Float64(Float64(Float64(Float64(-b) / Float64(c / d)) - a) / hypot(c, d));
	elseif (c <= -7.5e-140)
		tmp = t_0;
	elseif (c <= 3.5e-158)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	elseif (c <= 8.8e+23)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / 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));
	tmp = 0.0;
	if (c <= -1.9e+93)
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	elseif (c <= -7.5e-140)
		tmp = t_0;
	elseif (c <= 3.5e-158)
		tmp = (b / d) + (a / (d * (d / c)));
	elseif (c <= 8.8e+23)
		tmp = t_0;
	else
		tmp = (a + (b / (c / d))) / 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]}, If[LessEqual[c, -1.9e+93], N[(N[(N[((-b) / N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -7.5e-140], t$95$0, If[LessEqual[c, 3.5e-158], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.8e+23], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.8999999999999999e93

   1. Initial program 29.6%

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

      1. *-un-lft-identity 29.6%

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

      2. add-sqr-sqrt 29.6%

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

      3. times-frac 29.6%

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

      4. hypot-def 29.6%

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

      5. fma-def 29.6%

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

      6. hypot-def 50.4%

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

   4. Applied egg-rr 50.4%

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

      1. associate-*l/ 50.5%

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

      2. *-un-lft-identity 50.5%

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

   6. Applied egg-rr 50.5%

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

   7. Taylor expanded in c around -inf 80.8%

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

      1. neg-mul-1 80.8%

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

      2. +-commutative 80.8%

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

      3. unsub-neg 80.8%

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

      4. mul-1-neg 80.8%

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

      5. associate-/l* 86.2%

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

      6. distribute-neg-frac 86.2%

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

   9. Simplified 86.2%

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

   if -1.8999999999999999e93 < c < -7.4999999999999998e-140 or 3.50000000000000012e-158 < c < 8.80000000000000034e23

   1. Initial program 82.2%

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

   if -7.4999999999999998e-140 < c < 3.50000000000000012e-158

   1. Initial program 66.7%

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

   3. Taylor expanded in c around 0 85.5%

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

      1. associate-/l* 81.5%

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

   5. Simplified 81.5%

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

      1. pow2 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. times-frac 93.1%

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

   7. Applied egg-rr 93.1%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 93.1%

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

      2. clear-num 93.1%

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

      3. un-div-inv 93.1%

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

   9. Applied egg-rr 93.1%

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

       1. associate-/r/ 93.1%

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

   11. Simplified 93.1%

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

   if 8.80000000000000034e23 < c

   1. Initial program 35.5%

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

      1. *-un-lft-identity 35.5%

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

      2. add-sqr-sqrt 35.5%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. fma-def 35.5%

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

      6. hypot-def 51.1%

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

   4. Applied egg-rr 51.1%

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

      1. associate-*l/ 51.3%

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

      2. *-un-lft-identity 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in c around inf 72.5%

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

      1. associate-/l* 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-158}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{+23}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -5.5e+64)
     t_1
     (if (<= d -2.5e-128)
       t_0
       (if (<= d 3.9e-106)
         (+ (/ a c) (/ b (* c (/ c d))))
         (if (<= d 4.2e+110) t_0 t_1))))))

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 / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -5.5e+64) {
		tmp = t_1;
	} else if (d <= -2.5e-128) {
		tmp = t_0;
	} else if (d <= 3.9e-106) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (d <= 4.2e+110) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + ((c / d) * (a / d))
    if (d <= (-5.5d+64)) then
        tmp = t_1
    else if (d <= (-2.5d-128)) then
        tmp = t_0
    else if (d <= 3.9d-106) then
        tmp = (a / c) + (b / (c * (c / d)))
    else if (d <= 4.2d+110) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -5.5e+64) {
		tmp = t_1;
	} else if (d <= -2.5e-128) {
		tmp = t_0;
	} else if (d <= 3.9e-106) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (d <= 4.2e+110) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -5.5e+64:
		tmp = t_1
	elif d <= -2.5e-128:
		tmp = t_0
	elif d <= 3.9e-106:
		tmp = (a / c) + (b / (c * (c / d)))
	elif d <= 4.2e+110:
		tmp = t_0
	else:
		tmp = t_1
	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(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -5.5e+64)
		tmp = t_1;
	elseif (d <= -2.5e-128)
		tmp = t_0;
	elseif (d <= 3.9e-106)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (d <= 4.2e+110)
		tmp = t_0;
	else
		tmp = t_1;
	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 / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -5.5e+64)
		tmp = t_1;
	elseif (d <= -2.5e-128)
		tmp = t_0;
	elseif (d <= 3.9e-106)
		tmp = (a / c) + (b / (c * (c / d)));
	elseif (d <= 4.2e+110)
		tmp = t_0;
	else
		tmp = t_1;
	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[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.5e+64], t$95$1, If[LessEqual[d, -2.5e-128], t$95$0, If[LessEqual[d, 3.9e-106], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+110], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.4999999999999996e64 or 4.2000000000000003e110 < d

   1. Initial program 28.5%

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

   3. Taylor expanded in c around 0 68.9%

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

      1. associate-/l* 71.7%

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

   5. Simplified 71.7%

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

      1. pow2 71.7%

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

      2. *-un-lft-identity 71.7%

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

      3. times-frac 74.5%

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

   7. Applied egg-rr 74.5%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 74.5%

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

      2. *-un-lft-identity 74.5%

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

      3. *-commutative 74.5%

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

      4. times-frac 78.6%

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

      5. clear-num 78.6%

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

   9. Applied egg-rr 78.6%

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

   if -5.4999999999999996e64 < d < -2.5000000000000001e-128 or 3.9000000000000001e-106 < d < 4.2000000000000003e110

   1. Initial program 82.2%

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

   if -2.5000000000000001e-128 < d < 3.9000000000000001e-106

   1. Initial program 66.5%

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

   3. Taylor expanded in c around inf 84.1%

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

      1. associate-/l* 84.3%

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

   5. Simplified 84.3%

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

      1. div-inv 84.3%

         \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{{c}^{2} \cdot \frac{1}{d}}} \]

      2. unpow2 84.3%

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

      3. associate-*l* 87.7%

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

      4. div-inv 87.7%

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

   7. Applied egg-rr 87.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+110}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-19} \lor \neg \left(d \leq 1.05 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.65e-19) (not (<= d 1.05e-9)))
   (/ b d)
   (+ (/ a c) (/ b (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.65e-19) || !(d <= 1.05e-9)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (b / (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 ((d <= (-1.65d-19)) .or. (.not. (d <= 1.05d-9))) then
        tmp = b / d
    else
        tmp = (a / c) + (b / (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 ((d <= -1.65e-19) || !(d <= 1.05e-9)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.65e-19) or not (d <= 1.05e-9):
		tmp = b / d
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.65e-19) || !(d <= 1.05e-9))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.65e-19) || ~((d <= 1.05e-9)))
		tmp = b / d;
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.65e-19], N[Not[LessEqual[d, 1.05e-9]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6499999999999999e-19 or 1.0500000000000001e-9 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in c around 0 65.3%

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

   if -1.6499999999999999e-19 < d < 1.0500000000000001e-9

   1. Initial program 72.2%

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

   3. Taylor expanded in c around inf 76.2%

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

      1. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. div-inv 76.4%

         \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{{c}^{2} \cdot \frac{1}{d}}} \]

      2. unpow2 76.4%

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

      3. associate-*l* 78.7%

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

      4. div-inv 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-19} \lor \neg \left(d \leq 1.05 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-19} \lor \neg \left(d \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.5e-19) (not (<= d 6e-10)))
   (+ (/ b d) (* (/ c d) (/ a d)))
   (+ (/ a c) (/ b (* c (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.5e-19) || !(d <= 6e-10)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (b / (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 ((d <= (-3.5d-19)) .or. (.not. (d <= 6d-10))) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + (b / (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 ((d <= -3.5e-19) || !(d <= 6e-10)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.5e-19) or not (d <= 6e-10):
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.5e-19) || !(d <= 6e-10))
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.5e-19) || ~((d <= 6e-10)))
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.5e-19], N[Not[LessEqual[d, 6e-10]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.50000000000000015e-19 or 6e-10 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in c around 0 69.1%

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

      1. associate-/l* 69.0%

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

   5. Simplified 69.0%

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

      1. pow2 69.0%

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

      2. *-un-lft-identity 69.0%

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

      3. times-frac 71.1%

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

   7. Applied egg-rr 71.1%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{d}{1} \cdot \frac{d}{c}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 71.1%

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

      2. *-un-lft-identity 71.1%

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

      3. *-commutative 71.1%

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

      4. times-frac 74.8%

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

      5. clear-num 75.3%

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

   9. Applied egg-rr 75.3%

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

   if -3.50000000000000015e-19 < d < 6e-10

   1. Initial program 72.2%

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

   3. Taylor expanded in c around inf 76.2%

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

      1. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. div-inv 76.4%

         \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{{c}^{2} \cdot \frac{1}{d}}} \]

      2. unpow2 76.4%

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

      3. associate-*l* 78.7%

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

      4. div-inv 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-19} \lor \neg \left(d \leq 6 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-30} \lor \neg \left(d \leq 2.1 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.6e-30) (not (<= d 2.1e-75))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e-30) || !(d <= 2.1e-75)) {
		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 <= (-1.6d-30)) .or. (.not. (d <= 2.1d-75))) 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 <= -1.6e-30) || !(d <= 2.1e-75)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.6e-30) or not (d <= 2.1e-75):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.6e-30) || !(d <= 2.1e-75))
		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 <= -1.6e-30) || ~((d <= 2.1e-75)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.6e-30], N[Not[LessEqual[d, 2.1e-75]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6e-30 or 2.1000000000000001e-75 < d

   1. Initial program 47.0%

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

   3. Taylor expanded in c around 0 63.2%

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

   if -1.6e-30 < d < 2.1000000000000001e-75

   1. Initial program 69.6%

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

   3. Taylor expanded in c around inf 67.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{-30} \lor \neg \left(d \leq 2.1 \cdot 10^{-75}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% 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 57.2%

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

3. Taylor expanded in c around inf 44.6%

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

4. Final simplification 44.6%

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

Developer target: 99.4% 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 2024020 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (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))))