Complex division, real part

Percentage Accurate: 61.7% → 81.5%

Time: 8.9s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* c (/ a d))) d)))
   (if (<= d -7e+78)
     t_0
     (if (<= d -4.4e-110)
       (/ (+ (* c a) (* d b)) (+ (* c c) (* d d)))
       (if (<= d 2.2e+20) (/ (fma b (/ d c) a) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -7e+78) {
		tmp = t_0;
	} else if (d <= -4.4e-110) {
		tmp = ((c * a) + (d * b)) / ((c * c) + (d * d));
	} else if (d <= 2.2e+20) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(c * Float64(a / d))) / d)
	tmp = 0.0
	if (d <= -7e+78)
		tmp = t_0;
	elseif (d <= -4.4e-110)
		tmp = Float64(Float64(Float64(c * a) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.2e+20)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7e+78], t$95$0, If[LessEqual[d, -4.4e-110], N[(N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.2e+20], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.0000000000000003e78 or 2.2e20 < d

   1. Initial program 48.3%

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

      1. fma-define 48.3%

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

      2. fma-define 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in d around inf 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 85.5%

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

   7. Applied egg-rr 85.5%

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

   if -7.0000000000000003e78 < d < -4.3999999999999999e-110

   1. Initial program 90.8%

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

   if -4.3999999999999999e-110 < d < 2.2e20

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in c around inf 84.9%

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

      1. +-commutative 84.9%

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

      2. associate-/l* 84.9%

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

      3. fma-define 84.9%

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

   7. Simplified 84.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* c (/ a d))) d)))
   (if (<= d -7e+78)
     t_0
     (if (<= d -7.8e-110)
       (/ (+ (* c a) (* d b)) (+ (* c c) (* d d)))
       (if (<= d 1.05e+18) (/ (+ a (* b (/ d c))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -7e+78) {
		tmp = t_0;
	} else if (d <= -7.8e-110) {
		tmp = ((c * a) + (d * b)) / ((c * c) + (d * d));
	} else if (d <= 1.05e+18) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b + (c * (a / d))) / d
    if (d <= (-7d+78)) then
        tmp = t_0
    else if (d <= (-7.8d-110)) then
        tmp = ((c * a) + (d * b)) / ((c * c) + (d * d))
    else if (d <= 1.05d+18) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -7e+78) {
		tmp = t_0;
	} else if (d <= -7.8e-110) {
		tmp = ((c * a) + (d * b)) / ((c * c) + (d * d));
	} else if (d <= 1.05e+18) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (c * (a / d))) / d
	tmp = 0
	if d <= -7e+78:
		tmp = t_0
	elif d <= -7.8e-110:
		tmp = ((c * a) + (d * b)) / ((c * c) + (d * d))
	elif d <= 1.05e+18:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(c * Float64(a / d))) / d)
	tmp = 0.0
	if (d <= -7e+78)
		tmp = t_0;
	elseif (d <= -7.8e-110)
		tmp = Float64(Float64(Float64(c * a) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.05e+18)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (c * (a / d))) / d;
	tmp = 0.0;
	if (d <= -7e+78)
		tmp = t_0;
	elseif (d <= -7.8e-110)
		tmp = ((c * a) + (d * b)) / ((c * c) + (d * d));
	elseif (d <= 1.05e+18)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7e+78], t$95$0, If[LessEqual[d, -7.8e-110], N[(N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e+18], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.0000000000000003e78 or 1.05e18 < d

   1. Initial program 48.3%

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

      1. fma-define 48.3%

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

      2. fma-define 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in d around inf 80.5%

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

      1. *-commutative 80.5%

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

      2. associate-/l* 85.5%

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

   7. Applied egg-rr 85.5%

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

   if -7.0000000000000003e78 < d < -7.8e-110

   1. Initial program 90.8%

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

   if -7.8e-110 < d < 1.05e18

   1. Initial program 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

   3. Simplified 69.9%

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

      1. fma-define 69.9%

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

      2. fma-define 69.9%

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

      3. div-inv 69.9%

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

      4. fma-define 69.9%

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

      5. fma-define 69.9%

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

      6. add-sqr-sqrt 69.9%

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

      7. pow2 69.9%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)}^{2}}} \]

      8. fma-define 69.9%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      9. hypot-define 69.9%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around inf 84.9%

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

      1. associate-/l* 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+78}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -7.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.00031 \lor \neg \left(d \leq 2.35 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.00031) (not (<= d 2.35e+21)))
   (/ (+ b (* c (/ a d))) d)
   (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.00031) || !(d <= 2.35e+21)) {
		tmp = (b + (c * (a / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-0.00031d0)) .or. (.not. (d <= 2.35d+21))) then
        tmp = (b + (c * (a / d))) / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.00031) || !(d <= 2.35e+21)) {
		tmp = (b + (c * (a / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.00031) or not (d <= 2.35e+21):
		tmp = (b + (c * (a / d))) / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.00031) || !(d <= 2.35e+21))
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -0.00031) || ~((d <= 2.35e+21)))
		tmp = (b + (c * (a / d))) / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -0.00031], N[Not[LessEqual[d, 2.35e+21]], $MachinePrecision]], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.1e-4 or 2.35e21 < d

   1. Initial program 53.6%

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

      1. fma-define 53.6%

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

      2. fma-define 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in d around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-/l* 84.5%

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

   7. Applied egg-rr 84.5%

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

   if -3.1e-4 < d < 2.35e21

   1. Initial program 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

   3. Simplified 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

      3. div-inv 72.2%

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

      4. fma-define 72.2%

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

      5. fma-define 72.3%

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

      6. add-sqr-sqrt 72.2%

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

      7. pow2 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)}^{2}}} \]

      8. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      9. hypot-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around inf 81.2%

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

      1. associate-/l* 81.2%

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

   9. Simplified 81.2%

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

       1. clear-num 81.2%

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

       2. un-div-inv 81.2%

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

   11. Applied egg-rr 81.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.00031 \lor \neg \left(d \leq 2.35 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.112 \lor \neg \left(d \leq 5.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.112) (not (<= d 5.5e+20)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.112) || !(d <= 5.5e+20)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-0.112d0)) .or. (.not. (d <= 5.5d+20))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.112) || !(d <= 5.5e+20)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.112) or not (d <= 5.5e+20):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.112) || !(d <= 5.5e+20))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -0.112) || ~((d <= 5.5e+20)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -0.112], N[Not[LessEqual[d, 5.5e+20]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -0.112000000000000002 or 5.5e20 < d

   1. Initial program 53.6%

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

      1. fma-define 53.6%

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

      2. fma-define 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in d around inf 80.0%

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

      1. associate-/l* 82.1%

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

   7. Simplified 82.1%

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

   if -0.112000000000000002 < d < 5.5e20

   1. Initial program 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

   3. Simplified 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

      3. div-inv 72.2%

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

      4. fma-define 72.2%

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

      5. fma-define 72.3%

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

      6. add-sqr-sqrt 72.2%

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

      7. pow2 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)}^{2}}} \]

      8. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      9. hypot-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around inf 81.2%

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

      1. associate-/l* 81.2%

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

   9. Simplified 81.2%

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

       1. clear-num 81.2%

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

       2. un-div-inv 81.2%

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

   11. Applied egg-rr 81.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.112 \lor \neg \left(d \leq 5.5 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \lor \neg \left(d \leq 6 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.0) (not (<= d 6e+19))) (/ b d) (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.0) || !(d <= 6e+19)) {
		tmp = b / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-7.0d0)) .or. (.not. (d <= 6d+19))) then
        tmp = b / d
    else
        tmp = (a + (b / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.0) || !(d <= 6e+19)) {
		tmp = b / d;
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.0) or not (d <= 6e+19):
		tmp = b / d
	else:
		tmp = (a + (b / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.0) || !(d <= 6e+19))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.0) || ~((d <= 6e+19)))
		tmp = b / d;
	else
		tmp = (a + (b / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.0], N[Not[LessEqual[d, 6e+19]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7 or 6e19 < d

   1. Initial program 53.6%

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

      1. fma-define 53.6%

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

      2. fma-define 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in c around 0 68.6%

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

   if -7 < d < 6e19

   1. Initial program 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

   3. Simplified 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

      3. div-inv 72.2%

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

      4. fma-define 72.2%

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

      5. fma-define 72.3%

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

      6. add-sqr-sqrt 72.2%

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

      7. pow2 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)}^{2}}} \]

      8. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      9. hypot-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around inf 81.2%

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

      1. associate-/l* 81.2%

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

   9. Simplified 81.2%

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

       1. clear-num 81.2%

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

       2. un-div-inv 81.2%

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

   11. Applied egg-rr 81.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \lor \neg \left(d \leq 6 \cdot 10^{+19}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.46 \lor \neg \left(d \leq 1.1 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.46) (not (<= d 1.1e+20))) (/ b d) (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.46) || !(d <= 1.1e+20)) {
		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 <= (-0.46d0)) .or. (.not. (d <= 1.1d+20))) 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 <= -0.46) || !(d <= 1.1e+20)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.46) or not (d <= 1.1e+20):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.46) || !(d <= 1.1e+20))
		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 <= -0.46) || ~((d <= 1.1e+20)))
		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, -0.46], N[Not[LessEqual[d, 1.1e+20]], $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 < -0.46000000000000002 or 1.1e20 < d

   1. Initial program 53.6%

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

      1. fma-define 53.6%

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

      2. fma-define 53.6%

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

   3. Simplified 53.6%

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

   5. Taylor expanded in c around 0 68.6%

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

   if -0.46000000000000002 < d < 1.1e20

   1. Initial program 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

   3. Simplified 72.3%

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

      1. fma-define 72.3%

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

      2. fma-define 72.3%

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

      3. div-inv 72.2%

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

      4. fma-define 72.2%

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

      5. fma-define 72.3%

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

      6. add-sqr-sqrt 72.2%

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

      7. pow2 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)}^{2}}} \]

      8. fma-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      9. hypot-define 72.2%

         \[\leadsto \mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around inf 81.2%

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

      1. associate-/l* 81.2%

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

   9. Simplified 81.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.46 \lor \neg \left(d \leq 1.1 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{-24}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.3e-16) (/ a c) (if (<= c 6.6e-24) (/ b d) (/ 1.0 (/ c a)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.3e-16) {
		tmp = a / c;
	} else if (c <= 6.6e-24) {
		tmp = b / d;
	} else {
		tmp = 1.0 / (c / a);
	}
	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 <= (-4.3d-16)) then
        tmp = a / c
    else if (c <= 6.6d-24) then
        tmp = b / d
    else
        tmp = 1.0d0 / (c / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.3e-16) {
		tmp = a / c;
	} else if (c <= 6.6e-24) {
		tmp = b / d;
	} else {
		tmp = 1.0 / (c / a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.3e-16:
		tmp = a / c
	elif c <= 6.6e-24:
		tmp = b / d
	else:
		tmp = 1.0 / (c / a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.3e-16)
		tmp = Float64(a / c);
	elseif (c <= 6.6e-24)
		tmp = Float64(b / d);
	else
		tmp = Float64(1.0 / Float64(c / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.3e-16)
		tmp = a / c;
	elseif (c <= 6.6e-24)
		tmp = b / d;
	else
		tmp = 1.0 / (c / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.3e-16], N[(a / c), $MachinePrecision], If[LessEqual[c, 6.6e-24], N[(b / d), $MachinePrecision], N[(1.0 / N[(c / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.2999999999999999e-16

   1. Initial program 55.9%

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

      1. fma-define 55.9%

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

      2. fma-define 55.9%

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

   3. Simplified 55.9%

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

   5. Taylor expanded in c around inf 73.6%

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

   if -4.2999999999999999e-16 < c < 6.59999999999999968e-24

   1. Initial program 78.3%

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

      1. fma-define 78.3%

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

      2. fma-define 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in c around 0 70.7%

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

   if 6.59999999999999968e-24 < c

   1. Initial program 45.6%

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

      1. fma-define 45.6%

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

      2. fma-define 45.6%

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

   3. Simplified 45.6%

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

   5. Taylor expanded in c around inf 56.2%

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

      1. clear-num 56.8%

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

      2. inv-pow 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. unpow-1 56.8%

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

   9. Simplified 56.8%

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

Alternative 8: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-16} \lor \neg \left(c \leq 6.6 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.5e-16) (not (<= c 6.6e-24))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e-16) || !(c <= 6.6e-24)) {
		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 <= (-4.5d-16)) .or. (.not. (c <= 6.6d-24))) 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 <= -4.5e-16) || !(c <= 6.6e-24)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.5e-16) or not (c <= 6.6e-24):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.5e-16) || !(c <= 6.6e-24))
		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 <= -4.5e-16) || ~((c <= 6.6e-24)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.5e-16], N[Not[LessEqual[c, 6.6e-24]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.5000000000000002e-16 or 6.59999999999999968e-24 < c

   1. Initial program 51.0%

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

      1. fma-define 51.0%

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

      2. fma-define 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in c around inf 65.3%

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

   if -4.5000000000000002e-16 < c < 6.59999999999999968e-24

   1. Initial program 78.3%

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

      1. fma-define 78.3%

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

      2. fma-define 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in c around 0 70.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-16} \lor \neg \left(c \leq 6.6 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 63.9%

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

   1. fma-define 63.9%

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

   2. fma-define 63.9%

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

3. Simplified 63.9%

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

5. Taylor expanded in c around inf 40.6%

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

Developer Target 1: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024129 
(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))))