Complex division, real part

Percentage Accurate: 62.5% → 86.1%

Time: 9.3s

Alternatives: 11

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: 62.5% 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: 86.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 -2e+278)
     (/ (+ a (* b (/ d c))) c)
     (if (<= t_0 2e+279)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (/ (+ b (* a (/ c d))) d)))))

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 (t_0 <= -2e+278) {
		tmp = (a + (b * (d / c))) / c;
	} else if (t_0 <= 2e+279) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	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 (t_0 <= -2e+278)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (t_0 <= 2e+279)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return 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[t$95$0, -2e+278], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[t$95$0, 2e+279], 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[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -1.99999999999999993e278

   1. Initial program 38.6%

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

   3. Taylor expanded in c around inf 79.9%

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

      1. +-commutative 79.9%

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

      2. associate-/l* 86.6%

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

      3. fma-define 86.6%

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

   5. Simplified 86.6%

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

      1. fma-undefine 86.6%

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

   7. Applied egg-rr 86.6%

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

   if -1.99999999999999993e278 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.00000000000000012e279

   1. Initial program 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. times-frac 82.9%

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

      4. hypot-define 82.9%

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

      5. fma-define 82.9%

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

      6. hypot-define 99.0%

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

   4. Applied egg-rr 99.0%

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

   if 2.00000000000000012e279 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.6%

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

   3. Taylor expanded in d around inf 52.1%

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

      1. associate-/l* 65.6%

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

   5. Simplified 65.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -2 \cdot 10^{+278}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+279}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{b + t\_1}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 8.7 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(\left(t\_1 - a \cdot {\left(\frac{c}{d}\right)}^{3}\right) - b \cdot {\left(\frac{c}{d}\right)}^{2}\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))) (t_1 (* a (/ c d))))
   (if (<= d -3.6e+113)
     (/ (+ b t_1) d)
     (if (<= d -9e-101)
       t_0
       (if (<= d 1.5e-129)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 8.7e+89)
           t_0
           (/
            (+ b (- (- t_1 (* a (pow (/ c d) 3.0))) (* b (pow (/ c d) 2.0))))
            d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = a * (c / d);
	double tmp;
	if (d <= -3.6e+113) {
		tmp = (b + t_1) / d;
	} else if (d <= -9e-101) {
		tmp = t_0;
	} else if (d <= 1.5e-129) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 8.7e+89) {
		tmp = t_0;
	} else {
		tmp = (b + ((t_1 - (a * pow((c / d), 3.0))) - (b * pow((c / d), 2.0)))) / 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) + (b * d)) / ((c * c) + (d * d))
    t_1 = a * (c / d)
    if (d <= (-3.6d+113)) then
        tmp = (b + t_1) / d
    else if (d <= (-9d-101)) then
        tmp = t_0
    else if (d <= 1.5d-129) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 8.7d+89) then
        tmp = t_0
    else
        tmp = (b + ((t_1 - (a * ((c / d) ** 3.0d0))) - (b * ((c / d) ** 2.0d0)))) / 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) + (b * d)) / ((c * c) + (d * d));
	double t_1 = a * (c / d);
	double tmp;
	if (d <= -3.6e+113) {
		tmp = (b + t_1) / d;
	} else if (d <= -9e-101) {
		tmp = t_0;
	} else if (d <= 1.5e-129) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 8.7e+89) {
		tmp = t_0;
	} else {
		tmp = (b + ((t_1 - (a * Math.pow((c / d), 3.0))) - (b * Math.pow((c / d), 2.0)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = a * (c / d)
	tmp = 0
	if d <= -3.6e+113:
		tmp = (b + t_1) / d
	elif d <= -9e-101:
		tmp = t_0
	elif d <= 1.5e-129:
		tmp = (a + (b * (d / c))) / c
	elif d <= 8.7e+89:
		tmp = t_0
	else:
		tmp = (b + ((t_1 - (a * math.pow((c / d), 3.0))) - (b * math.pow((c / d), 2.0)))) / d
	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)))
	t_1 = Float64(a * Float64(c / d))
	tmp = 0.0
	if (d <= -3.6e+113)
		tmp = Float64(Float64(b + t_1) / d);
	elseif (d <= -9e-101)
		tmp = t_0;
	elseif (d <= 1.5e-129)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 8.7e+89)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(Float64(t_1 - Float64(a * (Float64(c / d) ^ 3.0))) - Float64(b * (Float64(c / d) ^ 2.0)))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = a * (c / d);
	tmp = 0.0;
	if (d <= -3.6e+113)
		tmp = (b + t_1) / d;
	elseif (d <= -9e-101)
		tmp = t_0;
	elseif (d <= 1.5e-129)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 8.7e+89)
		tmp = t_0;
	else
		tmp = (b + ((t_1 - (a * ((c / d) ^ 3.0))) - (b * ((c / d) ^ 2.0)))) / d;
	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]}, Block[{t$95$1 = N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+113], N[(N[(b + t$95$1), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -9e-101], t$95$0, If[LessEqual[d, 1.5e-129], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 8.7e+89], t$95$0, N[(N[(b + N[(N[(t$95$1 - N[(a * N[Power[N[(c / d), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[Power[N[(c / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.59999999999999992e113

   1. Initial program 26.9%

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

   3. Taylor expanded in d around inf 76.2%

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

      1. associate-/l* 89.6%

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

   5. Simplified 89.6%

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

   if -3.59999999999999992e113 < d < -8.9999999999999997e-101 or 1.4999999999999999e-129 < d < 8.70000000000000019e89

   1. Initial program 87.8%

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

   if -8.9999999999999997e-101 < d < 1.4999999999999999e-129

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-/l* 89.9%

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

      3. fma-define 89.9%

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

   5. Simplified 89.9%

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

      1. fma-undefine 89.9%

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

   7. Applied egg-rr 89.9%

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

   if 8.70000000000000019e89 < d

   1. Initial program 27.5%

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

      1. *-un-lft-identity 27.5%

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

      2. add-sqr-sqrt 27.5%

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

      3. times-frac 27.5%

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

      4. hypot-define 27.5%

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

      5. fma-define 27.5%

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

      6. hypot-define 52.3%

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

   4. Applied egg-rr 52.3%

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

   5. Taylor expanded in d around inf 51.1%

      \[\leadsto \color{blue}{\frac{\left(b + \left(-1 \cdot \frac{a \cdot {c}^{3}}{{d}^{3}} + \frac{a \cdot c}{d}\right)\right) - \frac{b \cdot {c}^{2}}{{d}^{2}}}{d}} \]
   6. Step-by-step derivation

   7. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{b + \left(\left(a \cdot \frac{c}{d} - a \cdot {\left(\frac{c}{d}\right)}^{3}\right) - b \cdot {\left(\frac{c}{d}\right)}^{2}\right)}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+113}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-101}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 8.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \left(\left(a \cdot \frac{c}{d} - a \cdot {\left(\frac{c}{d}\right)}^{3}\right) - b \cdot {\left(\frac{c}{d}\right)}^{2}\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-131}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -8.2e+118)
     (/ (+ b (* a (/ c d))) d)
     (if (<= d -4.7e-101)
       t_0
       (if (<= d 1.85e-131)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 1.5e+70) 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) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -8.2e+118) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -4.7e-101) {
		tmp = t_0;
	} else if (d <= 1.85e-131) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.5e+70) {
		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) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-8.2d+118)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-4.7d-101)) then
        tmp = t_0
    else if (d <= 1.85d-131) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 1.5d+70) 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) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -8.2e+118) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -4.7e-101) {
		tmp = t_0;
	} else if (d <= 1.85e-131) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.5e+70) {
		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) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -8.2e+118:
		tmp = (b + (a * (c / d))) / d
	elif d <= -4.7e-101:
		tmp = t_0
	elif d <= 1.85e-131:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1.5e+70:
		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(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -8.2e+118)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -4.7e-101)
		tmp = t_0;
	elseif (d <= 1.85e-131)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1.5e+70)
		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) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -8.2e+118)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -4.7e-101)
		tmp = t_0;
	elseif (d <= 1.85e-131)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1.5e+70)
		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[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.2e+118], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -4.7e-101], t$95$0, If[LessEqual[d, 1.85e-131], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.5e+70], 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 < -8.1999999999999994e118

   1. Initial program 26.9%

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

   3. Taylor expanded in d around inf 76.2%

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

      1. associate-/l* 89.6%

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

   5. Simplified 89.6%

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

   if -8.1999999999999994e118 < d < -4.6999999999999999e-101 or 1.8500000000000001e-131 < d < 1.49999999999999988e70

   1. Initial program 88.8%

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

   if -4.6999999999999999e-101 < d < 1.8500000000000001e-131

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf 89.9%

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

      1. +-commutative 89.9%

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

      2. associate-/l* 89.9%

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

      3. fma-define 89.9%

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

   5. Simplified 89.9%

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

      1. fma-undefine 89.9%

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

   7. Applied egg-rr 89.9%

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

   if 1.49999999999999988e70 < d

   1. Initial program 27.0%

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

   3. Taylor expanded in d around inf 79.0%

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

      1. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

      1. clear-num 86.7%

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

      2. un-div-inv 86.8%

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

   7. Applied egg-rr 86.8%

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

      1. clear-num 86.8%

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

      2. inv-pow 86.8%

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

   9. Applied egg-rr 86.8%

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

       1. unpow-1 86.8%

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

   11. Simplified 86.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+118}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-101}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-131}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.5e-98)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 2.5e-76)
     (/ (+ a (* b (/ d c))) c)
     (if (<= d 3.6e+47)
       (/ (* b d) (+ (* c c) (* d d)))
       (/ (+ b (/ a (/ 1.0 (/ c d)))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.5e-98) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.5e-76) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 3.6e+47) {
		tmp = (b * d) / ((c * c) + (d * d));
	} 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) :: tmp
    if (d <= (-4.5d-98)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 2.5d-76) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 3.6d+47) then
        tmp = (b * d) / ((c * c) + (d * d))
    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 tmp;
	if (d <= -4.5e-98) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.5e-76) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 3.6e+47) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else {
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.5e-98:
		tmp = (b + (a * (c / d))) / d
	elif d <= 2.5e-76:
		tmp = (a + (b * (d / c))) / c
	elif d <= 3.6e+47:
		tmp = (b * d) / ((c * c) + (d * d))
	else:
		tmp = (b + (a / (1.0 / (c / d)))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.5e-98)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 2.5e-76)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 3.6e+47)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	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)
	tmp = 0.0;
	if (d <= -4.5e-98)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 2.5e-76)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 3.6e+47)
		tmp = (b * d) / ((c * c) + (d * d));
	else
		tmp = (b + (a / (1.0 / (c / d)))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.5e-98], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.5e-76], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.6e+47], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 < -4.49999999999999997e-98

   1. Initial program 61.2%

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

   3. Taylor expanded in d around inf 71.2%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -4.49999999999999997e-98 < d < 2.4999999999999999e-76

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. associate-/l* 87.9%

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

      3. fma-define 87.9%

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

   5. Simplified 87.9%

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

      1. fma-undefine 87.9%

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

   7. Applied egg-rr 87.9%

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

   if 2.4999999999999999e-76 < d < 3.60000000000000008e47

   1. Initial program 84.4%

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

   3. Taylor expanded in a around 0 77.2%

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

      1. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   if 3.60000000000000008e47 < d

   1. Initial program 31.4%

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

   3. Taylor expanded in d around inf 78.3%

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

      1. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. clear-num 85.6%

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

      2. inv-pow 85.6%

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

   9. Applied egg-rr 85.6%

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

       1. unpow-1 85.6%

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

   11. Simplified 85.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{1}{\frac{c}{d}}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+48}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.5e-98)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 1.55e-77)
     (/ (+ a (* b (/ d c))) c)
     (if (<= d 1e+48)
       (/ (* b d) (+ (* c c) (* d d)))
       (/ (+ b (/ a (/ d c))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.5e-98) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.55e-77) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1e+48) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else {
		tmp = (b + (a / (d / 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 (d <= (-4.5d-98)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 1.55d-77) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 1d+48) then
        tmp = (b * d) / ((c * c) + (d * d))
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.5e-98) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.55e-77) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1e+48) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.5e-98:
		tmp = (b + (a * (c / d))) / d
	elif d <= 1.55e-77:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1e+48:
		tmp = (b * d) / ((c * c) + (d * d))
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.5e-98)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 1.55e-77)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1e+48)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4.5e-98)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 1.55e-77)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1e+48)
		tmp = (b * d) / ((c * c) + (d * d));
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.5e-98], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.55e-77], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1e+48], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.49999999999999997e-98

   1. Initial program 61.2%

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

   3. Taylor expanded in d around inf 71.2%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -4.49999999999999997e-98 < d < 1.55000000000000004e-77

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf 87.9%

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

      1. +-commutative 87.9%

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

      2. associate-/l* 87.9%

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

      3. fma-define 87.9%

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

   5. Simplified 87.9%

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

      1. fma-undefine 87.9%

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

   7. Applied egg-rr 87.9%

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

   if 1.55000000000000004e-77 < d < 1.00000000000000004e48

   1. Initial program 84.4%

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

   3. Taylor expanded in a around 0 77.2%

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

      1. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   if 1.00000000000000004e48 < d

   1. Initial program 31.4%

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

   3. Taylor expanded in d around inf 78.3%

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

      1. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+48}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98} \lor \neg \left(d \leq 6.8 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.5e-98) (not (<= d 6.8e+50)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.5e-98) || !(d <= 6.8e+50)) {
		tmp = (b + (a * (c / d))) / 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 <= (-4.5d-98)) .or. (.not. (d <= 6.8d+50))) then
        tmp = (b + (a * (c / d))) / 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 <= -4.5e-98) || !(d <= 6.8e+50)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.5e-98) or not (d <= 6.8e+50):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.5e-98) || !(d <= 6.8e+50))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.5e-98) || ~((d <= 6.8e+50)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.5e-98], N[Not[LessEqual[d, 6.8e+50]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.49999999999999997e-98 or 6.7999999999999997e50 < d

   1. Initial program 49.9%

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

   3. Taylor expanded in d around inf 73.9%

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

      1. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

   if -4.49999999999999997e-98 < d < 6.7999999999999997e50

   1. Initial program 76.2%

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

   3. Taylor expanded in c around inf 81.8%

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

      1. *-commutative 81.8%

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

   5. Simplified 81.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-98} \lor \neg \left(d \leq 6.8 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-15} \lor \neg \left(d \leq 3.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5e-15) (not (<= d 3.7e+54)))
   (/ b d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5e-15) || !(d <= 3.7e+54)) {
		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 <= (-5d-15)) .or. (.not. (d <= 3.7d+54))) 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 <= -5e-15) || !(d <= 3.7e+54)) {
		tmp = b / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5e-15) or not (d <= 3.7e+54):
		tmp = b / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5e-15) || !(d <= 3.7e+54))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5e-15) || ~((d <= 3.7e+54)))
		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, -5e-15], N[Not[LessEqual[d, 3.7e+54]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.99999999999999999e-15 or 3.7000000000000002e54 < d

   1. Initial program 41.6%

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

   3. Taylor expanded in c around 0 70.4%

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

   if -4.99999999999999999e-15 < d < 3.7000000000000002e54

   1. Initial program 78.7%

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

   3. Taylor expanded in c around inf 76.7%

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

      1. *-commutative 76.7%

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

   5. Simplified 76.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{-15} \lor \neg \left(d \leq 3.7 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.3e-99)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 5.6e+50) (/ (+ a (* b (/ d c))) c) (/ (+ b (/ a (/ d c))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.3e-99) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 5.6e+50) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / 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 (d <= (-3.3d-99)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 5.6d+50) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.3e-99) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 5.6e+50) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.3e-99:
		tmp = (b + (a * (c / d))) / d
	elif d <= 5.6e+50:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.3e-99)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 5.6e+50)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.3e-99)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 5.6e+50)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.3e-99], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 5.6e+50], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.29999999999999986e-99

   1. Initial program 61.2%

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

   3. Taylor expanded in d around inf 71.2%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -3.29999999999999986e-99 < d < 5.5999999999999996e50

   1. Initial program 76.2%

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

   3. Taylor expanded in c around inf 81.8%

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

      1. +-commutative 81.8%

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

      2. associate-/l* 81.8%

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

      3. fma-define 81.8%

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

   5. Simplified 81.8%

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

      1. fma-undefine 81.8%

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

   7. Applied egg-rr 81.8%

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

   if 5.5999999999999996e50 < d

   1. Initial program 31.4%

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

   3. Taylor expanded in d around inf 78.3%

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

      1. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+50}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7.8e-99)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 2.6e+47) (/ (+ a (/ (* b d) c)) c) (/ (+ b (/ a (/ d c))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7.8e-99) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.6e+47) {
		tmp = (a + ((b * d) / c)) / c;
	} else {
		tmp = (b + (a / (d / 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 (d <= (-7.8d-99)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 2.6d+47) then
        tmp = (a + ((b * d) / c)) / c
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7.8e-99) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.6e+47) {
		tmp = (a + ((b * d) / c)) / c;
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -7.8e-99:
		tmp = (b + (a * (c / d))) / d
	elif d <= 2.6e+47:
		tmp = (a + ((b * d) / c)) / c
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7.8e-99)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 2.6e+47)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -7.8e-99)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 2.6e+47)
		tmp = (a + ((b * d) / c)) / c;
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7.8e-99], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.6e+47], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.79999999999999975e-99

   1. Initial program 61.2%

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

   3. Taylor expanded in d around inf 71.2%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -7.79999999999999975e-99 < d < 2.60000000000000003e47

   1. Initial program 76.2%

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

   3. Taylor expanded in c around inf 81.8%

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

      1. *-commutative 81.8%

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

   5. Simplified 81.8%

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

   if 2.60000000000000003e47 < d

   1. Initial program 31.4%

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

   3. Taylor expanded in d around inf 78.3%

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

      1. associate-/l* 85.6%

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

   5. Simplified 85.6%

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

      1. clear-num 85.6%

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

      2. un-div-inv 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-99}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+47}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-99} \lor \neg \left(d \leq 3.45 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.45e-99) (not (<= d 3.45e-71))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.45e-99) || !(d <= 3.45e-71)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.45d-99)) .or. (.not. (d <= 3.45d-71))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.45e-99) || !(d <= 3.45e-71)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.45e-99) or not (d <= 3.45e-71):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.45e-99) || !(d <= 3.45e-71))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.45e-99) || ~((d <= 3.45e-71)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.45e-99], N[Not[LessEqual[d, 3.45e-71]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.44999999999999993e-99 or 3.4500000000000002e-71 < d

   1. Initial program 54.8%

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

   3. Taylor expanded in c around 0 62.8%

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

   if -1.44999999999999993e-99 < d < 3.4500000000000002e-71

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf 65.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{-99} \lor \neg \left(d \leq 3.45 \cdot 10^{-71}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.1% 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 62.9%

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

3. Taylor expanded in c around inf 37.7%

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

Developer Target 1: 99.4% 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 2024191 
(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))))