Complex division, real part

Percentage Accurate: 61.7% → 80.7%

Time: 9.8s

Alternatives: 10

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+66}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(d, b, c \cdot a\right) \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.3e+66)
   (/ (+ a (/ 1.0 (/ (/ c b) d))) c)
   (if (<= c -4.8e-41)
     (* (fma d b (* c a)) (pow (hypot d c) -2.0))
     (if (<= c 4.9e+41)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c 7.6e+92)
         (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))
         (if (<= c 1.5e+196)
           (/ (/ a (hypot d c)) (/ (hypot d c) c))
           (/ (+ a (* b (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.3e+66) {
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	} else if (c <= -4.8e-41) {
		tmp = fma(d, b, (c * a)) * pow(hypot(d, c), -2.0);
	} else if (c <= 4.9e+41) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 7.6e+92) {
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.5e+196) {
		tmp = (a / hypot(d, c)) / (hypot(d, c) / c);
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.3e+66)
		tmp = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c);
	elseif (c <= -4.8e-41)
		tmp = Float64(fma(d, b, Float64(c * a)) * (hypot(d, c) ^ -2.0));
	elseif (c <= 4.9e+41)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 7.6e+92)
		tmp = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.5e+196)
		tmp = Float64(Float64(a / hypot(d, c)) / Float64(hypot(d, c) / c));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.3e+66], N[(N[(a + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -4.8e-41], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.9e+41], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.6e+92], N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.5e+196], N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -1.30000000000000006e66

   1. Initial program 41.5%

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

   3. Taylor expanded in c around inf 92.2%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-/r* 92.6%

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

   7. Simplified 92.6%

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

   if -1.30000000000000006e66 < c < -4.80000000000000044e-41

   1. Initial program 95.2%

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

      1. fma-define 95.2%

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

      2. fma-define 95.3%

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

   3. Simplified 95.3%

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

      1. fma-define 95.3%

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

      2. +-commutative 95.3%

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

   6. Applied egg-rr 95.3%

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

      1. fma-define 95.2%

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

      2. div-inv 95.1%

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

      3. fma-define 95.1%

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

      4. +-commutative 95.1%

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

      5. add-sqr-sqrt 95.2%

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

      6. hypot-undefine 95.2%

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

      7. hypot-undefine 95.2%

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

      8. unpow2 95.2%

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

      9. pow-flip 95.3%

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

      10. metadata-eval 95.3%

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

   8. Applied egg-rr 95.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, d, a \cdot c\right) \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}} \]
   9. Step-by-step derivation

      1. fma-undefine 95.3%

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

      2. *-commutative 95.3%

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

      3. fma-undefine 95.3%

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

   10. Simplified 95.3%

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

   if -4.80000000000000044e-41 < c < 4.8999999999999999e41

   1. Initial program 68.0%

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

   3. Taylor expanded in d around inf 86.5%

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

      1. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

   if 4.8999999999999999e41 < c < 7.6000000000000001e92

   1. Initial program 84.3%

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

   if 7.6000000000000001e92 < c < 1.4999999999999999e196

   1. Initial program 40.9%

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

   3. Taylor expanded in a around inf 41.0%

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

      1. *-commutative 41.0%

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

   5. Simplified 41.0%

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

      1. +-commutative 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. hypot-undefine 41.0%

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

      4. hypot-undefine 41.0%

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

      5. times-frac 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. *-commutative 91.0%

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

      2. clear-num 91.0%

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

      3. un-div-inv 91.1%

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

   9. Applied egg-rr 91.1%

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

   if 1.4999999999999999e196 < c

   1. Initial program 33.5%

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

   3. Taylor expanded in c around inf 94.9%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a + b \cdot \frac{d}{c}}{c}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+66}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(d, b, c \cdot a\right) \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+196}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot a + b \cdot d\\ \mathbf{if}\;c \leq -8 \cdot 10^{+65}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.56 \cdot 10^{+92}:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.26 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c a) (* b d))))
   (if (<= c -8e+65)
     (/ (+ a (/ 1.0 (/ (/ c b) d))) c)
     (if (<= c -1.5e-40)
       (/ t_0 (fma c c (* d d)))
       (if (<= c 3.1e+41)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 1.56e+92)
           (/ t_0 (+ (* c c) (* d d)))
           (if (<= c 1.26e+196)
             (* (/ a (hypot d c)) (/ c (hypot d c)))
             (/ (+ a (* b (/ d c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * a) + (b * d);
	double tmp;
	if (c <= -8e+65) {
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	} else if (c <= -1.5e-40) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (c <= 3.1e+41) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.56e+92) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1.26e+196) {
		tmp = (a / hypot(d, c)) * (c / hypot(d, c));
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * a) + Float64(b * d))
	tmp = 0.0
	if (c <= -8e+65)
		tmp = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c);
	elseif (c <= -1.5e-40)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (c <= 3.1e+41)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 1.56e+92)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.26e+196)
		tmp = Float64(Float64(a / hypot(d, c)) * Float64(c / hypot(d, c)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8e+65], N[(N[(a + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.5e-40], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.1e+41], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.56e+92], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.26e+196], N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -7.9999999999999999e65

   1. Initial program 41.5%

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

   3. Taylor expanded in c around inf 92.2%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-/r* 92.6%

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

   7. Simplified 92.6%

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

   if -7.9999999999999999e65 < c < -1.5000000000000001e-40

   1. Initial program 95.2%

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

      1. fma-define 95.2%

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

      2. fma-define 95.3%

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

   3. Simplified 95.3%

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

      1. fma-define 95.3%

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

      2. +-commutative 95.3%

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

   6. Applied egg-rr 95.3%

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

   if -1.5000000000000001e-40 < c < 3.1e41

   1. Initial program 68.0%

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

   3. Taylor expanded in d around inf 86.5%

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

      1. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

   if 3.1e41 < c < 1.55999999999999987e92

   1. Initial program 84.3%

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

   if 1.55999999999999987e92 < c < 1.26e196

   1. Initial program 40.9%

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

   3. Taylor expanded in a around inf 41.0%

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

      1. *-commutative 41.0%

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

   5. Simplified 41.0%

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

      1. +-commutative 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. hypot-undefine 41.0%

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

      4. hypot-undefine 41.0%

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

      5. times-frac 91.0%

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

   7. Applied egg-rr 91.0%

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

   if 1.26e196 < c

   1. Initial program 33.5%

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

   3. Taylor expanded in c around inf 94.9%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a + b \cdot \frac{d}{c}}{c}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+65}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.56 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.26 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{c}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot a + b \cdot d\\ \mathbf{if}\;c \leq -4.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-43}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c a) (* b d))))
   (if (<= c -4.6e+66)
     (/ (+ a (/ 1.0 (/ (/ c b) d))) c)
     (if (<= c -9e-43)
       (/ t_0 (fma c c (* d d)))
       (if (<= c 3.1e+41)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 7.2e+92)
           (/ t_0 (+ (* c c) (* d d)))
           (if (<= c 9.5e+195)
             (/ (/ a (hypot d c)) (/ (hypot d c) c))
             (/ (+ a (* b (/ d c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * a) + (b * d);
	double tmp;
	if (c <= -4.6e+66) {
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	} else if (c <= -9e-43) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (c <= 3.1e+41) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 7.2e+92) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 9.5e+195) {
		tmp = (a / hypot(d, c)) / (hypot(d, c) / c);
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * a) + Float64(b * d))
	tmp = 0.0
	if (c <= -4.6e+66)
		tmp = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c);
	elseif (c <= -9e-43)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (c <= 3.1e+41)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 7.2e+92)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 9.5e+195)
		tmp = Float64(Float64(a / hypot(d, c)) / Float64(hypot(d, c) / c));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.6e+66], N[(N[(a + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -9e-43], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.1e+41], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.2e+92], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.5e+195], N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -4.6e66

   1. Initial program 41.5%

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

   3. Taylor expanded in c around inf 92.2%

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

      1. clear-num 92.2%

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

      2. inv-pow 92.2%

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

   5. Applied egg-rr 92.2%

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

      1. unpow-1 92.2%

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

      2. associate-/r* 92.6%

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

   7. Simplified 92.6%

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

   if -4.6e66 < c < -9.0000000000000005e-43

   1. Initial program 95.2%

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

      1. fma-define 95.2%

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

      2. fma-define 95.3%

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

   3. Simplified 95.3%

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

      1. fma-define 95.3%

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

      2. +-commutative 95.3%

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

   6. Applied egg-rr 95.3%

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

   if -9.0000000000000005e-43 < c < 3.1e41

   1. Initial program 68.0%

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

   3. Taylor expanded in d around inf 86.5%

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

      1. associate-/l* 87.1%

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

   5. Simplified 87.1%

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

   if 3.1e41 < c < 7.2e92

   1. Initial program 84.3%

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

   if 7.2e92 < c < 9.5000000000000004e195

   1. Initial program 40.9%

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

   3. Taylor expanded in a around inf 41.0%

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

      1. *-commutative 41.0%

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

   5. Simplified 41.0%

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

      1. +-commutative 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. hypot-undefine 41.0%

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

      4. hypot-undefine 41.0%

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

      5. times-frac 91.0%

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

   7. Applied egg-rr 91.0%

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

      1. *-commutative 91.0%

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

      2. clear-num 91.0%

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

      3. un-div-inv 91.1%

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

   9. Applied egg-rr 91.1%

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

   if 9.5000000000000004e195 < c

   1. Initial program 33.5%

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

   3. Taylor expanded in c around inf 94.9%

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

      1. associate-/l* 100.0%

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{a + b \cdot \frac{d}{c}}{c}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+69}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (/ 1.0 (/ (/ c b) d))) c)))
   (if (<= c -1.3e+65)
     t_0
     (if (<= c -5.2e-41)
       (/ (+ (* c a) (* b d)) (fma c c (* d d)))
       (if (<= c 8e+69) (/ (+ b (* a (/ c d))) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (1.0 / ((c / b) / d))) / c;
	double tmp;
	if (c <= -1.3e+65) {
		tmp = t_0;
	} else if (c <= -5.2e-41) {
		tmp = ((c * a) + (b * d)) / fma(c, c, (d * d));
	} else if (c <= 8e+69) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c)
	tmp = 0.0
	if (c <= -1.3e+65)
		tmp = t_0;
	elseif (c <= -5.2e-41)
		tmp = Float64(Float64(Float64(c * a) + Float64(b * d)) / fma(c, c, Float64(d * d)));
	elseif (c <= 8e+69)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.3e+65], t$95$0, If[LessEqual[c, -5.2e-41], N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e+69], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.30000000000000001e65 or 8.0000000000000006e69 < c

   1. Initial program 42.9%

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

   3. Taylor expanded in c around inf 85.9%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. associate-/r* 87.8%

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

   7. Simplified 87.8%

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

   if -1.30000000000000001e65 < c < -5.1999999999999999e-41

   1. Initial program 95.2%

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

      1. fma-define 95.2%

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

      2. fma-define 95.3%

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

   3. Simplified 95.3%

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

      1. fma-define 95.3%

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

      2. +-commutative 95.3%

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

   6. Applied egg-rr 95.3%

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

   if -5.1999999999999999e-41 < c < 8.0000000000000006e69

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf 85.9%

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

      1. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+65}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+69}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (/ 1.0 (/ (/ c b) d))) c)))
   (if (<= c -3.1e+65)
     t_0
     (if (<= c -6.6e-43)
       (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 1.7e+71) (/ (+ b (* a (/ c d))) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (1.0 / ((c / b) / d))) / c;
	double tmp;
	if (c <= -3.1e+65) {
		tmp = t_0;
	} else if (c <= -6.6e-43) {
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.7e+71) {
		tmp = (b + (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 + (1.0d0 / ((c / b) / d))) / c
    if (c <= (-3.1d+65)) then
        tmp = t_0
    else if (c <= (-6.6d-43)) then
        tmp = ((c * a) + (b * d)) / ((c * c) + (d * d))
    else if (c <= 1.7d+71) then
        tmp = (b + (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 + (1.0 / ((c / b) / d))) / c;
	double tmp;
	if (c <= -3.1e+65) {
		tmp = t_0;
	} else if (c <= -6.6e-43) {
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 1.7e+71) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (1.0 / ((c / b) / d))) / c
	tmp = 0
	if c <= -3.1e+65:
		tmp = t_0
	elif c <= -6.6e-43:
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d))
	elif c <= 1.7e+71:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c)
	tmp = 0.0
	if (c <= -3.1e+65)
		tmp = t_0;
	elseif (c <= -6.6e-43)
		tmp = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.7e+71)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (1.0 / ((c / b) / d))) / c;
	tmp = 0.0;
	if (c <= -3.1e+65)
		tmp = t_0;
	elseif (c <= -6.6e-43)
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	elseif (c <= 1.7e+71)
		tmp = (b + (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 + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.1e+65], t$95$0, If[LessEqual[c, -6.6e-43], N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e+71], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.09999999999999991e65 or 1.6999999999999999e71 < c

   1. Initial program 42.9%

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

   3. Taylor expanded in c around inf 85.9%

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

      1. clear-num 85.9%

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

      2. inv-pow 85.9%

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

   5. Applied egg-rr 85.9%

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

      1. unpow-1 85.9%

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

      2. associate-/r* 87.8%

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

   7. Simplified 87.8%

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

   if -3.09999999999999991e65 < c < -6.60000000000000031e-43

   1. Initial program 95.2%

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

   if -6.60000000000000031e-43 < c < 1.6999999999999999e71

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf 85.9%

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

      1. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{+65}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{elif}\;c \leq -6.6 \cdot 10^{-43}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+71}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-39} \lor \neg \left(c \leq 9 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.2e-39) (not (<= c 9e+69)))
   (/ (+ a (/ 1.0 (/ (/ c b) d))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.2e-39) || !(c <= 9e+69)) {
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.2d-39)) .or. (.not. (c <= 9d+69))) then
        tmp = (a + (1.0d0 / ((c / b) / d))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.2e-39) || !(c <= 9e+69)) {
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.2e-39) or not (c <= 9e+69):
		tmp = (a + (1.0 / ((c / b) / d))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.2e-39) || !(c <= 9e+69))
		tmp = Float64(Float64(a + Float64(1.0 / Float64(Float64(c / b) / d))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.2e-39) || ~((c <= 9e+69)))
		tmp = (a + (1.0 / ((c / b) / d))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.2e-39], N[Not[LessEqual[c, 9e+69]], $MachinePrecision]], N[(N[(a + N[(1.0 / N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.20000000000000008e-39 or 8.9999999999999999e69 < c

   1. Initial program 51.5%

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

   3. Taylor expanded in c around inf 83.3%

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

      1. clear-num 83.3%

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

      2. inv-pow 83.3%

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

   5. Applied egg-rr 83.3%

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

      1. unpow-1 83.3%

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

      2. associate-/r* 84.9%

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

   7. Simplified 84.9%

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

   if -1.20000000000000008e-39 < c < 8.9999999999999999e69

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf 85.9%

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

      1. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-39} \lor \neg \left(c \leq 9 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{a + \frac{1}{\frac{\frac{c}{b}}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{-36} \lor \neg \left(c \leq 1.72 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.8e-36) (not (<= c 1.72e+69)))
   (/ (+ a (* b (/ d c))) c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.8e-36) || !(c <= 1.72e+69)) {
		tmp = (a + (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 <= (-1.8d-36)) .or. (.not. (c <= 1.72d+69))) then
        tmp = (a + (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 <= -1.8e-36) || !(c <= 1.72e+69)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.8e-36) or not (c <= 1.72e+69):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.8e-36) || !(c <= 1.72e+69))
		tmp = Float64(Float64(a + Float64(b * 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 <= -1.8e-36) || ~((c <= 1.72e+69)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.8e-36], N[Not[LessEqual[c, 1.72e+69]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.80000000000000016e-36 or 1.72e69 < c

   1. Initial program 51.5%

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

   3. Taylor expanded in c around inf 83.3%

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

      1. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   if -1.80000000000000016e-36 < c < 1.72e69

   1. Initial program 69.0%

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

   3. Taylor expanded in c around 0 69.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{-36} \lor \neg \left(c \leq 1.72 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-40} \lor \neg \left(c \leq 3.9 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4e-40) (not (<= c 3.9e+68)))
   (/ (+ a (* b (/ d c))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-40) || !(c <= 3.9e+68)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4d-40)) .or. (.not. (c <= 3.9d+68))) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-40) || !(c <= 3.9e+68)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4e-40) or not (c <= 3.9e+68):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4e-40) || !(c <= 3.9e+68))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4e-40) || ~((c <= 3.9e+68)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4e-40], N[Not[LessEqual[c, 3.9e+68]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.9999999999999997e-40 or 3.90000000000000019e68 < c

   1. Initial program 51.5%

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

   3. Taylor expanded in c around inf 83.3%

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

      1. associate-/l* 84.2%

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

   5. Simplified 84.2%

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

   if -3.9999999999999997e-40 < c < 3.90000000000000019e68

   1. Initial program 69.0%

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

   3. Taylor expanded in d around inf 85.9%

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

      1. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-40} \lor \neg \left(c \leq 3.9 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{-37} \lor \neg \left(c \leq 2.85 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.5e-37) (not (<= c 2.85e+76))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.5e-37) || !(c <= 2.85e+76)) {
		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 <= (-2.5d-37)) .or. (.not. (c <= 2.85d+76))) 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 <= -2.5e-37) || !(c <= 2.85e+76)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.5e-37) or not (c <= 2.85e+76):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.5e-37) || !(c <= 2.85e+76))
		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 <= -2.5e-37) || ~((c <= 2.85e+76)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.5e-37], N[Not[LessEqual[c, 2.85e+76]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.4999999999999999e-37 or 2.85000000000000002e76 < c

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 76.8%

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

   if -2.4999999999999999e-37 < c < 2.85000000000000002e76

   1. Initial program 68.9%

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

   3. Taylor expanded in c around 0 68.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{-37} \lor \neg \left(c \leq 2.85 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in c around inf 45.1%

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

4. Final simplification 45.1%

   \[\leadsto \frac{a}{c} \]
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 2024060 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (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))))