Complex division, real part

Percentage Accurate: 62.2% → 80.7%

Time: 12.4s

Alternatives: 12

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% 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: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+27}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2150000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.9e+27)
     (* (+ a (/ b (/ c d))) (/ -1.0 (hypot c d)))
     (if (<= c -9.5e-79)
       t_0
       (if (<= c 9.2e-154)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 1.85e-56)
           t_0
           (if (<= c 2150000000.0)
             (+ (/ b d) (* (/ c d) (/ a d)))
             (/ (+ a (* b (/ d c))) (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+27) {
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	} else if (c <= -9.5e-79) {
		tmp = t_0;
	} else if (c <= 9.2e-154) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.85e-56) {
		tmp = t_0;
	} else if (c <= 2150000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a + (b * (d / c))) / 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+27) {
		tmp = (a + (b / (c / d))) * (-1.0 / Math.hypot(c, d));
	} else if (c <= -9.5e-79) {
		tmp = t_0;
	} else if (c <= 9.2e-154) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.85e-56) {
		tmp = t_0;
	} else if (c <= 2150000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a + (b * (d / c))) / 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+27:
		tmp = (a + (b / (c / d))) * (-1.0 / math.hypot(c, d))
	elif c <= -9.5e-79:
		tmp = t_0
	elif c <= 9.2e-154:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 1.85e-56:
		tmp = t_0
	elif c <= 2150000000.0:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a + (b * (d / c))) / 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+27)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -9.5e-79)
		tmp = t_0;
	elseif (c <= 9.2e-154)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 1.85e-56)
		tmp = t_0;
	elseif (c <= 2150000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / 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+27)
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	elseif (c <= -9.5e-79)
		tmp = t_0;
	elseif (c <= 9.2e-154)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 1.85e-56)
		tmp = t_0;
	elseif (c <= 2150000000.0)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a + (b * (d / c))) / 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+27], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -9.5e-79], t$95$0, If[LessEqual[c, 9.2e-154], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.85e-56], t$95$0, If[LessEqual[c, 2150000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.90000000000000011e27

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. add-sqr-sqrt 39.5%

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

      3. times-frac 39.6%

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

      4. hypot-def 39.6%

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

      5. fma-def 39.6%

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

      6. hypot-def 59.7%

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

   4. Applied egg-rr 59.7%

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

   5. Taylor expanded in c around -inf 74.8%

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

      1. neg-mul-1 74.8%

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

      2. +-commutative 74.8%

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

      3. unsub-neg 74.8%

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

      4. mul-1-neg 74.8%

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

      5. associate-/l* 80.4%

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

      6. distribute-neg-frac 80.4%

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

   7. Simplified 80.4%

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

   if -1.90000000000000011e27 < c < -9.4999999999999997e-79 or 9.1999999999999999e-154 < c < 1.8500000000000001e-56

   1. Initial program 89.8%

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

   if -9.4999999999999997e-79 < c < 9.1999999999999999e-154

   1. Initial program 73.8%

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

   3. Taylor expanded in c around 0 86.4%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 80.9%

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

   5. Simplified 80.9%

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

      1. pow2 80.9%

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

      2. associate-*l/ 86.4%

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

      3. associate-/r* 93.2%

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

   7. Applied egg-rr 93.2%

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

   if 1.8500000000000001e-56 < c < 2.15e9

   1. Initial program 50.2%

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

   3. Taylor expanded in c around 0 76.5%

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

      1. associate-/l* 64.2%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. pow2 76.5%

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

      2. associate-*l/ 76.5%

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

      3. associate-/r* 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 2.15e9 < c

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. add-sqr-sqrt 42.3%

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

      3. times-frac 42.4%

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

      4. hypot-def 42.4%

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

      5. fma-def 42.4%

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

      6. hypot-def 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in c around inf 82.1%

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

      1. *-lft-identity 82.1%

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

      2. times-frac 84.0%

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

      3. /-rgt-identity 84.0%

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

   9. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{a + b \cdot \frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+27}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{-56}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2150000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+290)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (+ (/ b d) (* (/ c d) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+290) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (b / d) + ((c / d) * (a / 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))) <= 5e+290)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / 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], 5e+290], 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[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $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))) < 4.9999999999999998e290

   1. Initial program 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.9%

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

      4. hypot-def 78.9%

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

      5. fma-def 78.9%

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

      6. hypot-def 95.1%

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

   4. Applied egg-rr 95.1%

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

      1. associate-*l/ 95.3%

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

      2. *-un-lft-identity 95.3%

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

   6. Applied egg-rr 95.3%

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

   if 4.9999999999999998e290 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 6.3%

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

   3. Taylor expanded in c around 0 48.3%

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

      1. associate-/l* 50.2%

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

      2. associate-/r/ 50.2%

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

   5. Simplified 50.2%

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

      1. pow2 50.2%

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

      2. associate-*l/ 48.3%

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

      3. associate-/r* 51.5%

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

   7. Applied egg-rr 51.5%

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

      1. associate-/l/ 48.3%

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

      2. *-commutative 48.3%

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

      3. times-frac 55.9%

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

   9. Applied egg-rr 55.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+290}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -3.7 \cdot 10^{+78}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.72 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 880000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -3.7e+78)
     (+ (/ a c) (* d (* (/ 1.0 c) (/ b c))))
     (if (<= c -2.65e-79)
       t_0
       (if (<= c 1.6e-153)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 1.72e-56)
           t_0
           (if (<= c 880000000.0)
             (+ (/ b d) (* (/ c d) (/ a d)))
             (/ (+ a (* b (/ d c))) (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 <= -3.7e+78) {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	} else if (c <= -2.65e-79) {
		tmp = t_0;
	} else if (c <= 1.6e-153) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.72e-56) {
		tmp = t_0;
	} else if (c <= 880000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a + (b * (d / c))) / 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 <= -3.7e+78) {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	} else if (c <= -2.65e-79) {
		tmp = t_0;
	} else if (c <= 1.6e-153) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.72e-56) {
		tmp = t_0;
	} else if (c <= 880000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a + (b * (d / c))) / 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 <= -3.7e+78:
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)))
	elif c <= -2.65e-79:
		tmp = t_0
	elif c <= 1.6e-153:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 1.72e-56:
		tmp = t_0
	elif c <= 880000000.0:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a + (b * (d / c))) / 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 <= -3.7e+78)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(1.0 / c) * Float64(b / c))));
	elseif (c <= -2.65e-79)
		tmp = t_0;
	elseif (c <= 1.6e-153)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 1.72e-56)
		tmp = t_0;
	elseif (c <= 880000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / 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 <= -3.7e+78)
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	elseif (c <= -2.65e-79)
		tmp = t_0;
	elseif (c <= 1.6e-153)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 1.72e-56)
		tmp = t_0;
	elseif (c <= 880000000.0)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a + (b * (d / c))) / 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, -3.7e+78], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.65e-79], t$95$0, If[LessEqual[c, 1.6e-153], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.72e-56], t$95$0, If[LessEqual[c, 880000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.69999999999999985e78

   1. Initial program 34.2%

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

   3. Taylor expanded in c around inf 71.7%

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

      1. associate-/l* 75.4%

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

      2. associate-/r/ 78.7%

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

   5. Simplified 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. pow2 78.7%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -3.69999999999999985e78 < c < -2.6499999999999999e-79 or 1.6e-153 < c < 1.72000000000000009e-56

   1. Initial program 83.0%

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

   if -2.6499999999999999e-79 < c < 1.6e-153

   1. Initial program 73.8%

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

   3. Taylor expanded in c around 0 86.4%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 80.9%

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

   5. Simplified 80.9%

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

      1. pow2 80.9%

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

      2. associate-*l/ 86.4%

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

      3. associate-/r* 93.2%

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

   7. Applied egg-rr 93.2%

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

   if 1.72000000000000009e-56 < c < 8.8e8

   1. Initial program 50.2%

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

   3. Taylor expanded in c around 0 76.5%

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

      1. associate-/l* 64.2%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. pow2 76.5%

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

      2. associate-*l/ 76.5%

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

      3. associate-/r* 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 76.7%

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

   9. Applied egg-rr 76.7%

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

   if 8.8e8 < c

   1. Initial program 42.3%

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

      1. *-un-lft-identity 42.3%

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

      2. add-sqr-sqrt 42.3%

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

      3. times-frac 42.4%

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

      4. hypot-def 42.4%

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

      5. fma-def 42.4%

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

      6. hypot-def 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. associate-*l/ 63.3%

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

      2. *-un-lft-identity 63.3%

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

   6. Applied egg-rr 63.3%

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

   7. Taylor expanded in c around inf 82.1%

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

      1. *-lft-identity 82.1%

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

      2. times-frac 84.0%

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

      3. /-rgt-identity 84.0%

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

   9. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{a + b \cdot \frac{d}{c}}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+78}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-153}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.72 \cdot 10^{-56}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 880000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ t_1 := \frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ t_2 := \frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 480000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (* (/ c d) (/ a d))))
        (t_1 (+ (/ a c) (* b (/ (/ d c) c))))
        (t_2 (+ (/ a c) (/ (/ (* b d) c) c))))
   (if (<= c -3.2e+115)
     t_1
     (if (<= c -2.7e+47)
       t_0
       (if (<= c -6.5e-6)
         t_2
         (if (<= c -2.75e-36)
           t_0
           (if (<= c -3.5e-78) t_1 (if (<= c 480000000000.0) t_0 t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double t_1 = (a / c) + (b * ((d / c) / c));
	double t_2 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_1;
	} else if (c <= -2.7e+47) {
		tmp = t_0;
	} else if (c <= -6.5e-6) {
		tmp = t_2;
	} else if (c <= -2.75e-36) {
		tmp = t_0;
	} else if (c <= -3.5e-78) {
		tmp = t_1;
	} else if (c <= 480000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (b / d) + ((c / d) * (a / d))
    t_1 = (a / c) + (b * ((d / c) / c))
    t_2 = (a / c) + (((b * d) / c) / c)
    if (c <= (-3.2d+115)) then
        tmp = t_1
    else if (c <= (-2.7d+47)) then
        tmp = t_0
    else if (c <= (-6.5d-6)) then
        tmp = t_2
    else if (c <= (-2.75d-36)) then
        tmp = t_0
    else if (c <= (-3.5d-78)) then
        tmp = t_1
    else if (c <= 480000000000.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double t_1 = (a / c) + (b * ((d / c) / c));
	double t_2 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_1;
	} else if (c <= -2.7e+47) {
		tmp = t_0;
	} else if (c <= -6.5e-6) {
		tmp = t_2;
	} else if (c <= -2.75e-36) {
		tmp = t_0;
	} else if (c <= -3.5e-78) {
		tmp = t_1;
	} else if (c <= 480000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + ((c / d) * (a / d))
	t_1 = (a / c) + (b * ((d / c) / c))
	t_2 = (a / c) + (((b * d) / c) / c)
	tmp = 0
	if c <= -3.2e+115:
		tmp = t_1
	elif c <= -2.7e+47:
		tmp = t_0
	elif c <= -6.5e-6:
		tmp = t_2
	elif c <= -2.75e-36:
		tmp = t_0
	elif c <= -3.5e-78:
		tmp = t_1
	elif c <= 480000000000.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	t_1 = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)))
	t_2 = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c))
	tmp = 0.0
	if (c <= -3.2e+115)
		tmp = t_1;
	elseif (c <= -2.7e+47)
		tmp = t_0;
	elseif (c <= -6.5e-6)
		tmp = t_2;
	elseif (c <= -2.75e-36)
		tmp = t_0;
	elseif (c <= -3.5e-78)
		tmp = t_1;
	elseif (c <= 480000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + ((c / d) * (a / d));
	t_1 = (a / c) + (b * ((d / c) / c));
	t_2 = (a / c) + (((b * d) / c) / c);
	tmp = 0.0;
	if (c <= -3.2e+115)
		tmp = t_1;
	elseif (c <= -2.7e+47)
		tmp = t_0;
	elseif (c <= -6.5e-6)
		tmp = t_2;
	elseif (c <= -2.75e-36)
		tmp = t_0;
	elseif (c <= -3.5e-78)
		tmp = t_1;
	elseif (c <= 480000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.2e+115], t$95$1, If[LessEqual[c, -2.7e+47], t$95$0, If[LessEqual[c, -6.5e-6], t$95$2, If[LessEqual[c, -2.75e-36], t$95$0, If[LessEqual[c, -3.5e-78], t$95$1, If[LessEqual[c, 480000000000.0], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.2e115 or -2.74999999999999992e-36 < c < -3.4999999999999999e-78

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. associate-/l* 81.1%

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

      2. associate-/r/ 79.6%

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

   5. Simplified 79.6%

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

      1. expm1-log1p-u 77.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 78.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 79.3%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 79.3%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 81.1%

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

      3. *-commutative 81.1%

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

   9. Simplified 81.1%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 81.1%

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

       2. pow-flip 81.1%

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

       3. pow2 81.1%

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

       4. div-inv 81.1%

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

       5. associate-/r* 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -3.2e115 < c < -2.69999999999999996e47 or -6.4999999999999996e-6 < c < -2.74999999999999992e-36 or -3.4999999999999999e-78 < c < 4.8e11

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0 82.2%

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

      1. associate-/l* 81.2%

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

      2. associate-/r/ 77.8%

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

   5. Simplified 77.8%

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

      1. pow2 77.8%

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

      2. associate-*l/ 82.2%

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

      3. associate-/r* 86.9%

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

   7. Applied egg-rr 86.9%

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

      1. associate-/l/ 82.2%

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

      2. *-commutative 82.2%

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

      3. times-frac 86.1%

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

   9. Applied egg-rr 86.1%

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

   if -2.69999999999999996e47 < c < -6.4999999999999996e-6 or 4.8e11 < c

   1. Initial program 50.7%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   5. Simplified 78.3%

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

      1. pow2 78.3%

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

      2. associate-*l/ 78.2%

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

      3. associate-/r* 81.1%

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

   7. Applied egg-rr 81.1%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 480000000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ t_1 := \frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ t_2 := \frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 7000000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ (* c (/ a d)) d)))
        (t_1 (+ (/ a c) (* b (/ (/ d c) c))))
        (t_2 (+ (/ a c) (/ (/ (* b d) c) c))))
   (if (<= c -3.2e+115)
     t_1
     (if (<= c -2.5e+47)
       t_0
       (if (<= c -1.25e-7)
         t_2
         (if (<= c -2.4e-36)
           (+ (/ b d) (* (/ c d) (/ a d)))
           (if (<= c -3.5e-78) t_1 (if (<= c 7000000000.0) t_0 t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c * (a / d)) / d);
	double t_1 = (a / c) + (b * ((d / c) / c));
	double t_2 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_1;
	} else if (c <= -2.5e+47) {
		tmp = t_0;
	} else if (c <= -1.25e-7) {
		tmp = t_2;
	} else if (c <= -2.4e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3.5e-78) {
		tmp = t_1;
	} else if (c <= 7000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (b / d) + ((c * (a / d)) / d)
    t_1 = (a / c) + (b * ((d / c) / c))
    t_2 = (a / c) + (((b * d) / c) / c)
    if (c <= (-3.2d+115)) then
        tmp = t_1
    else if (c <= (-2.5d+47)) then
        tmp = t_0
    else if (c <= (-1.25d-7)) then
        tmp = t_2
    else if (c <= (-2.4d-36)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (c <= (-3.5d-78)) then
        tmp = t_1
    else if (c <= 7000000000.0d0) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c * (a / d)) / d);
	double t_1 = (a / c) + (b * ((d / c) / c));
	double t_2 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_1;
	} else if (c <= -2.5e+47) {
		tmp = t_0;
	} else if (c <= -1.25e-7) {
		tmp = t_2;
	} else if (c <= -2.4e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3.5e-78) {
		tmp = t_1;
	} else if (c <= 7000000000.0) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + ((c * (a / d)) / d)
	t_1 = (a / c) + (b * ((d / c) / c))
	t_2 = (a / c) + (((b * d) / c) / c)
	tmp = 0
	if c <= -3.2e+115:
		tmp = t_1
	elif c <= -2.5e+47:
		tmp = t_0
	elif c <= -1.25e-7:
		tmp = t_2
	elif c <= -2.4e-36:
		tmp = (b / d) + ((c / d) * (a / d))
	elif c <= -3.5e-78:
		tmp = t_1
	elif c <= 7000000000.0:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d))
	t_1 = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)))
	t_2 = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c))
	tmp = 0.0
	if (c <= -3.2e+115)
		tmp = t_1;
	elseif (c <= -2.5e+47)
		tmp = t_0;
	elseif (c <= -1.25e-7)
		tmp = t_2;
	elseif (c <= -2.4e-36)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (c <= -3.5e-78)
		tmp = t_1;
	elseif (c <= 7000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + ((c * (a / d)) / d);
	t_1 = (a / c) + (b * ((d / c) / c));
	t_2 = (a / c) + (((b * d) / c) / c);
	tmp = 0.0;
	if (c <= -3.2e+115)
		tmp = t_1;
	elseif (c <= -2.5e+47)
		tmp = t_0;
	elseif (c <= -1.25e-7)
		tmp = t_2;
	elseif (c <= -2.4e-36)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (c <= -3.5e-78)
		tmp = t_1;
	elseif (c <= 7000000000.0)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.2e+115], t$95$1, If[LessEqual[c, -2.5e+47], t$95$0, If[LessEqual[c, -1.25e-7], t$95$2, If[LessEqual[c, -2.4e-36], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.5e-78], t$95$1, If[LessEqual[c, 7000000000.0], t$95$0, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.2e115 or -2.4e-36 < c < -3.4999999999999999e-78

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. associate-/l* 81.1%

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

      2. associate-/r/ 79.6%

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

   5. Simplified 79.6%

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

      1. expm1-log1p-u 77.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 78.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 79.3%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 79.3%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 81.1%

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

      3. *-commutative 81.1%

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

   9. Simplified 81.1%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 81.1%

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

       2. pow-flip 81.1%

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

       3. pow2 81.1%

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

       4. div-inv 81.1%

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

       5. associate-/r* 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -3.2e115 < c < -2.50000000000000011e47 or -3.4999999999999999e-78 < c < 7e9

   1. Initial program 72.1%

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

   3. Taylor expanded in c around 0 81.4%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 76.9%

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

   5. Simplified 76.9%

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

      1. pow2 76.9%

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

      2. associate-*l/ 81.4%

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

      3. associate-/r* 86.3%

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

   7. Applied egg-rr 86.3%

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

      1. associate-/l* 87.1%

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

      2. associate-/r/ 85.6%

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

   9. Applied egg-rr 85.6%

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

   if -2.50000000000000011e47 < c < -1.24999999999999994e-7 or 7e9 < c

   1. Initial program 50.7%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   5. Simplified 78.3%

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

      1. pow2 78.3%

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

      2. associate-*l/ 78.2%

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

      3. associate-/r* 81.1%

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

   7. Applied egg-rr 81.1%

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

   if -1.24999999999999994e-7 < c < -2.4e-36

   1. Initial program 41.2%

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

   3. Taylor expanded in c around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Simplified 100.0%

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

      1. pow2 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/l/ 100.0%

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

      2. *-commutative 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.4 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 7000000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ t_1 := \frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -9.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 14500000000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (* b (/ (/ d c) c))))
        (t_1 (+ (/ a c) (/ (/ (* b d) c) c))))
   (if (<= c -3.2e+115)
     t_0
     (if (<= c -3e+47)
       (+ (/ b d) (/ (* c (/ a d)) d))
       (if (<= c -8.4e-7)
         t_1
         (if (<= c -9.2e-36)
           (+ (/ b d) (* (/ c d) (/ a d)))
           (if (<= c -3e-78)
             t_0
             (if (<= c 14500000000.0)
               (+ (/ b d) (/ (/ (* a c) d) d))
               t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (b * ((d / c) / c));
	double t_1 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_0;
	} else if (c <= -3e+47) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= -8.4e-7) {
		tmp = t_1;
	} else if (c <= -9.2e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3e-78) {
		tmp = t_0;
	} else if (c <= 14500000000.0) {
		tmp = (b / d) + (((a * c) / d) / d);
	} 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))
    t_1 = (a / c) + (((b * d) / c) / c)
    if (c <= (-3.2d+115)) then
        tmp = t_0
    else if (c <= (-3d+47)) then
        tmp = (b / d) + ((c * (a / d)) / d)
    else if (c <= (-8.4d-7)) then
        tmp = t_1
    else if (c <= (-9.2d-36)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (c <= (-3d-78)) then
        tmp = t_0
    else if (c <= 14500000000.0d0) then
        tmp = (b / d) + (((a * c) / d) / d)
    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));
	double t_1 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -3.2e+115) {
		tmp = t_0;
	} else if (c <= -3e+47) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= -8.4e-7) {
		tmp = t_1;
	} else if (c <= -9.2e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3e-78) {
		tmp = t_0;
	} else if (c <= 14500000000.0) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + (b * ((d / c) / c))
	t_1 = (a / c) + (((b * d) / c) / c)
	tmp = 0
	if c <= -3.2e+115:
		tmp = t_0
	elif c <= -3e+47:
		tmp = (b / d) + ((c * (a / d)) / d)
	elif c <= -8.4e-7:
		tmp = t_1
	elif c <= -9.2e-36:
		tmp = (b / d) + ((c / d) * (a / d))
	elif c <= -3e-78:
		tmp = t_0
	elif c <= 14500000000.0:
		tmp = (b / d) + (((a * c) / d) / d)
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)))
	t_1 = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c))
	tmp = 0.0
	if (c <= -3.2e+115)
		tmp = t_0;
	elseif (c <= -3e+47)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	elseif (c <= -8.4e-7)
		tmp = t_1;
	elseif (c <= -9.2e-36)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (c <= -3e-78)
		tmp = t_0;
	elseif (c <= 14500000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	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));
	t_1 = (a / c) + (((b * d) / c) / c);
	tmp = 0.0;
	if (c <= -3.2e+115)
		tmp = t_0;
	elseif (c <= -3e+47)
		tmp = (b / d) + ((c * (a / d)) / d);
	elseif (c <= -8.4e-7)
		tmp = t_1;
	elseif (c <= -9.2e-36)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (c <= -3e-78)
		tmp = t_0;
	elseif (c <= 14500000000.0)
		tmp = (b / d) + (((a * c) / d) / d);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.2e+115], t$95$0, If[LessEqual[c, -3e+47], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -8.4e-7], t$95$1, If[LessEqual[c, -9.2e-36], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3e-78], t$95$0, If[LessEqual[c, 14500000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.2e115 or -9.19999999999999986e-36 < c < -2.99999999999999988e-78

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. associate-/l* 81.1%

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

      2. associate-/r/ 79.6%

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

   5. Simplified 79.6%

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

      1. expm1-log1p-u 77.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 78.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 78.0%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 79.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 79.3%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 79.3%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 81.1%

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

      3. *-commutative 81.1%

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

   9. Simplified 81.1%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 81.1%

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

       2. pow-flip 81.1%

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

       3. pow2 81.1%

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

       4. div-inv 81.1%

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

       5. associate-/r* 82.7%

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

   11. Applied egg-rr 82.7%

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

   if -3.2e115 < c < -3.0000000000000001e47

   1. Initial program 43.0%

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

   3. Taylor expanded in c around 0 67.8%

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

      1. associate-/l* 67.8%

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

      2. associate-/r/ 68.0%

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

   5. Simplified 68.0%

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

      1. pow2 68.0%

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

      2. associate-*l/ 67.8%

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

      3. associate-/r* 68.1%

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

   7. Applied egg-rr 68.1%

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

      1. associate-/l* 76.1%

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

      2. associate-/r/ 76.1%

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

   9. Applied egg-rr 76.1%

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

   if -3.0000000000000001e47 < c < -8.4e-7 or 1.45e10 < c

   1. Initial program 50.7%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   5. Simplified 78.3%

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

      1. pow2 78.3%

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

      2. associate-*l/ 78.2%

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

      3. associate-/r* 81.1%

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

   7. Applied egg-rr 81.1%

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

   if -8.4e-7 < c < -9.19999999999999986e-36

   1. Initial program 41.2%

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

   3. Taylor expanded in c around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Simplified 100.0%

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

      1. pow2 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/l/ 100.0%

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

      2. *-commutative 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -2.99999999999999988e-78 < c < 1.45e10

   1. Initial program 75.2%

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

   3. Taylor expanded in c around 0 82.9%

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

      1. associate-/l* 81.7%

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

      2. associate-/r/ 77.9%

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

   5. Simplified 77.9%

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

      1. pow2 77.9%

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

      2. associate-*l/ 82.9%

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

      3. associate-/r* 88.3%

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

   7. Applied egg-rr 88.3%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -9.2 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 14500000000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -3.65 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 950000000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ a c) (* d (* (/ 1.0 c) (/ b c))))))
   (if (<= c -3.65e+78)
     t_1
     (if (<= c -2.6e-79)
       t_0
       (if (<= c 2.6e-154)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 2.8e-56)
           t_0
           (if (<= c 950000000000.0) (+ (/ b d) (* (/ c d) (/ a d))) 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 = (a / c) + (d * ((1.0 / c) * (b / c)));
	double tmp;
	if (c <= -3.65e+78) {
		tmp = t_1;
	} else if (c <= -2.6e-79) {
		tmp = t_0;
	} else if (c <= 2.6e-154) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.8e-56) {
		tmp = t_0;
	} else if (c <= 950000000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} 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 = (a / c) + (d * ((1.0d0 / c) * (b / c)))
    if (c <= (-3.65d+78)) then
        tmp = t_1
    else if (c <= (-2.6d-79)) then
        tmp = t_0
    else if (c <= 2.6d-154) then
        tmp = (b / d) + (((a * c) / d) / d)
    else if (c <= 2.8d-56) then
        tmp = t_0
    else if (c <= 950000000000.0d0) then
        tmp = (b / d) + ((c / d) * (a / d))
    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 = (a / c) + (d * ((1.0 / c) * (b / c)));
	double tmp;
	if (c <= -3.65e+78) {
		tmp = t_1;
	} else if (c <= -2.6e-79) {
		tmp = t_0;
	} else if (c <= 2.6e-154) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.8e-56) {
		tmp = t_0;
	} else if (c <= 950000000000.0) {
		tmp = (b / d) + ((c / d) * (a / d));
	} 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 = (a / c) + (d * ((1.0 / c) * (b / c)))
	tmp = 0
	if c <= -3.65e+78:
		tmp = t_1
	elif c <= -2.6e-79:
		tmp = t_0
	elif c <= 2.6e-154:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 2.8e-56:
		tmp = t_0
	elif c <= 950000000000.0:
		tmp = (b / d) + ((c / d) * (a / d))
	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(a / c) + Float64(d * Float64(Float64(1.0 / c) * Float64(b / c))))
	tmp = 0.0
	if (c <= -3.65e+78)
		tmp = t_1;
	elseif (c <= -2.6e-79)
		tmp = t_0;
	elseif (c <= 2.6e-154)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 2.8e-56)
		tmp = t_0;
	elseif (c <= 950000000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	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 = (a / c) + (d * ((1.0 / c) * (b / c)));
	tmp = 0.0;
	if (c <= -3.65e+78)
		tmp = t_1;
	elseif (c <= -2.6e-79)
		tmp = t_0;
	elseif (c <= 2.6e-154)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 2.8e-56)
		tmp = t_0;
	elseif (c <= 950000000000.0)
		tmp = (b / d) + ((c / d) * (a / d));
	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[(a / c), $MachinePrecision] + N[(d * N[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.65e+78], t$95$1, If[LessEqual[c, -2.6e-79], t$95$0, If[LessEqual[c, 2.6e-154], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.8e-56], t$95$0, If[LessEqual[c, 950000000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.65e78 or 9.5e11 < c

   1. Initial program 38.1%

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

   3. Taylor expanded in c around inf 74.9%

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

      1. associate-/l* 76.9%

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

      2. associate-/r/ 78.6%

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

   5. Simplified 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. pow2 78.6%

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

      3. times-frac 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -3.65e78 < c < -2.59999999999999994e-79 or 2.6e-154 < c < 2.79999999999999993e-56

   1. Initial program 83.0%

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

   if -2.59999999999999994e-79 < c < 2.6e-154

   1. Initial program 73.8%

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

   3. Taylor expanded in c around 0 86.4%

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

      1. associate-/l* 86.0%

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

      2. associate-/r/ 80.9%

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

   5. Simplified 80.9%

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

      1. pow2 80.9%

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

      2. associate-*l/ 86.4%

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

      3. associate-/r* 93.2%

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

   7. Applied egg-rr 93.2%

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

   if 2.79999999999999993e-56 < c < 9.5e11

   1. Initial program 50.2%

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

   3. Taylor expanded in c around 0 76.5%

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

      1. associate-/l* 64.2%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. pow2 76.5%

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

      2. associate-*l/ 76.5%

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

      3. associate-/r* 76.3%

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

   7. Applied egg-rr 76.3%

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

      1. associate-/l/ 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 76.7%

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

   9. Applied egg-rr 76.7%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{c}{d} \cdot \frac{a}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.65 \cdot 10^{+78}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 950000000000:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -0.00033:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 8200000000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (* d (* (/ 1.0 c) (/ b c))))))
   (if (<= c -0.00033)
     t_0
     (if (<= c -2.7e-36)
       (+ (/ b d) (* (/ c d) (/ a d)))
       (if (<= c -3.5e-78)
         (+ (/ a c) (* b (/ (/ d c) c)))
         (if (<= c 8200000000.0) (+ (/ b d) (/ (/ (* a c) d) d)) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (d * ((1.0 / c) * (b / c)));
	double tmp;
	if (c <= -0.00033) {
		tmp = t_0;
	} else if (c <= -2.7e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3.5e-78) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else if (c <= 8200000000.0) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else {
		tmp = t_0;
	}
	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) + (d * ((1.0d0 / c) * (b / c)))
    if (c <= (-0.00033d0)) then
        tmp = t_0
    else if (c <= (-2.7d-36)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (c <= (-3.5d-78)) then
        tmp = (a / c) + (b * ((d / c) / c))
    else if (c <= 8200000000.0d0) then
        tmp = (b / d) + (((a * c) / d) / d)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (d * ((1.0 / c) * (b / c)));
	double tmp;
	if (c <= -0.00033) {
		tmp = t_0;
	} else if (c <= -2.7e-36) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= -3.5e-78) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else if (c <= 8200000000.0) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + (d * ((1.0 / c) * (b / c)))
	tmp = 0
	if c <= -0.00033:
		tmp = t_0
	elif c <= -2.7e-36:
		tmp = (b / d) + ((c / d) * (a / d))
	elif c <= -3.5e-78:
		tmp = (a / c) + (b * ((d / c) / c))
	elif c <= 8200000000.0:
		tmp = (b / d) + (((a * c) / d) / d)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(d * Float64(Float64(1.0 / c) * Float64(b / c))))
	tmp = 0.0
	if (c <= -0.00033)
		tmp = t_0;
	elseif (c <= -2.7e-36)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (c <= -3.5e-78)
		tmp = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)));
	elseif (c <= 8200000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + (d * ((1.0 / c) * (b / c)));
	tmp = 0.0;
	if (c <= -0.00033)
		tmp = t_0;
	elseif (c <= -2.7e-36)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (c <= -3.5e-78)
		tmp = (a / c) + (b * ((d / c) / c));
	elseif (c <= 8200000000.0)
		tmp = (b / d) + (((a * c) / d) / d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -0.00033], t$95$0, If[LessEqual[c, -2.7e-36], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.5e-78], N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8200000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.3e-4 or 8.2e9 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf 72.6%

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

      1. associate-/l* 74.3%

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

      2. associate-/r/ 76.5%

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

   5. Simplified 76.5%

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

      1. *-un-lft-identity 76.5%

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

      2. pow2 76.5%

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

      3. times-frac 79.5%

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

   7. Applied egg-rr 79.5%

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

   if -3.3e-4 < c < -2.70000000000000007e-36

   1. Initial program 41.2%

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

   3. Taylor expanded in c around 0 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate-/r/ 100.0%

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

   5. Simplified 100.0%

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

      1. pow2 100.0%

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

      2. associate-*l/ 100.0%

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

      3. associate-/r* 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-/l/ 100.0%

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

      2. *-commutative 100.0%

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

      3. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -2.70000000000000007e-36 < c < -3.4999999999999999e-78

   1. Initial program 99.6%

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

   3. Taylor expanded in c around inf 77.0%

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

      1. associate-/l* 77.0%

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

      2. associate-/r/ 68.7%

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

   5. Simplified 68.7%

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

      1. expm1-log1p-u 68.3%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 68.3%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 68.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 68.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 75.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 75.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 75.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 75.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 75.9%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 75.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 77.1%

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

      3. *-commutative 77.1%

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

   9. Simplified 77.1%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 77.1%

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

       2. pow-flip 77.1%

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

       3. pow2 77.1%

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

       4. div-inv 77.1%

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

       5. associate-/r* 77.1%

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

   11. Applied egg-rr 77.1%

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

   if -3.4999999999999999e-78 < c < 8.2e9

   1. Initial program 75.2%

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

   3. Taylor expanded in c around 0 82.9%

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

      1. associate-/l* 81.7%

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

      2. associate-/r/ 77.9%

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

   5. Simplified 77.9%

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

      1. pow2 77.9%

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

      2. associate-*l/ 82.9%

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

      3. associate-/r* 88.3%

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

   7. Applied egg-rr 88.3%

      \[\leadsto \frac{b}{d} + \color{blue}{\frac{\frac{a \cdot c}{d}}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.00033:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 8200000000:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+86} \lor \neg \left(c \leq -3.2 \cdot 10^{+47} \lor \neg \left(c \leq -3 \cdot 10^{-78}\right) \land c \leq 3 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.35e+86)
         (not (or (<= c -3.2e+47) (and (not (<= c -3e-78)) (<= c 3e-153)))))
   (+ (/ a c) (* b (/ (/ d c) c)))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.35e+86) || !((c <= -3.2e+47) || (!(c <= -3e-78) && (c <= 3e-153)))) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else {
		tmp = b / 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 <= (-2.35d+86)) .or. (.not. (c <= (-3.2d+47)) .or. (.not. (c <= (-3d-78))) .and. (c <= 3d-153))) then
        tmp = (a / c) + (b * ((d / c) / c))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.35e+86) || !((c <= -3.2e+47) || (!(c <= -3e-78) && (c <= 3e-153)))) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.35e+86) or not ((c <= -3.2e+47) or (not (c <= -3e-78) and (c <= 3e-153))):
		tmp = (a / c) + (b * ((d / c) / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.35e+86) || !((c <= -3.2e+47) || (!(c <= -3e-78) && (c <= 3e-153))))
		tmp = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.35e+86) || ~(((c <= -3.2e+47) || (~((c <= -3e-78)) && (c <= 3e-153)))))
		tmp = (a / c) + (b * ((d / c) / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.35e+86], N[Not[Or[LessEqual[c, -3.2e+47], And[N[Not[LessEqual[c, -3e-78]], $MachinePrecision], LessEqual[c, 3e-153]]]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.3500000000000001e86 or -3.2e47 < c < -2.99999999999999988e-78 or 3e-153 < c

   1. Initial program 53.1%

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

   3. Taylor expanded in c around inf 69.8%

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

      1. associate-/l* 70.6%

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

      2. associate-/r/ 69.5%

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

   5. Simplified 69.5%

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

      1. expm1-log1p-u 66.4%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 65.2%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 65.2%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 65.2%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 66.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 66.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 66.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 66.3%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 66.3%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 67.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 70.7%

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

      3. *-commutative 70.7%

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

   9. Simplified 70.7%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 70.7%

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

       2. pow-flip 70.7%

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

       3. pow2 70.7%

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

       4. div-inv 70.7%

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

       5. associate-/r* 71.3%

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

   11. Applied egg-rr 71.3%

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

   if -2.3500000000000001e86 < c < -3.2e47 or -2.99999999999999988e-78 < c < 3e-153

   1. Initial program 70.5%

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

   3. Taylor expanded in c around 0 84.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+86} \lor \neg \left(c \leq -3.2 \cdot 10^{+47} \lor \neg \left(c \leq -3 \cdot 10^{-78}\right) \land c \leq 3 \cdot 10^{-153}\right):\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (/ (* b d) c) c))))
   (if (<= c -2.35e+86)
     t_0
     (if (<= c -3.2e+47)
       (/ b d)
       (if (<= c -6e-79)
         (+ (/ a c) (* b (/ (/ d c) c)))
         (if (<= c 3e-153) (/ b d) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (((b * d) / c) / c);
	double tmp;
	if (c <= -2.35e+86) {
		tmp = t_0;
	} else if (c <= -3.2e+47) {
		tmp = b / d;
	} else if (c <= -6e-79) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else if (c <= 3e-153) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	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)
    if (c <= (-2.35d+86)) then
        tmp = t_0
    else if (c <= (-3.2d+47)) then
        tmp = b / d
    else if (c <= (-6d-79)) then
        tmp = (a / c) + (b * ((d / c) / c))
    else if (c <= 3d-153) then
        tmp = b / d
    else
        tmp = t_0
    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);
	double tmp;
	if (c <= -2.35e+86) {
		tmp = t_0;
	} else if (c <= -3.2e+47) {
		tmp = b / d;
	} else if (c <= -6e-79) {
		tmp = (a / c) + (b * ((d / c) / c));
	} else if (c <= 3e-153) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + (((b * d) / c) / c)
	tmp = 0
	if c <= -2.35e+86:
		tmp = t_0
	elif c <= -3.2e+47:
		tmp = b / d
	elif c <= -6e-79:
		tmp = (a / c) + (b * ((d / c) / c))
	elif c <= 3e-153:
		tmp = b / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c))
	tmp = 0.0
	if (c <= -2.35e+86)
		tmp = t_0;
	elseif (c <= -3.2e+47)
		tmp = Float64(b / d);
	elseif (c <= -6e-79)
		tmp = Float64(Float64(a / c) + Float64(b * Float64(Float64(d / c) / c)));
	elseif (c <= 3e-153)
		tmp = Float64(b / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + (((b * d) / c) / c);
	tmp = 0.0;
	if (c <= -2.35e+86)
		tmp = t_0;
	elseif (c <= -3.2e+47)
		tmp = b / d;
	elseif (c <= -6e-79)
		tmp = (a / c) + (b * ((d / c) / c));
	elseif (c <= 3e-153)
		tmp = b / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.35e+86], t$95$0, If[LessEqual[c, -3.2e+47], N[(b / d), $MachinePrecision], If[LessEqual[c, -6e-79], N[(N[(a / c), $MachinePrecision] + N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3e-153], N[(b / d), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.3500000000000001e86 or 3e-153 < c

   1. Initial program 46.5%

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

   3. Taylor expanded in c around inf 71.0%

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

      1. associate-/l* 71.9%

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

      2. associate-/r/ 71.3%

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

   5. Simplified 71.3%

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

      1. pow2 71.3%

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

      2. associate-*l/ 71.0%

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

      3. associate-/r* 74.2%

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

   7. Applied egg-rr 74.2%

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

   if -2.3500000000000001e86 < c < -3.2e47 or -5.99999999999999999e-79 < c < 3e-153

   1. Initial program 70.5%

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

   3. Taylor expanded in c around 0 84.8%

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

   if -3.2e47 < c < -5.99999999999999999e-79

   1. Initial program 83.0%

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

   3. Taylor expanded in c around inf 64.6%

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

      1. associate-/l* 64.5%

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

      2. associate-/r/ 61.5%

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

   5. Simplified 61.5%

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

      1. expm1-log1p-u 57.1%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)\right)} \]

      2. expm1-udef 53.9%

         \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{b}{{c}^{2}} \cdot d\right)} - 1\right)} \]

      3. pow2 53.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\frac{b}{\color{blue}{c \cdot c}} \cdot d\right)} - 1\right) \]

      4. div-inv 53.9%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot \frac{1}{c \cdot c}\right)} \cdot d\right)} - 1\right) \]

      5. associate-*l* 56.8%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(\color{blue}{b \cdot \left(\frac{1}{c \cdot c} \cdot d\right)}\right)} - 1\right) \]

      6. pow2 56.8%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\frac{1}{\color{blue}{{c}^{2}}} \cdot d\right)\right)} - 1\right) \]

      7. pow-flip 56.8%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left(\color{blue}{{c}^{\left(-2\right)}} \cdot d\right)\right)} - 1\right) \]

      8. metadata-eval 56.8%

         \[\leadsto \frac{a}{c} + \left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{\color{blue}{-2}} \cdot d\right)\right)} - 1\right) \]

   7. Applied egg-rr 56.8%

      \[\leadsto \frac{a}{c} + \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)} - 1\right)} \]
   8. Step-by-step derivation

      1. expm1-def 60.0%

         \[\leadsto \frac{a}{c} + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left({c}^{-2} \cdot d\right)\right)\right)} \]

      2. expm1-log1p 64.6%

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

      3. *-commutative 64.6%

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

   9. Simplified 64.6%

      \[\leadsto \frac{a}{c} + \color{blue}{b \cdot \left(d \cdot {c}^{-2}\right)} \]
   10. Step-by-step derivation

       1. metadata-eval 64.6%

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

       2. pow-flip 64.6%

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

       3. pow2 64.6%

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

       4. div-inv 64.6%

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

       5. associate-/r* 64.6%

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

   11. Applied egg-rr 64.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-153}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+99} \lor \neg \left(c \leq -5.8 \cdot 10^{-36}\right) \land \left(c \leq -3.5 \cdot 10^{-78} \lor \neg \left(c \leq 3.5 \cdot 10^{+53}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6e+99)
         (and (not (<= c -5.8e-36)) (or (<= c -3.5e-78) (not (<= c 3.5e+53)))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6e+99) || (!(c <= -5.8e-36) && ((c <= -3.5e-78) || !(c <= 3.5e+53)))) {
		tmp = a / c;
	} else {
		tmp = b / 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 <= (-6d+99)) .or. (.not. (c <= (-5.8d-36))) .and. (c <= (-3.5d-78)) .or. (.not. (c <= 3.5d+53))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6e+99) || (!(c <= -5.8e-36) && ((c <= -3.5e-78) || !(c <= 3.5e+53)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6e+99) or (not (c <= -5.8e-36) and ((c <= -3.5e-78) or not (c <= 3.5e+53))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6e+99) || (!(c <= -5.8e-36) && ((c <= -3.5e-78) || !(c <= 3.5e+53))))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -6e+99) || (~((c <= -5.8e-36)) && ((c <= -3.5e-78) || ~((c <= 3.5e+53)))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6e+99], And[N[Not[LessEqual[c, -5.8e-36]], $MachinePrecision], Or[LessEqual[c, -3.5e-78], N[Not[LessEqual[c, 3.5e+53]], $MachinePrecision]]]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.00000000000000029e99 or -5.80000000000000026e-36 < c < -3.4999999999999999e-78 or 3.50000000000000019e53 < c

   1. Initial program 41.9%

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

   3. Taylor expanded in c around inf 79.4%

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

   if -6.00000000000000029e99 < c < -5.80000000000000026e-36 or -3.4999999999999999e-78 < c < 3.50000000000000019e53

   1. Initial program 72.3%

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

   3. Taylor expanded in c around 0 69.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+99} \lor \neg \left(c \leq -5.8 \cdot 10^{-36}\right) \land \left(c \leq -3.5 \cdot 10^{-78} \lor \neg \left(c \leq 3.5 \cdot 10^{+53}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.5% 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 59.6%

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

3. Taylor expanded in c around inf 46.0%

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

4. Final simplification 46.0%

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

Developer target: 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 2024021 
(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))))