Complex division, real part

Percentage Accurate: 60.8% → 80.9%

Time: 10.4s

Alternatives: 8

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: 60.8% 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: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -5 \cdot 10^{+130}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -5e+130)
     (/ (+ a (* d (/ b c))) c)
     (if (<= c -2.3e-86)
       t_0
       (if (<= c 2.4e-141)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 6.8e-68)
           t_0
           (if (<= c 2.3e+30)
             (/ (+ b (* c (/ a d))) d)
             (if (<= c 2.7e+163)
               (* (/ c (hypot d c)) (/ a (hypot d c)))
               (/ (+ a (* b (/ d c))) c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -5e+130) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= -2.3e-86) {
		tmp = t_0;
	} else if (c <= 2.4e-141) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 6.8e-68) {
		tmp = t_0;
	} else if (c <= 2.3e+30) {
		tmp = (b + (c * (a / d))) / d;
	} else if (c <= 2.7e+163) {
		tmp = (c / hypot(d, c)) * (a / hypot(d, c));
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -5e+130) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= -2.3e-86) {
		tmp = t_0;
	} else if (c <= 2.4e-141) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 6.8e-68) {
		tmp = t_0;
	} else if (c <= 2.3e+30) {
		tmp = (b + (c * (a / d))) / d;
	} else if (c <= 2.7e+163) {
		tmp = (c / Math.hypot(d, c)) * (a / Math.hypot(d, c));
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -5e+130:
		tmp = (a + (d * (b / c))) / c
	elif c <= -2.3e-86:
		tmp = t_0
	elif c <= 2.4e-141:
		tmp = (b + (a * (c / d))) / d
	elif c <= 6.8e-68:
		tmp = t_0
	elif c <= 2.3e+30:
		tmp = (b + (c * (a / d))) / d
	elif c <= 2.7e+163:
		tmp = (c / math.hypot(d, c)) * (a / math.hypot(d, c))
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -5e+130)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	elseif (c <= -2.3e-86)
		tmp = t_0;
	elseif (c <= 2.4e-141)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 6.8e-68)
		tmp = t_0;
	elseif (c <= 2.3e+30)
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	elseif (c <= 2.7e+163)
		tmp = Float64(Float64(c / hypot(d, c)) * Float64(a / hypot(d, c)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -5e+130)
		tmp = (a + (d * (b / c))) / c;
	elseif (c <= -2.3e-86)
		tmp = t_0;
	elseif (c <= 2.4e-141)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 6.8e-68)
		tmp = t_0;
	elseif (c <= 2.3e+30)
		tmp = (b + (c * (a / d))) / d;
	elseif (c <= 2.7e+163)
		tmp = (c / hypot(d, c)) * (a / hypot(d, c));
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5e+130], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -2.3e-86], t$95$0, If[LessEqual[c, 2.4e-141], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.8e-68], t$95$0, If[LessEqual[c, 2.3e+30], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.7e+163], N[(N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -4.9999999999999996e130

   1. Initial program 36.2%

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

   3. Taylor expanded in c around inf 73.8%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. associate-/r/ 85.7%

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

   9. Applied egg-rr 85.7%

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

   if -4.9999999999999996e130 < c < -2.29999999999999996e-86 or 2.4000000000000001e-141 < c < 6.80000000000000037e-68

   1. Initial program 88.9%

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

   if -2.29999999999999996e-86 < c < 2.4000000000000001e-141

   1. Initial program 74.2%

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

   3. Taylor expanded in d around inf 95.9%

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

      1. associate-/l* 96.3%

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

   5. Simplified 96.3%

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

   if 6.80000000000000037e-68 < c < 2.3e30

   1. Initial program 61.1%

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

   3. Taylor expanded in d around inf 83.9%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

      1. clear-num 79.0%

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

      2. un-div-inv 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. associate-/r/ 84.0%

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

   9. Applied egg-rr 84.0%

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

   if 2.3e30 < c < 2.69999999999999999e163

   1. Initial program 60.8%

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

   3. Taylor expanded in a around inf 54.1%

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

      1. *-commutative 54.1%

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

   5. Simplified 54.1%

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

      1. *-un-lft-identity 54.1%

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

      2. add-sqr-sqrt 54.1%

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

      3. hypot-undefine 54.1%

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

      4. hypot-undefine 54.1%

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

      5. times-frac 62.1%

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

      6. hypot-undefine 54.1%

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

      7. +-commutative 54.1%

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

      8. hypot-define 62.1%

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

      9. hypot-undefine 54.1%

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

      10. +-commutative 54.1%

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

      11. hypot-define 62.1%

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

   7. Applied egg-rr 62.1%

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

      1. associate-*l/ 62.2%

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

      2. *-lft-identity 62.2%

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

      3. associate-/l* 79.7%

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

      4. associate-*l/ 79.7%

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

   9. Simplified 79.7%

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

   if 2.69999999999999999e163 < c

   1. Initial program 27.5%

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

   3. Taylor expanded in c around inf 84.1%

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

      1. associate-/l* 90.5%

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

   5. Simplified 90.5%

      \[\leadsto \color{blue}{\frac{a + b \cdot \frac{d}{c}}{c}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+130}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-141}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-68}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+292)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+292) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+292)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+292], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 4.9999999999999996e292

   1. Initial program 78.2%

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

      1. *-un-lft-identity 78.2%

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

      2. add-sqr-sqrt 78.1%

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

      3. times-frac 78.1%

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

      4. hypot-define 78.1%

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

      5. fma-define 78.1%

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

      6. hypot-define 95.3%

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

   4. Applied egg-rr 95.3%

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

   if 4.9999999999999996e292 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 20.4%

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

   3. Taylor expanded in c around inf 54.6%

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

      1. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. clear-num 64.6%

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

      2. un-div-inv 64.7%

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

   7. Applied egg-rr 64.7%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.62 \cdot 10^{-136}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.88 \cdot 10^{+31}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.9e+130)
     (/ (+ a (* d (/ b c))) c)
     (if (<= c -2e-86)
       t_0
       (if (<= c 1.62e-136)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 1.7e-66)
           t_0
           (if (<= c 1.88e+31)
             (/ (+ b (* c (/ a d))) d)
             (/ (+ a (* b (/ d c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.9e+130) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= -2e-86) {
		tmp = t_0;
	} else if (c <= 1.62e-136) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.7e-66) {
		tmp = t_0;
	} else if (c <= 1.88e+31) {
		tmp = (b + (c * (a / 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-1.9d+130)) then
        tmp = (a + (d * (b / c))) / c
    else if (c <= (-2d-86)) then
        tmp = t_0
    else if (c <= 1.62d-136) then
        tmp = (b + (a * (c / d))) / d
    else if (c <= 1.7d-66) then
        tmp = t_0
    else if (c <= 1.88d+31) then
        tmp = (b + (c * (a / 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 t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.9e+130) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= -2e-86) {
		tmp = t_0;
	} else if (c <= 1.62e-136) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.7e-66) {
		tmp = t_0;
	} else if (c <= 1.88e+31) {
		tmp = (b + (c * (a / d))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.9e+130:
		tmp = (a + (d * (b / c))) / c
	elif c <= -2e-86:
		tmp = t_0
	elif c <= 1.62e-136:
		tmp = (b + (a * (c / d))) / d
	elif c <= 1.7e-66:
		tmp = t_0
	elif c <= 1.88e+31:
		tmp = (b + (c * (a / d))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.9e+130)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	elseif (c <= -2e-86)
		tmp = t_0;
	elseif (c <= 1.62e-136)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 1.7e-66)
		tmp = t_0;
	elseif (c <= 1.88e+31)
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / 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)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.9e+130)
		tmp = (a + (d * (b / c))) / c;
	elseif (c <= -2e-86)
		tmp = t_0;
	elseif (c <= 1.62e-136)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 1.7e-66)
		tmp = t_0;
	elseif (c <= 1.88e+31)
		tmp = (b + (c * (a / d))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.9e+130], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -2e-86], t$95$0, If[LessEqual[c, 1.62e-136], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.7e-66], t$95$0, If[LessEqual[c, 1.88e+31], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.9000000000000001e130

   1. Initial program 36.2%

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

   3. Taylor expanded in c around inf 73.8%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. un-div-inv 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. associate-/r/ 85.7%

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

   9. Applied egg-rr 85.7%

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

   if -1.9000000000000001e130 < c < -2.00000000000000017e-86 or 1.6200000000000001e-136 < c < 1.69999999999999999e-66

   1. Initial program 88.9%

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

   if -2.00000000000000017e-86 < c < 1.6200000000000001e-136

   1. Initial program 74.2%

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

   3. Taylor expanded in d around inf 95.9%

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

      1. associate-/l* 96.3%

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

   5. Simplified 96.3%

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

   if 1.69999999999999999e-66 < c < 1.87999999999999996e31

   1. Initial program 61.1%

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

   3. Taylor expanded in d around inf 83.9%

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

      1. associate-/l* 83.8%

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

   5. Simplified 83.8%

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

      1. clear-num 79.0%

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

      2. un-div-inv 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. associate-/r/ 84.0%

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

   9. Applied egg-rr 84.0%

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

   if 1.87999999999999996e31 < c

   1. Initial program 42.0%

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

   3. Taylor expanded in c around inf 74.0%

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

      1. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+130}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-86}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.62 \cdot 10^{-136}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-66}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.88 \cdot 10^{+31}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-80} \lor \neg \left(c \leq 1.08 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.5e-80) (not (<= c 1.08e+30)))
   (/ (+ a (* b (/ d c))) c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e-80) || !(c <= 1.08e+30)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-7.5d-80)) .or. (.not. (c <= 1.08d+30))) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e-80) || !(c <= 1.08e+30)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.5e-80) or not (c <= 1.08e+30):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.5e-80) || !(c <= 1.08e+30))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -7.5e-80) || ~((c <= 1.08e+30)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.5e-80], N[Not[LessEqual[c, 1.08e+30]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.49999999999999999e-80 or 1.08e30 < c

   1. Initial program 52.1%

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

   3. Taylor expanded in c around inf 69.4%

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

      1. associate-/l* 73.6%

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

   5. Simplified 73.6%

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

   if -7.49999999999999999e-80 < c < 1.08e30

   1. Initial program 75.7%

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

   3. Taylor expanded in c around 0 76.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-80} \lor \neg \left(c \leq 1.08 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.4e-80)
   (/ (+ a (* d (/ b c))) c)
   (if (<= c 1.75e+30) (/ b d) (/ (+ a (* b (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.4e-80) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 1.75e+30) {
		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 (c <= (-6.4d-80)) then
        tmp = (a + (d * (b / c))) / c
    else if (c <= 1.75d+30) 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 (c <= -6.4e-80) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 1.75e+30) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.4e-80:
		tmp = (a + (d * (b / c))) / c
	elif c <= 1.75e+30:
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.4e-80)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	elseif (c <= 1.75e+30)
		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 (c <= -6.4e-80)
		tmp = (a + (d * (b / c))) / c;
	elseif (c <= 1.75e+30)
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.4e-80], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.75e+30], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.3999999999999998e-80

   1. Initial program 60.6%

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

   3. Taylor expanded in c around inf 65.6%

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

      1. associate-/l* 70.3%

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

   5. Simplified 70.3%

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

      1. clear-num 70.3%

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

      2. un-div-inv 70.3%

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

   7. Applied egg-rr 70.3%

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

      1. associate-/r/ 71.6%

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

   9. Applied egg-rr 71.6%

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

   if -6.3999999999999998e-80 < c < 1.75000000000000011e30

   1. Initial program 75.7%

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

   3. Taylor expanded in c around 0 76.4%

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

   if 1.75000000000000011e30 < c

   1. Initial program 42.0%

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

   3. Taylor expanded in c around inf 74.0%

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

      1. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e+33)
   (/ (+ a (* d (/ b c))) c)
   (if (<= c 2.6e+30) (/ (+ b (* a (/ c d))) d) (/ (+ a (* b (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.8e+33) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 2.6e+30) {
		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 (c <= (-6.8d+33)) then
        tmp = (a + (d * (b / c))) / c
    else if (c <= 2.6d+30) 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 (c <= -6.8e+33) {
		tmp = (a + (d * (b / c))) / c;
	} else if (c <= 2.6e+30) {
		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 c <= -6.8e+33:
		tmp = (a + (d * (b / c))) / c
	elif c <= 2.6e+30:
		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 (c <= -6.8e+33)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	elseif (c <= 2.6e+30)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / 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 (c <= -6.8e+33)
		tmp = (a + (d * (b / c))) / c;
	elseif (c <= 2.6e+30)
		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[LessEqual[c, -6.8e+33], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.6e+30], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.7999999999999999e33

   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 inf 71.3%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

      1. clear-num 77.3%

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

      2. un-div-inv 77.4%

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

   7. Applied egg-rr 77.4%

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

      1. associate-/r/ 79.0%

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

   9. Applied egg-rr 79.0%

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

   if -6.7999999999999999e33 < c < 2.59999999999999988e30

   1. Initial program 76.4%

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

   3. Taylor expanded in d around inf 86.6%

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

      1. associate-/l* 86.8%

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

   5. Simplified 86.8%

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

   if 2.59999999999999988e30 < c

   1. Initial program 42.0%

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

   3. Taylor expanded in c around inf 74.0%

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

      1. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+33}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-80} \lor \neg \left(c \leq 3.55 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.1e-80) (not (<= c 3.55e+31))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.1e-80) || !(c <= 3.55e+31)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.1d-80)) .or. (.not. (c <= 3.55d+31))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.1e-80) || !(c <= 3.55e+31)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.1e-80) or not (c <= 3.55e+31):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.1e-80) || !(c <= 3.55e+31))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.1e-80) || ~((c <= 3.55e+31)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.1e-80], N[Not[LessEqual[c, 3.55e+31]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.10000000000000001e-80 or 3.5499999999999998e31 < c

   1. Initial program 52.1%

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

   3. Taylor expanded in c around inf 61.7%

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

   if -2.10000000000000001e-80 < c < 3.5499999999999998e31

   1. Initial program 75.7%

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

   3. Taylor expanded in c around 0 76.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-80} \lor \neg \left(c \leq 3.55 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in c around inf 40.0%

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

4. Final simplification 40.0%

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

Developer target: 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 2024080 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

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