Complex division, real part

Percentage Accurate: 62.0% → 85.4%

Time: 7.5s

Alternatives: 9

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+231}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+231)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1e+231) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+231)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+231], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.0000000000000001e231

   1. Initial program 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. add-sqr-sqrt 76.1%

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

      3. times-frac 76.1%

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

      4. hypot-define 76.1%

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

      5. fma-define 76.1%

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

      6. hypot-define 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 1.0000000000000001e231 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 20.3%

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

   3. Taylor expanded in c around inf 53.6%

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

      1. associate-/l* 65.7%

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

   5. Simplified 65.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+231}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.6e-44)
   (* (/ c (hypot c d)) (/ a (hypot c d)))
   (if (<= c 4.4e-104)
     (/ (+ b (/ (* a c) d)) d)
     (if (<= c 4.5e+81)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (/ (+ a (/ b (/ c d))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e-44) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (c <= 4.4e-104) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (c <= 4.5e+81) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e-44) {
		tmp = (c / Math.hypot(c, d)) * (a / Math.hypot(c, d));
	} else if (c <= 4.4e-104) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (c <= 4.5e+81) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.6e-44:
		tmp = (c / math.hypot(c, d)) * (a / math.hypot(c, d))
	elif c <= 4.4e-104:
		tmp = (b + ((a * c) / d)) / d
	elif c <= 4.5e+81:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.6e-44)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (c <= 4.4e-104)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (c <= 4.5e+81)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.6e-44)
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	elseif (c <= 4.4e-104)
		tmp = (b + ((a * c) / d)) / d;
	elseif (c <= 4.5e+81)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.6e-44], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.4e-104], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+81], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.59999999999999996e-44

   1. Initial program 50.0%

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

   3. Taylor expanded in a around inf 43.1%

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

      1. *-commutative 43.1%

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

   5. Simplified 43.1%

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

      1. add-sqr-sqrt 43.1%

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

      2. hypot-undefine 43.1%

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

      3. hypot-undefine 43.1%

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

      4. times-frac 82.4%

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

   7. Applied egg-rr 82.4%

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

   if -4.59999999999999996e-44 < c < 4.40000000000000023e-104

   1. Initial program 71.4%

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

   3. Taylor expanded in d around inf 89.9%

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

   if 4.40000000000000023e-104 < c < 4.50000000000000017e81

   1. Initial program 85.7%

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

   if 4.50000000000000017e81 < c

   1. Initial program 40.7%

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

   3. Taylor expanded in c around inf 78.0%

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

      1. associate-/l* 88.9%

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

   5. Simplified 88.9%

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

      1. clear-num 88.9%

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

      2. un-div-inv 88.9%

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

   7. Applied egg-rr 88.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-46}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+81}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.02e+103)
     (/ (+ a (* b (/ d c))) c)
     (if (<= c -1.05e-46)
       t_0
       (if (<= c 2.5e-106)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= c 8.5e+81) t_0 (/ (+ a (/ b (/ c d))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.02e+103) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= -1.05e-46) {
		tmp = t_0;
	} else if (c <= 2.5e-106) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (c <= 8.5e+81) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-1.02d+103)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= (-1.05d-46)) then
        tmp = t_0
    else if (c <= 2.5d-106) then
        tmp = (b + ((a * c) / d)) / d
    else if (c <= 8.5d+81) then
        tmp = t_0
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.02e+103) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= -1.05e-46) {
		tmp = t_0;
	} else if (c <= 2.5e-106) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (c <= 8.5e+81) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.02e+103:
		tmp = (a + (b * (d / c))) / c
	elif c <= -1.05e-46:
		tmp = t_0
	elif c <= 2.5e-106:
		tmp = (b + ((a * c) / d)) / d
	elif c <= 8.5e+81:
		tmp = t_0
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.02e+103)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= -1.05e-46)
		tmp = t_0;
	elseif (c <= 2.5e-106)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (c <= 8.5e+81)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.02e+103)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= -1.05e-46)
		tmp = t_0;
	elseif (c <= 2.5e-106)
		tmp = (b + ((a * c) / d)) / d;
	elseif (c <= 8.5e+81)
		tmp = t_0;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.02e+103], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.05e-46], t$95$0, If[LessEqual[c, 2.5e-106], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8.5e+81], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.01999999999999991e103

   1. Initial program 20.0%

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

   3. Taylor expanded in c around inf 78.7%

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

      1. associate-/l* 86.2%

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

   5. Simplified 86.2%

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

   if -1.01999999999999991e103 < c < -1.04999999999999994e-46 or 2.49999999999999991e-106 < c < 8.49999999999999986e81

   1. Initial program 85.5%

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

   if -1.04999999999999994e-46 < c < 2.49999999999999991e-106

   1. Initial program 71.4%

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

   3. Taylor expanded in d around inf 89.9%

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

   if 8.49999999999999986e81 < c

   1. Initial program 40.7%

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

   3. Taylor expanded in c around inf 78.0%

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

      1. associate-/l* 88.9%

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

   5. Simplified 88.9%

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

      1. clear-num 88.9%

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

      2. un-div-inv 88.9%

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

   7. Applied egg-rr 88.9%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{+103}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-46}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-66} \lor \neg \left(c \leq 2.35 \cdot 10^{+72}\right) \land c \leq 1.15 \cdot 10^{+118}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.2e-26)
   (/ (+ a (* b (/ d c))) c)
   (if (or (<= c 6.5e-66) (and (not (<= c 2.35e+72)) (<= c 1.15e+118)))
     (/ (+ b (* a (/ c d))) d)
     (/ (+ a (/ b (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-26) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((c <= 6.5e-66) || (!(c <= 2.35e+72) && (c <= 1.15e+118))) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / 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 <= (-6.2d-26)) then
        tmp = (a + (b * (d / c))) / c
    else if ((c <= 6.5d-66) .or. (.not. (c <= 2.35d+72)) .and. (c <= 1.15d+118)) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-26) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((c <= 6.5e-66) || (!(c <= 2.35e+72) && (c <= 1.15e+118))) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.2e-26:
		tmp = (a + (b * (d / c))) / c
	elif (c <= 6.5e-66) or (not (c <= 2.35e+72) and (c <= 1.15e+118)):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.2e-26)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif ((c <= 6.5e-66) || (!(c <= 2.35e+72) && (c <= 1.15e+118)))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6.2e-26)
		tmp = (a + (b * (d / c))) / c;
	elseif ((c <= 6.5e-66) || (~((c <= 2.35e+72)) && (c <= 1.15e+118)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.2e-26], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[c, 6.5e-66], And[N[Not[LessEqual[c, 2.35e+72]], $MachinePrecision], LessEqual[c, 1.15e+118]]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.19999999999999966e-26

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf 74.9%

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

      1. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

   if -6.19999999999999966e-26 < c < 6.50000000000000024e-66 or 2.35000000000000017e72 < c < 1.15000000000000008e118

   1. Initial program 72.6%

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

   3. Taylor expanded in d around inf 86.9%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if 6.50000000000000024e-66 < c < 2.35000000000000017e72 or 1.15000000000000008e118 < c

   1. Initial program 57.9%

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

   3. Taylor expanded in c around inf 78.1%

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

      1. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

      1. clear-num 85.9%

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

      2. un-div-inv 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-66} \lor \neg \left(c \leq 2.35 \cdot 10^{+72}\right) \land c \leq 1.15 \cdot 10^{+118}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{-44} \lor \neg \left(c \leq 1.6 \cdot 10^{-68} \lor \neg \left(c \leq 1.62 \cdot 10^{+69}\right) \land c \leq 9 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1e-44)
         (not (or (<= c 1.6e-68) (and (not (<= c 1.62e+69)) (<= c 9e+117)))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1e-44) || !((c <= 1.6e-68) || (!(c <= 1.62e+69) && (c <= 9e+117)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1d-44)) .or. (.not. (c <= 1.6d-68) .or. (.not. (c <= 1.62d+69)) .and. (c <= 9d+117))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1e-44) || !((c <= 1.6e-68) || (!(c <= 1.62e+69) && (c <= 9e+117)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1e-44) or not ((c <= 1.6e-68) or (not (c <= 1.62e+69) and (c <= 9e+117))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1e-44) || !((c <= 1.6e-68) || (!(c <= 1.62e+69) && (c <= 9e+117))))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1e-44) || ~(((c <= 1.6e-68) || (~((c <= 1.62e+69)) && (c <= 9e+117)))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1e-44], N[Not[Or[LessEqual[c, 1.6e-68], And[N[Not[LessEqual[c, 1.62e+69]], $MachinePrecision], LessEqual[c, 9e+117]]]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -9.99999999999999953e-45 or 1.5999999999999999e-68 < c < 1.62e69 or 9e117 < c

   1. Initial program 53.6%

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

   3. Taylor expanded in c around inf 68.2%

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

   if -9.99999999999999953e-45 < c < 1.5999999999999999e-68 or 1.62e69 < c < 9e117

   1. Initial program 71.3%

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

   3. Taylor expanded in c around 0 72.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{-44} \lor \neg \left(c \leq 1.6 \cdot 10^{-68} \lor \neg \left(c \leq 1.62 \cdot 10^{+69}\right) \land c \leq 9 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+80} \lor \neg \left(d \leq 5.8 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.2e+80) (not (<= d 5.8e+140)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.2e+80) || !(d <= 5.8e+140)) {
		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 <= (-5.2d+80)) .or. (.not. (d <= 5.8d+140))) 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 <= -5.2e+80) || !(d <= 5.8e+140)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.2e+80) or not (d <= 5.8e+140):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.2e+80) || !(d <= 5.8e+140))
		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 <= -5.2e+80) || ~((d <= 5.8e+140)))
		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, -5.2e+80], N[Not[LessEqual[d, 5.8e+140]], $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 < -5.19999999999999963e80 or 5.7999999999999998e140 < d

   1. Initial program 34.7%

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

   3. Taylor expanded in c around 0 75.5%

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

   if -5.19999999999999963e80 < d < 5.7999999999999998e140

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf 72.8%

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

      1. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+80} \lor \neg \left(d \leq 5.8 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+80} \lor \neg \left(d \leq 6 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.2e+80) (not (<= d 6e+140)))
   (/ b d)
   (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.2e+80) || !(d <= 6e+140)) {
		tmp = b / d;
	} else {
		tmp = (a + (b / (c / d))) / 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 <= (-6.2d+80)) .or. (.not. (d <= 6d+140))) then
        tmp = b / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.2e+80) || !(d <= 6e+140)) {
		tmp = b / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.2e+80) or not (d <= 6e+140):
		tmp = b / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.2e+80) || !(d <= 6e+140))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -6.2e+80) || ~((d <= 6e+140)))
		tmp = b / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.2e+80], N[Not[LessEqual[d, 6e+140]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.19999999999999976e80 or 5.99999999999999993e140 < d

   1. Initial program 34.7%

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

   3. Taylor expanded in c around 0 75.5%

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

   if -6.19999999999999976e80 < d < 5.99999999999999993e140

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf 72.8%

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

      1. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

      1. clear-num 75.0%

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

      2. un-div-inv 75.0%

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

   7. Applied egg-rr 75.0%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+80} \lor \neg \left(d \leq 6 \cdot 10^{+140}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-45}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.4e-23)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 8e-45) (/ (+ b (/ (* a c) d)) d) (/ (+ a (/ b (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e-23) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 8e-45) {
		tmp = (b + ((a * c) / d)) / d;
	} else {
		tmp = (a + (b / (c / d))) / 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 <= (-3.4d-23)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 8d-45) then
        tmp = (b + ((a * c) / d)) / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e-23) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 8e-45) {
		tmp = (b + ((a * c) / d)) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.4e-23:
		tmp = (a + (b * (d / c))) / c
	elif c <= 8e-45:
		tmp = (b + ((a * c) / d)) / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.4e-23)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 8e-45)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.4e-23)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 8e-45)
		tmp = (b + ((a * c) / d)) / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.4e-23], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 8e-45], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.4000000000000001e-23

   1. Initial program 45.8%

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

   3. Taylor expanded in c around inf 74.9%

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

      1. associate-/l* 79.3%

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

   5. Simplified 79.3%

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

   if -3.4000000000000001e-23 < c < 7.99999999999999987e-45

   1. Initial program 73.7%

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

   3. Taylor expanded in d around inf 86.6%

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

   if 7.99999999999999987e-45 < c

   1. Initial program 57.2%

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

   3. Taylor expanded in c around inf 71.7%

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

      1. associate-/l* 78.8%

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

   5. Simplified 78.8%

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

      1. clear-num 78.8%

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

      2. un-div-inv 78.8%

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

   7. Applied egg-rr 78.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-45}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.0% 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 61.9%

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

3. Taylor expanded in c around inf 44.2%

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

4. Final simplification 44.2%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024079 
(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))))