Complex division, real part

Percentage Accurate: 61.4% → 84.5%

Time: 9.3s

Alternatives: 8

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: 61.4% 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: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -5 \cdot 10^{+206}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{b} \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq -2.55 \cdot 10^{-64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5.5 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))))
   (if (<= c -5e+206)
     (+ (/ a c) (/ 1.0 (* (/ c b) (/ c d))))
     (if (<= c -2.55e-64)
       t_0
       (if (<= c 4.5e-95)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 5.5e+119) t_0 (/ (+ a (* d (/ b c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	double tmp;
	if (c <= -5e+206) {
		tmp = (a / c) + (1.0 / ((c / b) * (c / d)));
	} else if (c <= -2.55e-64) {
		tmp = t_0;
	} else if (c <= 4.5e-95) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5.5e+119) {
		tmp = t_0;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)))
	tmp = 0.0
	if (c <= -5e+206)
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(Float64(c / b) * Float64(c / d))));
	elseif (c <= -2.55e-64)
		tmp = t_0;
	elseif (c <= 4.5e-95)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 5.5e+119)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = 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]}, If[LessEqual[c, -5e+206], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(N[(c / b), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.55e-64], t$95$0, If[LessEqual[c, 4.5e-95], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.5e+119], t$95$0, N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.0000000000000002e206

   1. Initial program 43.8%

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

   3. Taylor expanded in d around 0 88.3%

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

      1. associate-/l* 88.7%

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

   5. Simplified 88.7%

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

      1. *-un-lft-identity 88.7%

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

      2. pow2 88.7%

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

      3. times-frac 91.7%

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

   7. Applied egg-rr 91.7%

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

      1. associate-*r* 92.3%

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

      2. div-inv 92.3%

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

      3. clear-num 92.4%

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

      4. clear-num 92.4%

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

      5. frac-times 92.4%

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

      6. metadata-eval 92.4%

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

   9. Applied egg-rr 92.4%

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

   if -5.0000000000000002e206 < c < -2.54999999999999992e-64 or 4.5e-95 < c < 5.5000000000000003e119

   1. Initial program 68.4%

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

      1. *-un-lft-identity 68.4%

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

      2. add-sqr-sqrt 68.4%

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

      3. times-frac 68.5%

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

      4. hypot-define 68.5%

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

      5. fma-define 68.5%

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

      6. hypot-define 86.5%

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

   4. Applied egg-rr 86.5%

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

   if -2.54999999999999992e-64 < c < 4.5e-95

   1. Initial program 66.4%

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

   3. Taylor expanded in d around inf 93.7%

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

      1. associate-/l* 93.7%

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

   5. Simplified 93.7%

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

   if 5.5000000000000003e119 < c

   1. Initial program 34.0%

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

   3. Taylor expanded in c around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-/l* 94.5%

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

   5. Applied egg-rr 94.5%

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

Alternative 2: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ t_1 := \frac{b \cdot d + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -7 \cdot 10^{+59}:\\ \;\;\;\;\frac{a + d \cdot \left(b \cdot \frac{1}{c}\right)}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d))
        (t_1 (/ (+ (* b d) (* c a)) (+ (* c c) (* d d)))))
   (if (<= c -7e+59)
     (/ (+ a (* d (* b (/ 1.0 c)))) c)
     (if (<= c -1.7e+27)
       t_0
       (if (<= c -3.5e-64)
         t_1
         (if (<= c 6.5e-79)
           t_0
           (if (<= c 1.45e+119) t_1 (/ (+ a (* d (/ b c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double t_1 = ((b * d) + (c * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7e+59) {
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	} else if (c <= -1.7e+27) {
		tmp = t_0;
	} else if (c <= -3.5e-64) {
		tmp = t_1;
	} else if (c <= 6.5e-79) {
		tmp = t_0;
	} else if (c <= 1.45e+119) {
		tmp = t_1;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    t_1 = ((b * d) + (c * a)) / ((c * c) + (d * d))
    if (c <= (-7d+59)) then
        tmp = (a + (d * (b * (1.0d0 / c)))) / c
    else if (c <= (-1.7d+27)) then
        tmp = t_0
    else if (c <= (-3.5d-64)) then
        tmp = t_1
    else if (c <= 6.5d-79) then
        tmp = t_0
    else if (c <= 1.45d+119) then
        tmp = t_1
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double t_1 = ((b * d) + (c * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7e+59) {
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	} else if (c <= -1.7e+27) {
		tmp = t_0;
	} else if (c <= -3.5e-64) {
		tmp = t_1;
	} else if (c <= 6.5e-79) {
		tmp = t_0;
	} else if (c <= 1.45e+119) {
		tmp = t_1;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	t_1 = ((b * d) + (c * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -7e+59:
		tmp = (a + (d * (b * (1.0 / c)))) / c
	elif c <= -1.7e+27:
		tmp = t_0
	elif c <= -3.5e-64:
		tmp = t_1
	elif c <= 6.5e-79:
		tmp = t_0
	elif c <= 1.45e+119:
		tmp = t_1
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	t_1 = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -7e+59)
		tmp = Float64(Float64(a + Float64(d * Float64(b * Float64(1.0 / c)))) / c);
	elseif (c <= -1.7e+27)
		tmp = t_0;
	elseif (c <= -3.5e-64)
		tmp = t_1;
	elseif (c <= 6.5e-79)
		tmp = t_0;
	elseif (c <= 1.45e+119)
		tmp = t_1;
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	t_1 = ((b * d) + (c * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -7e+59)
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	elseif (c <= -1.7e+27)
		tmp = t_0;
	elseif (c <= -3.5e-64)
		tmp = t_1;
	elseif (c <= 6.5e-79)
		tmp = t_0;
	elseif (c <= 1.45e+119)
		tmp = t_1;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7e+59], N[(N[(a + N[(d * N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.7e+27], t$95$0, If[LessEqual[c, -3.5e-64], t$95$1, If[LessEqual[c, 6.5e-79], t$95$0, If[LessEqual[c, 1.45e+119], t$95$1, N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7e59

   1. Initial program 56.4%

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

   3. Taylor expanded in c around inf 82.0%

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

      1. div-inv 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 85.7%

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

   5. Applied egg-rr 85.7%

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

   if -7e59 < c < -1.7e27 or -3.5000000000000003e-64 < c < 6.5000000000000003e-79

   1. Initial program 63.6%

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

   3. Taylor expanded in d around inf 93.3%

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

      1. associate-/l* 93.3%

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

   5. Simplified 93.3%

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

   if -1.7e27 < c < -3.5000000000000003e-64 or 6.5000000000000003e-79 < c < 1.45000000000000004e119

   1. Initial program 73.8%

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

   if 1.45000000000000004e119 < c

   1. Initial program 34.0%

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

   3. Taylor expanded in c around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. associate-/l* 94.5%

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

   5. Applied egg-rr 94.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+59}:\\ \;\;\;\;\frac{a + d \cdot \left(b \cdot \frac{1}{c}\right)}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq -3.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-79}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+119}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -53000000 \lor \neg \left(d \leq -1.65 \cdot 10^{-9}\right) \land \left(d \leq -1.3 \cdot 10^{-41} \lor \neg \left(d \leq 1.15 \cdot 10^{+149}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -53000000.0)
         (and (not (<= d -1.65e-9))
              (or (<= d -1.3e-41) (not (<= d 1.15e+149)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -53000000.0) || (!(d <= -1.65e-9) && ((d <= -1.3e-41) || !(d <= 1.15e+149)))) {
		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 <= (-53000000.0d0)) .or. (.not. (d <= (-1.65d-9))) .and. (d <= (-1.3d-41)) .or. (.not. (d <= 1.15d+149))) 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 <= -53000000.0) || (!(d <= -1.65e-9) && ((d <= -1.3e-41) || !(d <= 1.15e+149)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -53000000.0) or (not (d <= -1.65e-9) and ((d <= -1.3e-41) or not (d <= 1.15e+149))):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -53000000.0) || (!(d <= -1.65e-9) && ((d <= -1.3e-41) || !(d <= 1.15e+149))))
		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 <= -53000000.0) || (~((d <= -1.65e-9)) && ((d <= -1.3e-41) || ~((d <= 1.15e+149)))))
		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, -53000000.0], And[N[Not[LessEqual[d, -1.65e-9]], $MachinePrecision], Or[LessEqual[d, -1.3e-41], N[Not[LessEqual[d, 1.15e+149]], $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 < -5.3e7 or -1.65000000000000009e-9 < d < -1.3e-41 or 1.1499999999999999e149 < d

   1. Initial program 48.1%

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

   3. Taylor expanded in c around 0 76.3%

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

   if -5.3e7 < d < -1.65000000000000009e-9 or -1.3e-41 < d < 1.1499999999999999e149

   1. Initial program 67.4%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 79.7%

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

   5. Simplified 79.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -53000000 \lor \neg \left(d \leq -1.65 \cdot 10^{-9}\right) \land \left(d \leq -1.3 \cdot 10^{-41} \lor \neg \left(d \leq 1.15 \cdot 10^{+149}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+61} \lor \neg \left(c \leq -1.4 \cdot 10^{-45}\right) \land \left(c \leq -1.65 \cdot 10^{-64} \lor \neg \left(c \leq 1.45 \cdot 10^{+66}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.45e+61)
         (and (not (<= c -1.4e-45))
              (or (<= c -1.65e-64) (not (<= c 1.45e+66)))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.45e+61) || (!(c <= -1.4e-45) && ((c <= -1.65e-64) || !(c <= 1.45e+66)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.45d+61)) .or. (.not. (c <= (-1.4d-45))) .and. (c <= (-1.65d-64)) .or. (.not. (c <= 1.45d+66))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.45e+61) || (!(c <= -1.4e-45) && ((c <= -1.65e-64) || !(c <= 1.45e+66)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.45e+61) or (not (c <= -1.4e-45) and ((c <= -1.65e-64) or not (c <= 1.45e+66))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.45e+61) || (!(c <= -1.4e-45) && ((c <= -1.65e-64) || !(c <= 1.45e+66))))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.45e+61) || (~((c <= -1.4e-45)) && ((c <= -1.65e-64) || ~((c <= 1.45e+66)))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.45e+61], And[N[Not[LessEqual[c, -1.4e-45]], $MachinePrecision], Or[LessEqual[c, -1.65e-64], N[Not[LessEqual[c, 1.45e+66]], $MachinePrecision]]]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.45000000000000013e61 or -1.4000000000000001e-45 < c < -1.65e-64 or 1.44999999999999993e66 < c

   1. Initial program 52.2%

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

   3. Taylor expanded in c around inf 80.2%

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

   if -2.45000000000000013e61 < c < -1.4000000000000001e-45 or -1.65e-64 < c < 1.44999999999999993e66

   1. Initial program 64.7%

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

   3. Taylor expanded in c around 0 67.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+61} \lor \neg \left(c \leq -1.4 \cdot 10^{-45}\right) \land \left(c \leq -1.65 \cdot 10^{-64} \lor \neg \left(c \leq 1.45 \cdot 10^{+66}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{a + d \cdot \left(b \cdot \frac{1}{c}\right)}{c}\\ \mathbf{elif}\;c \leq 1050000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.65e+69)
   (/ (+ a (* d (* b (/ 1.0 c)))) c)
   (if (<= c 1050000.0) (/ (+ b (* a (/ c d))) d) (/ (+ a (* d (/ b c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.65e+69) {
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	} else if (c <= 1050000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / 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 (c <= (-1.65d+69)) then
        tmp = (a + (d * (b * (1.0d0 / c)))) / c
    else if (c <= 1050000.0d0) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.65e+69) {
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	} else if (c <= 1050000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.65e+69:
		tmp = (a + (d * (b * (1.0 / c)))) / c
	elif c <= 1050000.0:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.65e+69)
		tmp = Float64(Float64(a + Float64(d * Float64(b * Float64(1.0 / c)))) / c);
	elseif (c <= 1050000.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.65e+69)
		tmp = (a + (d * (b * (1.0 / c)))) / c;
	elseif (c <= 1050000.0)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.65e+69], N[(N[(a + N[(d * N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1050000.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.6499999999999999e69

   1. Initial program 56.4%

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

   3. Taylor expanded in c around inf 82.0%

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

      1. div-inv 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-*l* 85.7%

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

   5. Applied egg-rr 85.7%

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

   if -1.6499999999999999e69 < c < 1.05e6

   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 d around inf 84.9%

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

      1. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if 1.05e6 < c

   1. Initial program 47.5%

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

   3. Taylor expanded in c around inf 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 81.5%

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

   5. Applied egg-rr 81.5%

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

Alternative 6: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.12 \cdot 10^{+66}:\\ \;\;\;\;\frac{a + \frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;c \leq 5000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.12e+66)
   (/ (+ a (/ d (/ c b))) c)
   (if (<= c 5000.0) (/ (+ b (* a (/ c d))) d) (/ (+ a (* d (/ b c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.12e+66) {
		tmp = (a + (d / (c / b))) / c;
	} else if (c <= 5000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / 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 (c <= (-1.12d+66)) then
        tmp = (a + (d / (c / b))) / c
    else if (c <= 5000.0d0) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.12e+66) {
		tmp = (a + (d / (c / b))) / c;
	} else if (c <= 5000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.12e+66:
		tmp = (a + (d / (c / b))) / c
	elif c <= 5000.0:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.12e+66)
		tmp = Float64(Float64(a + Float64(d / Float64(c / b))) / c);
	elseif (c <= 5000.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.12e+66)
		tmp = (a + (d / (c / b))) / c;
	elseif (c <= 5000.0)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.12e+66], N[(N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 5000.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.12e66

   1. Initial program 56.4%

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

   3. Taylor expanded in c around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 85.7%

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

   5. Applied egg-rr 85.7%

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

      1. clear-num 85.7%

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

      2. un-div-inv 85.7%

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

   7. Applied egg-rr 85.7%

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

   if -1.12e66 < c < 5e3

   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 d around inf 84.9%

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

      1. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if 5e3 < c

   1. Initial program 47.5%

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

   3. Taylor expanded in c around inf 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-/l* 81.5%

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

   5. Applied egg-rr 81.5%

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

Alternative 7: 43.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -2.5e+194) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.5e+194) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-2.5d+194)) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.5e+194) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.5e+194:
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.5e+194)
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.5e+194)
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.5e+194], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.49999999999999994e194

   1. Initial program 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. add-sqr-sqrt 38.6%

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

      3. times-frac 38.6%

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

      4. hypot-define 38.6%

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

      5. fma-define 38.6%

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

      6. hypot-define 64.2%

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

   4. Applied egg-rr 64.2%

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

   5. Taylor expanded in c around inf 30.7%

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

   6. Taylor expanded in c around 0 30.6%

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

   if -2.49999999999999994e194 < d

   1. Initial program 61.7%

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

   3. Taylor expanded in c around inf 50.6%

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

Alternative 8: 43.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in c around inf 46.3%

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

Developer target: 99.2% 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 2024085 
(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))))