Complex division, real part

Percentage Accurate: 61.6% → 82.9%

Time: 10.9s

Alternatives: 10

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: 61.6% 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: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\left(b + \frac{a}{\frac{d}{c}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.65 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.5e+137)
     (* (+ b (/ a (/ d c))) (/ -1.0 (hypot c d)))
     (if (<= d -2.65e-107)
       t_0
       (if (<= d 2.3e-77)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 4.1e+75)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (/ c (/ d a))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.5e+137) {
		tmp = (b + (a / (d / c))) * (-1.0 / hypot(c, d));
	} else if (d <= -2.65e-107) {
		tmp = t_0;
	} else if (d <= 2.3e-77) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4.1e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	}
	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 (d <= -1.5e+137) {
		tmp = (b + (a / (d / c))) * (-1.0 / Math.hypot(c, d));
	} else if (d <= -2.65e-107) {
		tmp = t_0;
	} else if (d <= 2.3e-77) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4.1e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (c / (d / a)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.5e+137:
		tmp = (b + (a / (d / c))) * (-1.0 / math.hypot(c, d))
	elif d <= -2.65e-107:
		tmp = t_0
	elif d <= 2.3e-77:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 4.1e+75:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (c / (d / a)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.5e+137)
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -2.65e-107)
		tmp = t_0;
	elseif (d <= 2.3e-77)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 4.1e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(c / Float64(d / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.5e+137)
		tmp = (b + (a / (d / c))) * (-1.0 / hypot(c, d));
	elseif (d <= -2.65e-107)
		tmp = t_0;
	elseif (d <= 2.3e-77)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 4.1e+75)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.5e+137], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.65e-107], t$95$0, If[LessEqual[d, 2.3e-77], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e+75], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.5e137

   1. Initial program 36.1%

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

      1. *-un-lft-identity 36.1%

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

      2. add-sqr-sqrt 36.1%

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

      3. times-frac 36.1%

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

      4. hypot-def 36.1%

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

      5. fma-def 36.1%

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

      6. hypot-def 56.6%

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

   4. Applied egg-rr 56.6%

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

   5. Taylor expanded in d around -inf 75.9%

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

      1. neg-mul-1 75.9%

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

      2. +-commutative 75.9%

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

      3. unsub-neg 75.9%

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

      4. mul-1-neg 75.9%

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

      5. associate-/l* 80.6%

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

      6. distribute-neg-frac 80.6%

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

   7. Simplified 80.6%

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

   if -1.5e137 < d < -2.65e-107 or 2.29999999999999999e-77 < d < 4.0999999999999998e75

   1. Initial program 84.1%

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

   if -2.65e-107 < d < 2.29999999999999999e-77

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. clear-num 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. associate-*l/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-/r* 92.8%

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

      4. *-commutative 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 4.0999999999999998e75 < d

   1. Initial program 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. add-sqr-sqrt 22.8%

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

      3. times-frac 22.9%

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

      4. hypot-def 22.9%

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

      5. fma-def 22.9%

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

      6. hypot-def 44.3%

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

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in c around 0 81.4%

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

      1. associate-/l* 90.2%

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

   7. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. associate-/r/ 90.2%

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

      3. clear-num 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-*l/ 81.4%

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

       2. associate-/l* 90.3%

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

   11. Applied egg-rr 90.3%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \color{blue}{\frac{c}{\frac{d}{a}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\left(b + \frac{a}{\frac{d}{c}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.65 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 2e+272)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 2e+272) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	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))) <= 2e+272)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	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], 2e+272], 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[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 2.0000000000000001e272

   1. Initial program 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.8%

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

      4. hypot-def 78.8%

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

      5. fma-def 78.8%

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

      6. hypot-def 96.2%

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

   4. Applied egg-rr 96.2%

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

   if 2.0000000000000001e272 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 8.5%

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

      1. *-un-lft-identity 8.5%

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

      2. add-sqr-sqrt 8.5%

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

      3. times-frac 8.5%

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

      4. hypot-def 8.5%

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

      5. fma-def 8.5%

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

      6. hypot-def 10.8%

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

   4. Applied egg-rr 10.8%

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

   5. Taylor expanded in c around 0 32.1%

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

      1. associate-/l* 38.2%

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

   7. Simplified 38.2%

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

      1. clear-num 38.2%

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

      2. associate-/r/ 38.2%

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

      3. clear-num 38.2%

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

   9. Applied egg-rr 38.2%

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

   10. Taylor expanded in c around 0 60.0%

       \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c}{d} \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.85e+137)
     (* (/ 1.0 d) (+ b (/ a (/ d c))))
     (if (<= d -1.65e-105)
       t_0
       (if (<= d 4.5e-78)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 4e+75) t_0 (* (/ 1.0 (hypot c d)) (+ b (* a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.85e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -1.65e-105) {
		tmp = t_0;
	} else if (d <= 4.5e-78) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	}
	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 (d <= -1.85e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -1.65e-105) {
		tmp = t_0;
	} else if (d <= 4.5e-78) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.85e+137:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	elif d <= -1.65e-105:
		tmp = t_0
	elif d <= 4.5e-78:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 4e+75:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.85e+137)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (d <= -1.65e-105)
		tmp = t_0;
	elseif (d <= 4.5e-78)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 4e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.85e+137)
		tmp = (1.0 / d) * (b + (a / (d / c)));
	elseif (d <= -1.65e-105)
		tmp = t_0;
	elseif (d <= 4.5e-78)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 4e+75)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.85e+137], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.65e-105], t$95$0, If[LessEqual[d, 4.5e-78], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e+75], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.8500000000000001e137

   1. Initial program 36.1%

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

      1. *-un-lft-identity 36.1%

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

      2. add-sqr-sqrt 36.1%

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

      3. times-frac 36.1%

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

      4. hypot-def 36.1%

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

      5. fma-def 36.1%

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

      6. hypot-def 56.6%

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

   4. Applied egg-rr 56.6%

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

   5. Taylor expanded in c around 0 28.3%

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

      1. associate-/l* 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in c around 0 77.7%

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

   if -1.8500000000000001e137 < d < -1.6499999999999999e-105 or 4.5e-78 < d < 3.99999999999999971e75

   1. Initial program 84.1%

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

   if -1.6499999999999999e-105 < d < 4.5e-78

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. clear-num 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. associate-*l/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-/r* 92.8%

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

      4. *-commutative 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 3.99999999999999971e75 < d

   1. Initial program 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. add-sqr-sqrt 22.8%

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

      3. times-frac 22.9%

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

      4. hypot-def 22.9%

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

      5. fma-def 22.9%

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

      6. hypot-def 44.3%

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

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in c around 0 81.4%

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

      1. associate-/l* 90.2%

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

   7. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. associate-/r/ 90.2%

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

      3. clear-num 90.3%

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

   9. Applied egg-rr 90.3%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \color{blue}{\frac{c}{d} \cdot a}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.66 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.66e+137)
     (* (/ 1.0 d) (+ b (/ a (/ d c))))
     (if (<= d -5.1e-107)
       t_0
       (if (<= d 2.5e-77)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 4.2e+75)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (/ c (/ d a))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.66e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -5.1e-107) {
		tmp = t_0;
	} else if (d <= 2.5e-77) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4.2e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	}
	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 (d <= -1.66e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -5.1e-107) {
		tmp = t_0;
	} else if (d <= 2.5e-77) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 4.2e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (c / (d / a)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.66e+137:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	elif d <= -5.1e-107:
		tmp = t_0
	elif d <= 2.5e-77:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 4.2e+75:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (c / (d / a)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.66e+137)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (d <= -5.1e-107)
		tmp = t_0;
	elseif (d <= 2.5e-77)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 4.2e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(c / Float64(d / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.66e+137)
		tmp = (1.0 / d) * (b + (a / (d / c)));
	elseif (d <= -5.1e-107)
		tmp = t_0;
	elseif (d <= 2.5e-77)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 4.2e+75)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.66e+137], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.1e-107], t$95$0, If[LessEqual[d, 2.5e-77], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+75], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.65999999999999991e137

   1. Initial program 36.1%

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

      1. *-un-lft-identity 36.1%

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

      2. add-sqr-sqrt 36.1%

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

      3. times-frac 36.1%

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

      4. hypot-def 36.1%

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

      5. fma-def 36.1%

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

      6. hypot-def 56.6%

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

   4. Applied egg-rr 56.6%

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

   5. Taylor expanded in c around 0 28.3%

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

      1. associate-/l* 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in c around 0 77.7%

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

   if -1.65999999999999991e137 < d < -5.1000000000000002e-107 or 2.49999999999999982e-77 < d < 4.19999999999999997e75

   1. Initial program 84.1%

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

   if -5.1000000000000002e-107 < d < 2.49999999999999982e-77

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. clear-num 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. associate-*l/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-/r* 92.8%

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

      4. *-commutative 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 4.19999999999999997e75 < d

   1. Initial program 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. add-sqr-sqrt 22.8%

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

      3. times-frac 22.9%

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

      4. hypot-def 22.9%

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

      5. fma-def 22.9%

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

      6. hypot-def 44.3%

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

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in c around 0 81.4%

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

      1. associate-/l* 90.2%

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

   7. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. associate-/r/ 90.2%

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

      3. clear-num 90.3%

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

   9. Applied egg-rr 90.3%

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

       1. associate-*l/ 81.4%

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

       2. associate-/l* 90.3%

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

   11. Applied egg-rr 90.3%

       \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \color{blue}{\frac{c}{\frac{d}{a}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.66 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.5e+137)
     (* (/ 1.0 d) (+ b (/ a (/ d c))))
     (if (<= d -8.5e-106)
       t_0
       (if (<= d 3.2e-78)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 3.7e+75) t_0 (* (/ 1.0 d) (+ b (* a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.5e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -8.5e-106) {
		tmp = t_0;
	} else if (d <= 3.2e-78) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 3.7e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / d) * (b + (a * (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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-1.5d+137)) then
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    else if (d <= (-8.5d-106)) then
        tmp = t_0
    else if (d <= 3.2d-78) then
        tmp = (a / c) + (((b * d) / c) / c)
    else if (d <= 3.7d+75) then
        tmp = t_0
    else
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.5e+137) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= -8.5e-106) {
		tmp = t_0;
	} else if (d <= 3.2e-78) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 3.7e+75) {
		tmp = t_0;
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.5e+137:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	elif d <= -8.5e-106:
		tmp = t_0
	elif d <= 3.2e-78:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 3.7e+75:
		tmp = t_0
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.5e+137)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (d <= -8.5e-106)
		tmp = t_0;
	elseif (d <= 3.2e-78)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 3.7e+75)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.5e+137)
		tmp = (1.0 / d) * (b + (a / (d / c)));
	elseif (d <= -8.5e-106)
		tmp = t_0;
	elseif (d <= 3.2e-78)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 3.7e+75)
		tmp = t_0;
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.5e+137], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8.5e-106], t$95$0, If[LessEqual[d, 3.2e-78], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.7e+75], t$95$0, N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.5e137

   1. Initial program 36.1%

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

      1. *-un-lft-identity 36.1%

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

      2. add-sqr-sqrt 36.1%

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

      3. times-frac 36.1%

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

      4. hypot-def 36.1%

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

      5. fma-def 36.1%

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

      6. hypot-def 56.6%

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

   4. Applied egg-rr 56.6%

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

   5. Taylor expanded in c around 0 28.3%

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

      1. associate-/l* 28.5%

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

   7. Simplified 28.5%

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

   8. Taylor expanded in c around 0 77.7%

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

   if -1.5e137 < d < -8.4999999999999998e-106 or 3.2e-78 < d < 3.70000000000000011e75

   1. Initial program 84.1%

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

   if -8.4999999999999998e-106 < d < 3.2e-78

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

      1. clear-num 83.2%

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

      2. associate-/r/ 83.2%

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

      3. clear-num 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. associate-*l/ 86.0%

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

      2. unpow2 86.0%

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

      3. associate-/r* 92.8%

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

      4. *-commutative 92.8%

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

   9. Applied egg-rr 92.8%

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

   if 3.70000000000000011e75 < d

   1. Initial program 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. add-sqr-sqrt 22.8%

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

      3. times-frac 22.9%

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

      4. hypot-def 22.9%

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

      5. fma-def 22.9%

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

      6. hypot-def 44.3%

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

   4. Applied egg-rr 44.3%

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

   5. Taylor expanded in c around 0 81.4%

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

      1. associate-/l* 90.2%

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

   7. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. associate-/r/ 90.2%

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

      3. clear-num 90.3%

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

   9. Applied egg-rr 90.3%

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

   10. Taylor expanded in c around 0 90.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{-6} \lor \neg \left(c \leq 1.25 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.3e-6) (not (<= c 1.25e+32)))
   (/ a c)
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.3e-6) || !(c <= 1.25e+32)) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a * (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 ((c <= (-1.3d-6)) .or. (.not. (c <= 1.25d+32))) then
        tmp = a / c
    else
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.3e-6) || !(c <= 1.25e+32)) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.3e-6) or not (c <= 1.25e+32):
		tmp = a / c
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.3e-6) || !(c <= 1.25e+32))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.3e-6) || ~((c <= 1.25e+32)))
		tmp = a / c;
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.3e-6], N[Not[LessEqual[c, 1.25e+32]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.30000000000000005e-6 or 1.2499999999999999e32 < c

   1. Initial program 51.3%

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

   3. Taylor expanded in c around inf 67.2%

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

   if -1.30000000000000005e-6 < c < 1.2499999999999999e32

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. add-sqr-sqrt 71.7%

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

      3. times-frac 71.7%

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

      4. hypot-def 71.7%

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

      5. fma-def 71.7%

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

      6. hypot-def 82.1%

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

   4. Applied egg-rr 82.1%

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

   5. Taylor expanded in c around 0 50.0%

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

      1. associate-/l* 50.0%

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

   7. Simplified 50.0%

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

      1. clear-num 50.0%

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

      2. associate-/r/ 50.0%

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

      3. clear-num 50.0%

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

   9. Applied egg-rr 50.0%

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

   10. Taylor expanded in c around 0 81.2%

       \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c}{d} \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{-6} \lor \neg \left(c \leq 1.25 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-8} \lor \neg \left(c \leq 3.6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.45e-8) (not (<= c 3.6e+23)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e-8) || !(c <= 3.6e+23)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (1.0 / d) * (b + (a * (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 ((c <= (-1.45d-8)) .or. (.not. (c <= 3.6d+23))) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e-8) || !(c <= 3.6e+23)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.45e-8) or not (c <= 3.6e+23):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.45e-8) || !(c <= 3.6e+23))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.45e-8) || ~((c <= 3.6e+23)))
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.45e-8], N[Not[LessEqual[c, 3.6e+23]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.4500000000000001e-8 or 3.5999999999999998e23 < c

   1. Initial program 51.7%

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

   3. Taylor expanded in c around inf 77.8%

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

      1. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

      1. pow2 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. times-frac 77.2%

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

   7. Applied egg-rr 77.2%

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

      1. /-rgt-identity 77.2%

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

      2. *-un-lft-identity 77.2%

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

      3. *-commutative 77.2%

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

      4. times-frac 82.4%

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

      5. clear-num 82.8%

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

   9. Applied egg-rr 82.8%

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

   if -1.4500000000000001e-8 < c < 3.5999999999999998e23

   1. Initial program 71.5%

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

      1. *-un-lft-identity 71.5%

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

      2. add-sqr-sqrt 71.5%

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

      3. times-frac 71.5%

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

      4. hypot-def 71.5%

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

      5. fma-def 71.5%

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

      6. hypot-def 82.0%

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

   4. Applied egg-rr 82.0%

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

   5. Taylor expanded in c around 0 50.4%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

      1. clear-num 50.4%

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

      2. associate-/r/ 50.3%

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

      3. clear-num 50.3%

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

   9. Applied egg-rr 50.3%

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

   10. Taylor expanded in c around 0 81.8%

       \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{c}{d} \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-8} \lor \neg \left(c \leq 3.6 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-32} \lor \neg \left(c \leq 1.6 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.5e-32) (not (<= c 1.6e-58)))
   (+ (/ a c) (/ (/ b (/ c d)) c))
   (* (/ 1.0 d) (+ b (/ a (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e-32) || !(c <= 1.6e-58)) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-7.5d-32)) .or. (.not. (c <= 1.6d-58))) then
        tmp = (a / c) + ((b / (c / d)) / c)
    else
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e-32) || !(c <= 1.6e-58)) {
		tmp = (a / c) + ((b / (c / d)) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.5e-32) or not (c <= 1.6e-58):
		tmp = (a / c) + ((b / (c / d)) / c)
	else:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.5e-32) || !(c <= 1.6e-58))
		tmp = Float64(Float64(a / c) + Float64(Float64(b / Float64(c / d)) / c));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -7.5e-32) || ~((c <= 1.6e-58)))
		tmp = (a / c) + ((b / (c / d)) / c);
	else
		tmp = (1.0 / d) * (b + (a / (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.5e-32], N[Not[LessEqual[c, 1.6e-58]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.49999999999999953e-32 or 1.6e-58 < c

   1. Initial program 53.2%

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

   3. Taylor expanded in c around inf 74.1%

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

      1. associate-/l* 72.3%

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

   5. Simplified 72.3%

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

      1. clear-num 72.3%

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

      2. associate-/r/ 72.3%

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

      3. clear-num 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. associate-*l/ 74.1%

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

      2. unpow2 74.1%

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

      3. times-frac 77.7%

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

      4. clear-num 77.5%

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

      5. times-frac 74.2%

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

      6. *-un-lft-identity 74.2%

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

      7. associate-/r* 78.9%

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

   9. Applied egg-rr 78.9%

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

   if -7.49999999999999953e-32 < c < 1.6e-58

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.7%

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

      3. times-frac 72.7%

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

      4. hypot-def 72.7%

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

      5. fma-def 72.7%

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

      6. hypot-def 84.2%

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

   4. Applied egg-rr 84.2%

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

   5. Taylor expanded in c around 0 53.1%

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

      1. associate-/l* 53.1%

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

   7. Simplified 53.1%

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

   8. Taylor expanded in c around 0 87.2%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot \left(b + \frac{a}{\frac{d}{c}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-32} \lor \neg \left(c \leq 1.6 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{-5} \lor \neg \left(c \leq 1.56 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.95e-5) (not (<= c 1.56e-58))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.95e-5) || !(c <= 1.56e-58)) {
		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 <= (-1.95d-5)) .or. (.not. (c <= 1.56d-58))) 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 <= -1.95e-5) || !(c <= 1.56e-58)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.95e-5) or not (c <= 1.56e-58):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.95e-5) || !(c <= 1.56e-58))
		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 <= -1.95e-5) || ~((c <= 1.56e-58)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.95e-5], N[Not[LessEqual[c, 1.56e-58]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.95e-5 or 1.56000000000000008e-58 < c

   1. Initial program 52.0%

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

   3. Taylor expanded in c around inf 65.6%

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

   if -1.95e-5 < c < 1.56000000000000008e-58

   1. Initial program 72.8%

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

   3. Taylor expanded in c around 0 66.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{-5} \lor \neg \left(c \leq 1.56 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in c around inf 43.6%

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

4. Final simplification 43.6%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024029 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (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))))