Complex division, real part

Percentage Accurate: 61.5% → 82.8%

Time: 9.1s

Alternatives: 7

Speedup: 1.2×

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: 61.5% 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: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + d \cdot b\\ t_1 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+134}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* d b))) (t_1 (+ (* c c) (* d d))))
   (if (<= d -6.2e+104)
     (/ (+ b (* a (/ c d))) d)
     (if (<= d -5.6e-117)
       (/ t_0 t_1)
       (if (<= d 3.3e-125)
         (/ (+ a (/ (* d b) c)) c)
         (if (<= d 2.6e+134)
           (/ 1.0 (/ t_1 t_0))
           (/ (+ b (/ a (/ 1.0 (/ c d)))) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (d * b);
	double t_1 = (c * c) + (d * d);
	double tmp;
	if (d <= -6.2e+104) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5.6e-117) {
		tmp = t_0 / t_1;
	} else if (d <= 3.3e-125) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.6e+134) {
		tmp = 1.0 / (t_1 / t_0);
	} else {
		tmp = (b + (a / (1.0 / (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a * c) + (d * b)
    t_1 = (c * c) + (d * d)
    if (d <= (-6.2d+104)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-5.6d-117)) then
        tmp = t_0 / t_1
    else if (d <= 3.3d-125) then
        tmp = (a + ((d * b) / c)) / c
    else if (d <= 2.6d+134) then
        tmp = 1.0d0 / (t_1 / t_0)
    else
        tmp = (b + (a / (1.0d0 / (c / d)))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (d * b);
	double t_1 = (c * c) + (d * d);
	double tmp;
	if (d <= -6.2e+104) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5.6e-117) {
		tmp = t_0 / t_1;
	} else if (d <= 3.3e-125) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.6e+134) {
		tmp = 1.0 / (t_1 / t_0);
	} else {
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (d * b)
	t_1 = (c * c) + (d * d)
	tmp = 0
	if d <= -6.2e+104:
		tmp = (b + (a * (c / d))) / d
	elif d <= -5.6e-117:
		tmp = t_0 / t_1
	elif d <= 3.3e-125:
		tmp = (a + ((d * b) / c)) / c
	elif d <= 2.6e+134:
		tmp = 1.0 / (t_1 / t_0)
	else:
		tmp = (b + (a / (1.0 / (c / d)))) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(d * b))
	t_1 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (d <= -6.2e+104)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -5.6e-117)
		tmp = Float64(t_0 / t_1);
	elseif (d <= 3.3e-125)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 2.6e+134)
		tmp = Float64(1.0 / Float64(t_1 / t_0));
	else
		tmp = Float64(Float64(b + Float64(a / Float64(1.0 / Float64(c / d)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (d * b);
	t_1 = (c * c) + (d * d);
	tmp = 0.0;
	if (d <= -6.2e+104)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -5.6e-117)
		tmp = t_0 / t_1;
	elseif (d <= 3.3e-125)
		tmp = (a + ((d * b) / c)) / c;
	elseif (d <= 2.6e+134)
		tmp = 1.0 / (t_1 / t_0);
	else
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e+104], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -5.6e-117], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[d, 3.3e-125], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.6e+134], N[(1.0 / N[(t$95$1 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a / N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.20000000000000033e104

   1. Initial program 49.2%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 91.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 91.5%

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

   if -6.20000000000000033e104 < d < -5.6e-117

   1. Initial program 73.8%

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

   if -5.6e-117 < d < 3.3000000000000001e-125

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 95.0%

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

   if 3.3000000000000001e-125 < d < 2.6000000000000002e134

   1. Initial program 83.7%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{c \cdot c + d \cdot d}{a \cdot c + b \cdot d}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(c \cdot c + d \cdot d\right), \color{blue}{\left(a \cdot c + b \cdot d\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot c\right), \left(d \cdot d\right)\right), \left(\color{blue}{a \cdot c} + b \cdot d\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(d \cdot d\right)\right), \left(\color{blue}{a} \cdot c + b \cdot d\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \left(a \cdot \color{blue}{c} + b \cdot d\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{+.f64}\left(\left(a \cdot c\right), \color{blue}{\left(b \cdot d\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \left(\color{blue}{b} \cdot d\right)\right)\right)\right) \]

      9. *-lowering-*.f64 83.8%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, c\right), \mathsf{*.f64}\left(b, \color{blue}{d}\right)\right)\right)\right) \]

   4. Applied egg-rr 83.8%

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

   if 2.6000000000000002e134 < d

   1. Initial program 34.2%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 92.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{1}{\frac{d}{c}}\right)\right), d\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a}{\frac{d}{c}}\right)\right), d\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{d}{c}\right)\right)\right), d\right) \]

      4. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(d, c\right)\right)\right), d\right) \]

   7. Applied egg-rr 92.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{c}{d}}\right)\right)\right), d\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(1, \left(\frac{c}{d}\right)\right)\right)\right), d\right) \]

      3. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(c, d\right)\right)\right)\right), d\right) \]

   9. Applied egg-rr 92.0%

      \[\leadsto \frac{b + \frac{a}{\color{blue}{\frac{1}{\frac{c}{d}}}}}{d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+104}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-125}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+134}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{a \cdot c + d \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+104}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+136}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* d b)) (+ (* c c) (* d d)))))
   (if (<= d -4e+104)
     (/ (+ b (* a (/ c d))) d)
     (if (<= d -5.6e-117)
       t_0
       (if (<= d 3.9e-125)
         (/ (+ a (/ (* d b) c)) c)
         (if (<= d 1.45e+136) t_0 (/ (+ b (/ a (/ 1.0 (/ c d)))) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4e+104) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5.6e-117) {
		tmp = t_0;
	} else if (d <= 3.9e-125) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 1.45e+136) {
		tmp = t_0;
	} else {
		tmp = (b + (a / (1.0 / (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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (d * b)) / ((c * c) + (d * d))
    if (d <= (-4d+104)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-5.6d-117)) then
        tmp = t_0
    else if (d <= 3.9d-125) then
        tmp = (a + ((d * b) / c)) / c
    else if (d <= 1.45d+136) then
        tmp = t_0
    else
        tmp = (b + (a / (1.0d0 / (c / d)))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4e+104) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5.6e-117) {
		tmp = t_0;
	} else if (d <= 3.9e-125) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 1.45e+136) {
		tmp = t_0;
	} else {
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (d * b)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -4e+104:
		tmp = (b + (a * (c / d))) / d
	elif d <= -5.6e-117:
		tmp = t_0
	elif d <= 3.9e-125:
		tmp = (a + ((d * b) / c)) / c
	elif d <= 1.45e+136:
		tmp = t_0
	else:
		tmp = (b + (a / (1.0 / (c / d)))) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -4e+104)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -5.6e-117)
		tmp = t_0;
	elseif (d <= 3.9e-125)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 1.45e+136)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(a / Float64(1.0 / Float64(c / d)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (d * b)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -4e+104)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -5.6e-117)
		tmp = t_0;
	elseif (d <= 3.9e-125)
		tmp = (a + ((d * b) / c)) / c;
	elseif (d <= 1.45e+136)
		tmp = t_0;
	else
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4e+104], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -5.6e-117], t$95$0, If[LessEqual[d, 3.9e-125], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.45e+136], t$95$0, N[(N[(b + N[(a / N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4e104

   1. Initial program 49.2%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 91.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 91.5%

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

   if -4e104 < d < -5.6e-117 or 3.89999999999999982e-125 < d < 1.44999999999999987e136

   1. Initial program 78.1%

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

   if -5.6e-117 < d < 3.89999999999999982e-125

   1. Initial program 78.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 95.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 95.0%

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

   if 1.44999999999999987e136 < d

   1. Initial program 34.2%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 92.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{1}{\frac{d}{c}}\right)\right), d\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a}{\frac{d}{c}}\right)\right), d\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{d}{c}\right)\right)\right), d\right) \]

      4. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(d, c\right)\right)\right), d\right) \]

   7. Applied egg-rr 92.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{c}{d}}\right)\right)\right), d\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(1, \left(\frac{c}{d}\right)\right)\right)\right), d\right) \]

      3. /-lowering-/.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(c, d\right)\right)\right)\right), d\right) \]

   9. Applied egg-rr 92.0%

      \[\leadsto \frac{b + \frac{a}{\color{blue}{\frac{1}{\frac{c}{d}}}}}{d} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+104}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-117}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-125}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+136}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.25e-34)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 1.52e-5)
     (/ (+ a (/ (* d b) c)) c)
     (* (/ 1.0 d) (+ b (* c (/ a d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e-34) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.52e-5) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.25d-34)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 1.52d-5) then
        tmp = (a + ((d * b) / c)) / c
    else
        tmp = (1.0d0 / d) * (b + (c * (a / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e-34) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.52e-5) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.25e-34:
		tmp = (b + (a * (c / d))) / d
	elif d <= 1.52e-5:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = (1.0 / d) * (b + (c * (a / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.25e-34)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 1.52e-5)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.25e-34)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 1.52e-5)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = (1.0 / d) * (b + (c * (a / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.25e-34], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.52e-5], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2500000000000001e-34

   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 d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 77.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 77.6%

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

   if -1.2500000000000001e-34 < d < 1.52e-5

   1. Initial program 81.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 85.7%

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

   if 1.52e-5 < d

   1. Initial program 52.5%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 79.6%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{d}\right), \color{blue}{\left(b + a \cdot \frac{c}{d}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \left(\color{blue}{b} + a \cdot \frac{c}{d}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(b, \color{blue}{\left(a \cdot \frac{c}{d}\right)}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{\color{blue}{d}}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot c\right), \color{blue}{d}\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(c \cdot a\right), d\right)\right)\right) \]

      9. *-lowering-*.f64 74.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(c, a\right), d\right)\right)\right) \]

   7. Applied egg-rr 74.7%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \left(\frac{c \cdot a}{d} + \color{blue}{b}\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(\left(\frac{c \cdot a}{d}\right), \color{blue}{b}\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(\left(c \cdot \frac{a}{d}\right), b\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \left(\frac{a}{d}\right)\right), b\right)\right) \]

      5. /-lowering-/.f64 80.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, d\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(c, \mathsf{/.f64}\left(a, d\right)\right), b\right)\right) \]

   9. Applied egg-rr 80.9%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -6.8e-36) t_0 (if (<= d 6.2e-7) (/ (+ a (/ (* d b) 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 <= -6.8e-36) {
		tmp = t_0;
	} else if (d <= 6.2e-7) {
		tmp = (a + ((d * b) / 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 <= (-6.8d-36)) then
        tmp = t_0
    else if (d <= 6.2d-7) then
        tmp = (a + ((d * b) / 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 <= -6.8e-36) {
		tmp = t_0;
	} else if (d <= 6.2e-7) {
		tmp = (a + ((d * b) / 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 <= -6.8e-36:
		tmp = t_0
	elif d <= 6.2e-7:
		tmp = (a + ((d * b) / 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 <= -6.8e-36)
		tmp = t_0;
	elseif (d <= 6.2e-7)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / 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 <= -6.8e-36)
		tmp = t_0;
	elseif (d <= 6.2e-7)
		tmp = (a + ((d * b) / 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, -6.8e-36], t$95$0, If[LessEqual[d, 6.2e-7], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.8000000000000005e-36 or 6.1999999999999999e-7 < d

   1. Initial program 55.0%

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

   3. Taylor expanded in d around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \frac{a \cdot c}{d}\right), \color{blue}{d}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{a \cdot c}{d}\right)\right), d\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \left(a \cdot \frac{c}{d}\right)\right), d\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), d\right) \]

      5. /-lowering-/.f64 78.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), d\right) \]

   5. Simplified 78.5%

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

   if -6.8000000000000005e-36 < d < 6.1999999999999999e-7

   1. Initial program 81.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 85.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 85.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{-8}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.8e-38)
   (/ b d)
   (if (<= d 1.22e-8) (/ (+ a (/ (* d b) c)) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.8e-38) {
		tmp = b / d;
	} else if (d <= 1.22e-8) {
		tmp = (a + ((d * b) / 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 (d <= (-5.8d-38)) then
        tmp = b / d
    else if (d <= 1.22d-8) then
        tmp = (a + ((d * b) / 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 (d <= -5.8e-38) {
		tmp = b / d;
	} else if (d <= 1.22e-8) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.8e-38:
		tmp = b / d
	elif d <= 1.22e-8:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.8e-38)
		tmp = Float64(b / d);
	elseif (d <= 1.22e-8)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -5.8e-38)
		tmp = b / d;
	elseif (d <= 1.22e-8)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.8e-38], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.22e-8], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.79999999999999988e-38 or 1.22e-8 < d

   1. Initial program 55.6%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 67.5%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{d}\right) \]

   5. Simplified 67.5%

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

   if -5.79999999999999988e-38 < d < 1.22e-8

   1. Initial program 80.7%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(a + \frac{b \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \left(\frac{b \cdot d}{c}\right)\right), c\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\left(b \cdot d\right), c\right)\right), c\right) \]

      4. *-lowering-*.f64 86.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(a, \mathsf{/.f64}\left(\mathsf{*.f64}\left(b, d\right), c\right)\right), c\right) \]

   5. Simplified 86.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{-8}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e-14) (/ a c) (if (<= c 1.2e-76) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-14) {
		tmp = a / c;
	} else if (c <= 1.2e-76) {
		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 (c <= (-5.6d-14)) then
        tmp = a / c
    else if (c <= 1.2d-76) 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 (c <= -5.6e-14) {
		tmp = a / c;
	} else if (c <= 1.2e-76) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e-14:
		tmp = a / c
	elif c <= 1.2e-76:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e-14)
		tmp = Float64(a / c);
	elseif (c <= 1.2e-76)
		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 (c <= -5.6e-14)
		tmp = a / c;
	elseif (c <= 1.2e-76)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e-14], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.2e-76], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.6000000000000001e-14 or 1.20000000000000007e-76 < c

   1. Initial program 57.4%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 65.4%

         \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{c}\right) \]

   5. Simplified 65.4%

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

   if -5.6000000000000001e-14 < c < 1.20000000000000007e-76

   1. Initial program 78.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 71.5%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{d}\right) \]

   5. Simplified 71.5%

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

Alternative 7: 42.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 43.9%

      \[\leadsto \mathsf{/.f64}\left(a, \color{blue}{c}\right) \]

5. Simplified 43.9%

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

Developer Target 1: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024161 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))