Complex division, real part

Percentage Accurate: 61.9% → 81.5%

Time: 9.1s

Alternatives: 8

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: 61.9% 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: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot a + b \cdot d\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{c}{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}}\\ \mathbf{elif}\;c \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -5.2:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 10^{-120}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c a) (* b d))))
   (if (<= c -2.4e+103)
     (/ 1.0 (/ c (fma b (/ d c) a)))
     (if (<= c -1.55e+77)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c -5.2)
         (/ t_0 (+ (* c c) (* d d)))
         (if (<= c 1e-120)
           (/ (+ b (/ a (/ d c))) d)
           (if (<= c 8.2e+92)
             (/ t_0 (fma c c (* d d)))
             (/ (+ 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 <= -2.4e+103) {
		tmp = 1.0 / (c / fma(b, (d / c), a));
	} else if (c <= -1.55e+77) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= -5.2) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1e-120) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 8.2e+92) {
		tmp = t_0 / fma(c, c, (d * d));
	} 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 <= -2.4e+103)
		tmp = Float64(1.0 / Float64(c / fma(b, Float64(d / c), a)));
	elseif (c <= -1.55e+77)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= -5.2)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1e-120)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (c <= 8.2e+92)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	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, -2.4e+103], N[(1.0 / N[(c / N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.55e+77], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -5.2], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e-120], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8.2e+92], N[(t$95$0 / N[(c * c + N[(d * d), $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 < -2.3999999999999998e103

   1. Initial program 35.6%

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

   3. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 92.9%

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

   5. Simplified 92.9%

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

      1. clear-num 92.9%

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

      2. un-div-inv 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. clear-num 92.9%

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

      2. inv-pow 92.9%

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

      3. +-commutative 92.9%

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

      4. associate-/r/ 93.0%

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

      5. fma-define 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. unpow-1 93.0%

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

       2. fma-undefine 93.0%

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

       3. associate-*l/ 81.8%

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

       4. associate-*r/ 92.9%

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

       5. fma-undefine 92.9%

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

   11. Simplified 92.9%

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

   if -2.3999999999999998e103 < c < -1.54999999999999999e77

   1. Initial program 30.8%

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

   3. Taylor expanded in d around inf 86.2%

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

      1. associate-/l* 86.2%

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

   5. Simplified 86.2%

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

   if -1.54999999999999999e77 < c < -5.20000000000000018

   1. Initial program 87.0%

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

   if -5.20000000000000018 < c < 9.99999999999999979e-121

   1. Initial program 61.0%

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

   3. Taylor expanded in d around inf 86.1%

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

      1. associate-/l* 87.0%

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

      2. *-commutative 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. clear-num 87.0%

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

      3. un-div-inv 87.0%

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

   7. Applied egg-rr 87.0%

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

   if 9.99999999999999979e-121 < c < 8.20000000000000047e92

   1. Initial program 82.6%

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

      1. fma-define 82.6%

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

      2. fma-define 82.6%

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

   3. Simplified 82.6%

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

      1. fma-define 82.6%

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

      2. +-commutative 82.6%

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

   6. Applied egg-rr 82.6%

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

   if 8.20000000000000047e92 < c

   1. Initial program 30.6%

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

   3. Taylor expanded in c around inf 84.4%

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

      1. associate-/l* 90.6%

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

   5. Simplified 90.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{c}{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}}\\ \mathbf{elif}\;c \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -5.2:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 10^{-120}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{c}{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}}\\ \mathbf{elif}\;c \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -7.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.4e+103)
     (/ 1.0 (/ c (fma b (/ d c) a)))
     (if (<= c -1.55e+77)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c -7.8)
         t_0
         (if (<= c 1.2e-119)
           (/ (+ b (/ a (/ d c))) d)
           (if (<= c 8.5e+92) t_0 (/ (+ a (* b (/ d c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.4e+103) {
		tmp = 1.0 / (c / fma(b, (d / c), a));
	} else if (c <= -1.55e+77) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= -7.8) {
		tmp = t_0;
	} else if (c <= 1.2e-119) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 8.5e+92) {
		tmp = t_0;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.4e+103)
		tmp = Float64(1.0 / Float64(c / fma(b, Float64(d / c), a)));
	elseif (c <= -1.55e+77)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= -7.8)
		tmp = t_0;
	elseif (c <= 1.2e-119)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (c <= 8.5e+92)
		tmp = t_0;
	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[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.4e+103], N[(1.0 / N[(c / N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.55e+77], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -7.8], t$95$0, If[LessEqual[c, 1.2e-119], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8.5e+92], t$95$0, N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.3999999999999998e103

   1. Initial program 35.6%

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

   3. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 92.9%

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

   5. Simplified 92.9%

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

      1. clear-num 92.9%

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

      2. un-div-inv 92.9%

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

   7. Applied egg-rr 92.9%

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

      1. clear-num 92.9%

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

      2. inv-pow 92.9%

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

      3. +-commutative 92.9%

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

      4. associate-/r/ 93.0%

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

      5. fma-define 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. unpow-1 93.0%

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

       2. fma-undefine 93.0%

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

       3. associate-*l/ 81.8%

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

       4. associate-*r/ 92.9%

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

       5. fma-undefine 92.9%

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

   11. Simplified 92.9%

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

   if -2.3999999999999998e103 < c < -1.54999999999999999e77

   1. Initial program 30.8%

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

   3. Taylor expanded in d around inf 86.2%

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

      1. associate-/l* 86.2%

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

   5. Simplified 86.2%

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

   if -1.54999999999999999e77 < c < -7.79999999999999982 or 1.20000000000000004e-119 < c < 8.5000000000000001e92

   1. Initial program 83.6%

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

   if -7.79999999999999982 < c < 1.20000000000000004e-119

   1. Initial program 61.0%

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

   3. Taylor expanded in d around inf 86.1%

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

      1. associate-/l* 87.0%

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

      2. *-commutative 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. clear-num 87.0%

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

      3. un-div-inv 87.0%

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

   7. Applied egg-rr 87.0%

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

   if 8.5000000000000001e92 < c

   1. Initial program 30.6%

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

   3. Taylor expanded in c around inf 84.4%

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

      1. associate-/l* 90.6%

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

   5. Simplified 90.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{\frac{c}{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}}\\ \mathbf{elif}\;c \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -7.8:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + b \cdot \frac{d}{c}}{c}\\ t_1 := \frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.12 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -16.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* b (/ d c))) c))
        (t_1 (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.4e+103)
     t_0
     (if (<= c -1.12e+77)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c -16.5)
         t_1
         (if (<= c 1.05e-119)
           (/ (+ b (/ a (/ d c))) d)
           (if (<= c 8.2e+92) t_1 t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (b * (d / c))) / c;
	double t_1 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.4e+103) {
		tmp = t_0;
	} else if (c <= -1.12e+77) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= -16.5) {
		tmp = t_1;
	} else if (c <= 1.05e-119) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 8.2e+92) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (a + (b * (d / c))) / c
    t_1 = ((c * a) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-2.4d+103)) then
        tmp = t_0
    else if (c <= (-1.12d+77)) then
        tmp = (b + (a * (c / d))) / d
    else if (c <= (-16.5d0)) then
        tmp = t_1
    else if (c <= 1.05d-119) then
        tmp = (b + (a / (d / c))) / d
    else if (c <= 8.2d+92) then
        tmp = t_1
    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 + (b * (d / c))) / c;
	double t_1 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.4e+103) {
		tmp = t_0;
	} else if (c <= -1.12e+77) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= -16.5) {
		tmp = t_1;
	} else if (c <= 1.05e-119) {
		tmp = (b + (a / (d / c))) / d;
	} else if (c <= 8.2e+92) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (b * (d / c))) / c
	t_1 = ((c * a) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.4e+103:
		tmp = t_0
	elif c <= -1.12e+77:
		tmp = (b + (a * (c / d))) / d
	elif c <= -16.5:
		tmp = t_1
	elif c <= 1.05e-119:
		tmp = (b + (a / (d / c))) / d
	elif c <= 8.2e+92:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(b * Float64(d / c))) / c)
	t_1 = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.4e+103)
		tmp = t_0;
	elseif (c <= -1.12e+77)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= -16.5)
		tmp = t_1;
	elseif (c <= 1.05e-119)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (c <= 8.2e+92)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (b * (d / c))) / c;
	t_1 = ((c * a) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -2.4e+103)
		tmp = t_0;
	elseif (c <= -1.12e+77)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= -16.5)
		tmp = t_1;
	elseif (c <= 1.05e-119)
		tmp = (b + (a / (d / c))) / d;
	elseif (c <= 8.2e+92)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = 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, -2.4e+103], t$95$0, If[LessEqual[c, -1.12e+77], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -16.5], t$95$1, If[LessEqual[c, 1.05e-119], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8.2e+92], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.3999999999999998e103 or 8.20000000000000047e92 < c

   1. Initial program 32.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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* 91.5%

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

   5. Simplified 91.5%

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

   if -2.3999999999999998e103 < c < -1.1199999999999999e77

   1. Initial program 30.8%

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

   3. Taylor expanded in d around inf 86.2%

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

      1. associate-/l* 86.2%

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

   5. Simplified 86.2%

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

   if -1.1199999999999999e77 < c < -16.5 or 1.05e-119 < c < 8.20000000000000047e92

   1. Initial program 83.6%

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

   if -16.5 < c < 1.05e-119

   1. Initial program 61.0%

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

   3. Taylor expanded in d around inf 86.1%

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

      1. associate-/l* 87.0%

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

      2. *-commutative 87.0%

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

   5. Applied egg-rr 87.0%

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

      1. *-commutative 87.0%

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

      2. clear-num 87.0%

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

      3. un-div-inv 87.0%

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

   7. Applied egg-rr 87.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.12 \cdot 10^{+77}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -16.5:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-119}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -9.6 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-70} \lor \neg \left(d \leq 1.9 \cdot 10^{-10}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -9.6e+83)
     t_0
     (if (<= d -1.9e-37)
       (/ (+ a (/ b (/ c d))) c)
       (if (or (<= d -1.75e-70) (not (<= d 1.9e-10)))
         t_0
         (/ (+ a (/ (* b d) c)) c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -9.6e+83) {
		tmp = t_0;
	} else if (d <= -1.9e-37) {
		tmp = (a + (b / (c / d))) / c;
	} else if ((d <= -1.75e-70) || !(d <= 1.9e-10)) {
		tmp = t_0;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    if (d <= (-9.6d+83)) then
        tmp = t_0
    else if (d <= (-1.9d-37)) then
        tmp = (a + (b / (c / d))) / c
    else if ((d <= (-1.75d-70)) .or. (.not. (d <= 1.9d-10))) then
        tmp = t_0
    else
        tmp = (a + ((b * d) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -9.6e+83) {
		tmp = t_0;
	} else if (d <= -1.9e-37) {
		tmp = (a + (b / (c / d))) / c;
	} else if ((d <= -1.75e-70) || !(d <= 1.9e-10)) {
		tmp = t_0;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -9.6e+83:
		tmp = t_0
	elif d <= -1.9e-37:
		tmp = (a + (b / (c / d))) / c
	elif (d <= -1.75e-70) or not (d <= 1.9e-10):
		tmp = t_0
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -9.6e+83)
		tmp = t_0;
	elseif (d <= -1.9e-37)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif ((d <= -1.75e-70) || !(d <= 1.9e-10))
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -9.6e+83)
		tmp = t_0;
	elseif (d <= -1.9e-37)
		tmp = (a + (b / (c / d))) / c;
	elseif ((d <= -1.75e-70) || ~((d <= 1.9e-10)))
		tmp = t_0;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.6e+83], t$95$0, If[LessEqual[d, -1.9e-37], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, -1.75e-70], N[Not[LessEqual[d, 1.9e-10]], $MachinePrecision]], t$95$0, N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.59999999999999965e83 or -1.9000000000000002e-37 < d < -1.74999999999999987e-70 or 1.8999999999999999e-10 < d

   1. Initial program 43.1%

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

   3. Taylor expanded in d around inf 78.4%

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

      1. associate-/l* 80.2%

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

   5. Simplified 80.2%

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

   if -9.59999999999999965e83 < d < -1.9000000000000002e-37

   1. Initial program 63.6%

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

   3. Taylor expanded in c around inf 59.8%

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

      1. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

      1. clear-num 64.8%

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

      2. un-div-inv 64.9%

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

   7. Applied egg-rr 64.9%

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

   if -1.74999999999999987e-70 < d < 1.8999999999999999e-10

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 90.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-70} \lor \neg \left(d \leq 1.9 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.06 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-37}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-71}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-10}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -1.06e+85)
     t_0
     (if (<= d -1.8e-37)
       (/ (+ a (/ b (/ c d))) c)
       (if (<= d -5e-71)
         (/ (+ b (/ a (/ d c))) d)
         (if (<= d 1.85e-10) (/ (+ a (/ (* b d) c)) c) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.06e+85) {
		tmp = t_0;
	} else if (d <= -1.8e-37) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= -5e-71) {
		tmp = (b + (a / (d / c))) / d;
	} else if (d <= 1.85e-10) {
		tmp = (a + ((b * d) / c)) / c;
	} 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 = (b + (a * (c / d))) / d
    if (d <= (-1.06d+85)) then
        tmp = t_0
    else if (d <= (-1.8d-37)) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= (-5d-71)) then
        tmp = (b + (a / (d / c))) / d
    else if (d <= 1.85d-10) then
        tmp = (a + ((b * d) / c)) / c
    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 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.06e+85) {
		tmp = t_0;
	} else if (d <= -1.8e-37) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= -5e-71) {
		tmp = (b + (a / (d / c))) / d;
	} else if (d <= 1.85e-10) {
		tmp = (a + ((b * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -1.06e+85:
		tmp = t_0
	elif d <= -1.8e-37:
		tmp = (a + (b / (c / d))) / c
	elif d <= -5e-71:
		tmp = (b + (a / (d / c))) / d
	elif d <= 1.85e-10:
		tmp = (a + ((b * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -1.06e+85)
		tmp = t_0;
	elseif (d <= -1.8e-37)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= -5e-71)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (d <= 1.85e-10)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -1.06e+85)
		tmp = t_0;
	elseif (d <= -1.8e-37)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= -5e-71)
		tmp = (b + (a / (d / c))) / d;
	elseif (d <= 1.85e-10)
		tmp = (a + ((b * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.06e+85], t$95$0, If[LessEqual[d, -1.8e-37], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, -5e-71], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.85e-10], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.0600000000000001e85 or 1.85000000000000007e-10 < d

   1. Initial program 39.1%

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

   3. Taylor expanded in d around inf 77.7%

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

      1. associate-/l* 79.6%

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

   5. Simplified 79.6%

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

   if -1.0600000000000001e85 < d < -1.80000000000000004e-37

   1. Initial program 63.6%

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

   3. Taylor expanded in c around inf 59.8%

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

      1. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

      1. clear-num 64.8%

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

      2. un-div-inv 64.9%

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

   7. Applied egg-rr 64.9%

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

   if -1.80000000000000004e-37 < d < -4.99999999999999998e-71

   1. Initial program 99.4%

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

   3. Taylor expanded in d around inf 88.3%

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

      1. associate-/l* 88.3%

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

      2. *-commutative 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. *-commutative 88.3%

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

      2. clear-num 88.3%

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

      3. un-div-inv 88.3%

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

   7. Applied egg-rr 88.3%

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

   if -4.99999999999999998e-71 < d < 1.85000000000000007e-10

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 90.3%

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

Alternative 6: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+87} \lor \neg \left(d \leq 5 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.6e+87) (not (<= d 5e+93)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+87) || !(d <= 5e+93)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.6d+87)) .or. (.not. (d <= 5d+93))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+87) || !(d <= 5e+93)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.6e+87) or not (d <= 5e+93):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.6e+87) || !(d <= 5e+93))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.6e+87) || ~((d <= 5e+93)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.6e+87], N[Not[LessEqual[d, 5e+93]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6e87 or 5.0000000000000001e93 < d

   1. Initial program 32.7%

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

   3. Taylor expanded in c around 0 80.7%

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

   if -1.6e87 < d < 5.0000000000000001e93

   1. Initial program 68.6%

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

   3. Taylor expanded in c around inf 77.0%

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

      1. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+87} \lor \neg \left(d \leq 5 \cdot 10^{+93}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+84} \lor \neg \left(d \leq 1.76 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4e+84) (not (<= d 1.76e-10))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+84) || !(d <= 1.76e-10)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4d+84)) .or. (.not. (d <= 1.76d-10))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+84) || !(d <= 1.76e-10)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4e+84) or not (d <= 1.76e-10):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4e+84) || !(d <= 1.76e-10))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4e+84) || ~((d <= 1.76e-10)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4e+84], N[Not[LessEqual[d, 1.76e-10]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.00000000000000023e84 or 1.7600000000000001e-10 < d

   1. Initial program 39.1%

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

   3. Taylor expanded in c around 0 71.9%

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

   if -4.00000000000000023e84 < d < 1.7600000000000001e-10

   1. Initial program 70.7%

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

   3. Taylor expanded in c around inf 67.4%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+84} \lor \neg \left(d \leq 1.76 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.8% 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 56.8%

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

3. Taylor expanded in c around inf 45.7%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
4. 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 2024084 
(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))))