Complex division, real part

Percentage Accurate: 61.6% → 89.7%

Time: 13.9s

Alternatives: 13

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: 89.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a + b \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;t\_0 \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ a (* b (/ d c)))))
   (if (<= c -9.2e+132)
     (* t_0 (/ -1.0 (hypot c d)))
     (if (<= c 4.3e+98)
       (/ (fma a (/ c d) b) (* (hypot c d) (/ (hypot c d) d)))
       (* (/ 1.0 (hypot c d)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a + (b * (d / c));
	double tmp;
	if (c <= -9.2e+132) {
		tmp = t_0 * (-1.0 / hypot(c, d));
	} else if (c <= 4.3e+98) {
		tmp = fma(a, (c / d), b) / (hypot(c, d) * (hypot(c, d) / d));
	} else {
		tmp = (1.0 / hypot(c, d)) * t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a + Float64(b * Float64(d / c)))
	tmp = 0.0
	if (c <= -9.2e+132)
		tmp = Float64(t_0 * Float64(-1.0 / hypot(c, d)));
	elseif (c <= 4.3e+98)
		tmp = Float64(fma(a, Float64(c / d), b) / Float64(hypot(c, d) * Float64(hypot(c, d) / d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.2e+132], N[(t$95$0 * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.3e+98], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.2000000000000006e132

   1. Initial program 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. associate-*r/ 41.4%

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

      3. fma-define 41.4%

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

      4. add-sqr-sqrt 41.4%

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

      5. times-frac 41.4%

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

      6. fma-define 41.4%

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

      7. hypot-define 41.4%

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

      8. fma-define 41.4%

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

      9. fma-define 41.4%

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

      10. hypot-define 57.5%

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

   4. Applied egg-rr 57.5%

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

   5. Taylor expanded in c around -inf 77.8%

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

      1. distribute-lft-out 77.8%

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

      2. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

   if -9.2000000000000006e132 < c < 4.3000000000000001e98

   1. Initial program 69.4%

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

   3. Taylor expanded in d around inf 65.4%

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

      1. associate-/l* 63.8%

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

   5. Simplified 63.8%

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

      1. *-commutative 63.8%

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

      2. add-sqr-sqrt 63.8%

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

      3. hypot-undefine 63.8%

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

      4. hypot-undefine 63.8%

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

      5. times-frac 87.7%

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

      6. +-commutative 87.7%

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

      7. fma-define 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. clear-num 87.6%

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

      2. frac-times 91.5%

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

      3. *-commutative 91.5%

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

      4. *-un-lft-identity 91.5%

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

   9. Applied egg-rr 91.5%

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

   if 4.3000000000000001e98 < c

   1. Initial program 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. associate-*r/ 38.6%

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

      3. fma-define 38.6%

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

      4. add-sqr-sqrt 38.6%

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

      5. times-frac 38.6%

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

      6. fma-define 38.6%

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

      7. hypot-define 38.6%

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

      8. fma-define 38.6%

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

      9. fma-define 38.6%

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

      10. hypot-define 57.9%

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

   4. Applied egg-rr 57.9%

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

   5. Taylor expanded in c around inf 89.9%

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

      1. associate-/l* 93.7%

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

   7. Simplified 93.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.2 \cdot 10^{+132}:\\ \;\;\;\;\left(a + b \cdot \frac{d}{c}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e+77)
   (/ (- (- b) (* c (/ a d))) (hypot c d))
   (if (<= d -1.45e-65)
     (/ (+ (* d b) (* c a)) (+ (* c c) (* d d)))
     (if (<= d 1.5e-67)
       (/ (+ a (/ (* d b) c)) c)
       (if (<= d 5.3e+100)
         (/ (fma a c (* d b)) (fma c c (* d d)))
         (* (/ d (hypot d c)) (/ b (hypot d c))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+77) {
		tmp = (-b - (c * (a / d))) / hypot(c, d);
	} else if (d <= -1.45e-65) {
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d));
	} else if (d <= 1.5e-67) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 5.3e+100) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e+77)
		tmp = Float64(Float64(Float64(-b) - Float64(c * Float64(a / d))) / hypot(c, d));
	elseif (d <= -1.45e-65)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.5e-67)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 5.3e+100)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e+77], N[(N[((-b) - N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.45e-65], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.5e-67], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.3e+100], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.54999999999999999e77

   1. Initial program 44.6%

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

   3. Taylor expanded in d around inf 44.6%

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

      1. associate-/l* 42.8%

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

   5. Simplified 42.8%

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

      1. *-commutative 42.8%

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

      2. add-sqr-sqrt 42.8%

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

      3. hypot-undefine 42.8%

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

      4. hypot-undefine 42.8%

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

      5. times-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. fma-define 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. fma-undefine 92.2%

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

      2. +-commutative 92.2%

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

      3. clear-num 92.2%

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

      4. un-div-inv 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. associate-/r/ 94.1%

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

   11. Simplified 94.1%

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

   12. Taylor expanded in d around -inf 89.5%

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

   if -1.54999999999999999e77 < d < -1.4499999999999999e-65

   1. Initial program 91.9%

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

   if -1.4499999999999999e-65 < d < 1.50000000000000016e-67

   1. Initial program 61.5%

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

   3. Taylor expanded in c around inf 83.1%

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

   if 1.50000000000000016e-67 < d < 5.2999999999999998e100

   1. Initial program 81.1%

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

      1. fma-define 81.1%

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

      2. fma-define 81.2%

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

   3. Simplified 81.2%

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

   if 5.2999999999999998e100 < d

   1. Initial program 31.9%

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

   3. Taylor expanded in a around 0 31.9%

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

      1. *-commutative 31.9%

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

      2. fma-define 31.9%

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

      3. add-sqr-sqrt 31.9%

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

      4. fma-define 31.9%

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

      5. hypot-undefine 31.9%

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

      6. fma-define 31.9%

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

      7. hypot-undefine 31.9%

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

      8. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. hypot-undefine 36.5%

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

      2. unpow2 36.5%

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

      3. unpow2 36.5%

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

      4. +-commutative 36.5%

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

      5. unpow2 36.5%

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

      6. unpow2 36.5%

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

      7. hypot-define 90.9%

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

      8. hypot-undefine 36.5%

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

      9. unpow2 36.5%

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

      10. unpow2 36.5%

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

      11. +-commutative 36.5%

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

      12. unpow2 36.5%

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

      13. unpow2 36.5%

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

      14. hypot-define 90.9%

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

   7. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-65}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 5.3 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* c c) (* d d)))))
   (if (<= d -7.8e+76)
     (/ (- (- b) (* c (/ a d))) (hypot c d))
     (if (<= d -1.2e-65)
       t_0
       (if (<= d 1.1e-67)
         (/ (+ a (/ (* d b) c)) c)
         (if (<= d 2.25e+104) t_0 (* (/ d (hypot d c)) (/ b (hypot d c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -7.8e+76) {
		tmp = (-b - (c * (a / d))) / hypot(c, d);
	} else if (d <= -1.2e-65) {
		tmp = t_0;
	} else if (d <= 1.1e-67) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.25e+104) {
		tmp = t_0;
	} else {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -7.8e+76) {
		tmp = (-b - (c * (a / d))) / Math.hypot(c, d);
	} else if (d <= -1.2e-65) {
		tmp = t_0;
	} else if (d <= 1.1e-67) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.25e+104) {
		tmp = t_0;
	} else {
		tmp = (d / Math.hypot(d, c)) * (b / Math.hypot(d, c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -7.8e+76:
		tmp = (-b - (c * (a / d))) / math.hypot(c, d)
	elif d <= -1.2e-65:
		tmp = t_0
	elif d <= 1.1e-67:
		tmp = (a + ((d * b) / c)) / c
	elif d <= 2.25e+104:
		tmp = t_0
	else:
		tmp = (d / math.hypot(d, c)) * (b / math.hypot(d, c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -7.8e+76)
		tmp = Float64(Float64(Float64(-b) - Float64(c * Float64(a / d))) / hypot(c, d));
	elseif (d <= -1.2e-65)
		tmp = t_0;
	elseif (d <= 1.1e-67)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 2.25e+104)
		tmp = t_0;
	else
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -7.8e+76)
		tmp = (-b - (c * (a / d))) / hypot(c, d);
	elseif (d <= -1.2e-65)
		tmp = t_0;
	elseif (d <= 1.1e-67)
		tmp = (a + ((d * b) / c)) / c;
	elseif (d <= 2.25e+104)
		tmp = t_0;
	else
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.8e+76], N[(N[((-b) - N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.2e-65], t$95$0, If[LessEqual[d, 1.1e-67], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.25e+104], t$95$0, N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.79999999999999979e76

   1. Initial program 44.6%

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

   3. Taylor expanded in d around inf 44.6%

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

      1. associate-/l* 42.8%

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

   5. Simplified 42.8%

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

      1. *-commutative 42.8%

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

      2. add-sqr-sqrt 42.8%

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

      3. hypot-undefine 42.8%

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

      4. hypot-undefine 42.8%

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

      5. times-frac 92.2%

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

      6. +-commutative 92.2%

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

      7. fma-define 92.2%

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

   7. Applied egg-rr 92.2%

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

      1. fma-undefine 92.2%

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

      2. +-commutative 92.2%

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

      3. clear-num 92.2%

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

      4. un-div-inv 92.2%

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

   9. Applied egg-rr 92.2%

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

       1. associate-/r/ 94.1%

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

   11. Simplified 94.1%

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

   12. Taylor expanded in d around -inf 89.5%

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

   if -7.79999999999999979e76 < d < -1.2000000000000001e-65 or 1.1000000000000001e-67 < d < 2.2499999999999999e104

   1. Initial program 86.1%

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

   if -1.2000000000000001e-65 < d < 1.1000000000000001e-67

   1. Initial program 61.5%

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

   3. Taylor expanded in c around inf 83.1%

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

   if 2.2499999999999999e104 < d

   1. Initial program 31.9%

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

   3. Taylor expanded in a around 0 31.9%

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

      1. *-commutative 31.9%

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

      2. fma-define 31.9%

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

      3. add-sqr-sqrt 31.9%

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

      4. fma-define 31.9%

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

      5. hypot-undefine 31.9%

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

      6. fma-define 31.9%

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

      7. hypot-undefine 31.9%

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

      8. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

      1. hypot-undefine 36.5%

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

      2. unpow2 36.5%

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

      3. unpow2 36.5%

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

      4. +-commutative 36.5%

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

      5. unpow2 36.5%

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

      6. unpow2 36.5%

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

      7. hypot-define 90.9%

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

      8. hypot-undefine 36.5%

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

      9. unpow2 36.5%

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

      10. unpow2 36.5%

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

      11. +-commutative 36.5%

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

      12. unpow2 36.5%

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

      13. unpow2 36.5%

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

      14. hypot-define 90.9%

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

   7. Simplified 90.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -7.9 \cdot 10^{-22}:\\ \;\;\;\;t\_0 \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-116}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* c (/ a d))) (hypot c d))))
   (if (<= d -7.9e-22)
     (* t_0 (/ d (hypot c d)))
     (if (<= d 1.75e-116)
       (/ (+ a (/ (* d b) c)) c)
       (* t_0 (/ 1.0 (/ (hypot d c) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / hypot(c, d);
	double tmp;
	if (d <= -7.9e-22) {
		tmp = t_0 * (d / hypot(c, d));
	} else if (d <= 1.75e-116) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = t_0 * (1.0 / (hypot(d, c) / d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / Math.hypot(c, d);
	double tmp;
	if (d <= -7.9e-22) {
		tmp = t_0 * (d / Math.hypot(c, d));
	} else if (d <= 1.75e-116) {
		tmp = (a + ((d * b) / c)) / c;
	} else {
		tmp = t_0 * (1.0 / (Math.hypot(d, c) / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (c * (a / d))) / math.hypot(c, d)
	tmp = 0
	if d <= -7.9e-22:
		tmp = t_0 * (d / math.hypot(c, d))
	elif d <= 1.75e-116:
		tmp = (a + ((d * b) / c)) / c
	else:
		tmp = t_0 * (1.0 / (math.hypot(d, c) / d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(c * Float64(a / d))) / hypot(c, d))
	tmp = 0.0
	if (d <= -7.9e-22)
		tmp = Float64(t_0 * Float64(d / hypot(c, d)));
	elseif (d <= 1.75e-116)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	else
		tmp = Float64(t_0 * Float64(1.0 / Float64(hypot(d, c) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (c * (a / d))) / hypot(c, d);
	tmp = 0.0;
	if (d <= -7.9e-22)
		tmp = t_0 * (d / hypot(c, d));
	elseif (d <= 1.75e-116)
		tmp = (a + ((d * b) / c)) / c;
	else
		tmp = t_0 * (1.0 / (hypot(d, c) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.9e-22], N[(t$95$0 * N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-116], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(t$95$0 * N[(1.0 / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.8999999999999997e-22

   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 d around inf 59.4%

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

      1. associate-/l* 58.0%

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

   5. Simplified 58.0%

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

      1. *-commutative 58.0%

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

      2. add-sqr-sqrt 58.0%

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

      3. hypot-undefine 58.0%

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

      4. hypot-undefine 58.0%

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

      5. times-frac 93.2%

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

      6. +-commutative 93.2%

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

      7. fma-define 93.2%

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

   7. Applied egg-rr 93.2%

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

      1. fma-undefine 93.2%

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

      2. +-commutative 93.2%

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

      3. clear-num 93.2%

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

      4. un-div-inv 93.3%

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

   9. Applied egg-rr 93.3%

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

       1. associate-/r/ 94.6%

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

   11. Simplified 94.6%

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

   if -7.8999999999999997e-22 < d < 1.74999999999999992e-116

   1. Initial program 65.3%

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

   3. Taylor expanded in c around inf 84.4%

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

   if 1.74999999999999992e-116 < d

   1. Initial program 58.2%

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

   3. Taylor expanded in d around inf 57.0%

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

      1. associate-/l* 55.9%

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

   5. Simplified 55.9%

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

      1. *-commutative 55.9%

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

      2. add-sqr-sqrt 55.9%

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

      3. hypot-undefine 55.9%

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

      4. hypot-undefine 55.9%

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

      5. times-frac 90.8%

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

      6. +-commutative 90.8%

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

      7. fma-define 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. fma-undefine 90.8%

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

      2. +-commutative 90.8%

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

      3. clear-num 90.8%

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

      4. un-div-inv 90.9%

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

   9. Applied egg-rr 90.9%

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

       1. associate-/r/ 92.0%

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

   11. Simplified 92.0%

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

       1. clear-num 92.0%

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

       2. inv-pow 92.0%

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

   13. Applied egg-rr 92.0%

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

       1. unpow-1 92.0%

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

       2. hypot-undefine 62.2%

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

       3. unpow2 62.2%

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

       4. unpow2 62.2%

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

       5. +-commutative 62.2%

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

       6. unpow2 62.2%

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

       7. unpow2 62.2%

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

       8. hypot-define 92.0%

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

   15. Simplified 92.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-116}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-22} \lor \neg \left(d \leq 10^{-116}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.8e-22) (not (<= d 1e-116)))
   (* (/ (+ b (* c (/ a d))) (hypot c d)) (/ d (hypot c d)))
   (/ (+ a (/ (* d b) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.8e-22) || !(d <= 1e-116)) {
		tmp = ((b + (c * (a / d))) / hypot(c, d)) * (d / hypot(c, d));
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.8e-22) || !(d <= 1e-116)) {
		tmp = ((b + (c * (a / d))) / Math.hypot(c, d)) * (d / Math.hypot(c, d));
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.8e-22) or not (d <= 1e-116):
		tmp = ((b + (c * (a / d))) / math.hypot(c, d)) * (d / math.hypot(c, d))
	else:
		tmp = (a + ((d * b) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.8e-22) || !(d <= 1e-116))
		tmp = Float64(Float64(Float64(b + Float64(c * Float64(a / d))) / hypot(c, d)) * Float64(d / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.8e-22) || ~((d <= 1e-116)))
		tmp = ((b + (c * (a / d))) / hypot(c, d)) * (d / hypot(c, d));
	else
		tmp = (a + ((d * b) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.8e-22], N[Not[LessEqual[d, 1e-116]], $MachinePrecision]], N[(N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.79999999999999996e-22 or 9.9999999999999999e-117 < d

   1. Initial program 59.3%

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

   3. Taylor expanded in d around inf 58.1%

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

      1. associate-/l* 56.8%

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

   5. Simplified 56.8%

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

      1. *-commutative 56.8%

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

      2. add-sqr-sqrt 56.8%

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

      3. hypot-undefine 56.8%

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

      4. hypot-undefine 56.8%

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

      5. times-frac 91.9%

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

      6. +-commutative 91.9%

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

      7. fma-define 91.9%

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

   7. Applied egg-rr 91.9%

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

      1. fma-undefine 91.9%

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

      2. +-commutative 91.9%

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

      3. clear-num 91.9%

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

      4. un-div-inv 92.0%

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

   9. Applied egg-rr 92.0%

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

       1. associate-/r/ 93.2%

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

   11. Simplified 93.2%

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

   if -7.79999999999999996e-22 < d < 9.9999999999999999e-117

   1. Initial program 65.3%

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

   3. Taylor expanded in c around inf 84.4%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{-22} \lor \neg \left(d \leq 10^{-116}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := a + b \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+95}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* c c) (* d d))))
        (t_1 (+ a (* b (/ d c)))))
   (if (<= c -1.25e+95)
     (* t_1 (/ -1.0 (hypot c d)))
     (if (<= c -5e-26)
       t_0
       (if (<= c 1.25e-72)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 1.9e+67) t_0 (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = a + (b * (d / c));
	double tmp;
	if (c <= -1.25e+95) {
		tmp = t_1 * (-1.0 / hypot(c, d));
	} else if (c <= -5e-26) {
		tmp = t_0;
	} else if (c <= 1.25e-72) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.9e+67) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = a + (b * (d / c));
	double tmp;
	if (c <= -1.25e+95) {
		tmp = t_1 * (-1.0 / Math.hypot(c, d));
	} else if (c <= -5e-26) {
		tmp = t_0;
	} else if (c <= 1.25e-72) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.9e+67) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d))
	t_1 = a + (b * (d / c))
	tmp = 0
	if c <= -1.25e+95:
		tmp = t_1 * (-1.0 / math.hypot(c, d))
	elif c <= -5e-26:
		tmp = t_0
	elif c <= 1.25e-72:
		tmp = (b + (a * (c / d))) / d
	elif c <= 1.9e+67:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(a + Float64(b * Float64(d / c)))
	tmp = 0.0
	if (c <= -1.25e+95)
		tmp = Float64(t_1 * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -5e-26)
		tmp = t_0;
	elseif (c <= 1.25e-72)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 1.9e+67)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	t_1 = a + (b * (d / c));
	tmp = 0.0;
	if (c <= -1.25e+95)
		tmp = t_1 * (-1.0 / hypot(c, d));
	elseif (c <= -5e-26)
		tmp = t_0;
	elseif (c <= 1.25e-72)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 1.9e+67)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+95], N[(t$95$1 * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5e-26], t$95$0, If[LessEqual[c, 1.25e-72], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.9e+67], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.25000000000000006e95

   1. Initial program 38.6%

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

      1. *-un-lft-identity 38.6%

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

      2. associate-*r/ 38.6%

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

      3. fma-define 38.6%

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

      4. add-sqr-sqrt 38.6%

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

      5. times-frac 38.6%

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

      6. fma-define 38.6%

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

      7. hypot-define 38.6%

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

      8. fma-define 38.6%

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

      9. fma-define 38.6%

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

      10. hypot-define 52.3%

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

   4. Applied egg-rr 52.3%

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

   5. Taylor expanded in c around -inf 71.0%

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

      1. distribute-lft-out 71.0%

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

      2. associate-/l* 76.9%

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

   7. Simplified 76.9%

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

   if -1.25000000000000006e95 < c < -5.00000000000000019e-26 or 1.2499999999999999e-72 < c < 1.9000000000000001e67

   1. Initial program 81.7%

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

   if -5.00000000000000019e-26 < c < 1.2499999999999999e-72

   1. Initial program 69.6%

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

   3. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 89.9%

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

   5. Simplified 89.9%

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

   if 1.9000000000000001e67 < c

   1. Initial program 45.3%

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

      1. *-un-lft-identity 45.3%

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

      2. associate-*r/ 45.3%

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

      3. fma-define 45.3%

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

      4. add-sqr-sqrt 45.3%

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

      5. times-frac 45.4%

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

      6. fma-define 45.4%

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

      7. hypot-define 45.4%

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

      8. fma-define 45.4%

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

      9. fma-define 45.4%

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

      10. hypot-define 60.2%

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

   4. Applied egg-rr 60.2%

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

   5. Taylor expanded in c around inf 85.3%

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

      1. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+95}:\\ \;\;\;\;\left(a + b \cdot \frac{d}{c}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-26}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := a + b \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -9.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{t\_1}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* c c) (* d d))))
        (t_1 (+ a (* b (/ d c)))))
   (if (<= c -9.5e+94)
     (/ t_1 c)
     (if (<= c -5e-27)
       t_0
       (if (<= c 2e-72)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 2.35e+67) t_0 (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = a + (b * (d / c));
	double tmp;
	if (c <= -9.5e+94) {
		tmp = t_1 / c;
	} else if (c <= -5e-27) {
		tmp = t_0;
	} else if (c <= 2e-72) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 2.35e+67) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = a + (b * (d / c));
	double tmp;
	if (c <= -9.5e+94) {
		tmp = t_1 / c;
	} else if (c <= -5e-27) {
		tmp = t_0;
	} else if (c <= 2e-72) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 2.35e+67) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d))
	t_1 = a + (b * (d / c))
	tmp = 0
	if c <= -9.5e+94:
		tmp = t_1 / c
	elif c <= -5e-27:
		tmp = t_0
	elif c <= 2e-72:
		tmp = (b + (a * (c / d))) / d
	elif c <= 2.35e+67:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(a + Float64(b * Float64(d / c)))
	tmp = 0.0
	if (c <= -9.5e+94)
		tmp = Float64(t_1 / c);
	elseif (c <= -5e-27)
		tmp = t_0;
	elseif (c <= 2e-72)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 2.35e+67)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	t_1 = a + (b * (d / c));
	tmp = 0.0;
	if (c <= -9.5e+94)
		tmp = t_1 / c;
	elseif (c <= -5e-27)
		tmp = t_0;
	elseif (c <= 2e-72)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 2.35e+67)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.5e+94], N[(t$95$1 / c), $MachinePrecision], If[LessEqual[c, -5e-27], t$95$0, If[LessEqual[c, 2e-72], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.35e+67], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.4999999999999998e94

   1. Initial program 38.6%

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

   3. Taylor expanded in c around inf 70.9%

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

      1. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

   if -9.4999999999999998e94 < c < -5.0000000000000002e-27 or 1.9999999999999999e-72 < c < 2.35000000000000009e67

   1. Initial program 81.7%

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

   if -5.0000000000000002e-27 < c < 1.9999999999999999e-72

   1. Initial program 69.6%

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

   3. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 89.9%

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

   5. Simplified 89.9%

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

   if 2.35000000000000009e67 < c

   1. Initial program 45.3%

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

      1. *-un-lft-identity 45.3%

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

      2. associate-*r/ 45.3%

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

      3. fma-define 45.3%

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

      4. add-sqr-sqrt 45.3%

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

      5. times-frac 45.4%

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

      6. fma-define 45.4%

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

      7. hypot-define 45.4%

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

      8. fma-define 45.4%

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

      9. fma-define 45.4%

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

      10. hypot-define 60.2%

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

   4. Applied egg-rr 60.2%

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

   5. Taylor expanded in c around inf 85.3%

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

      1. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{+94}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-27}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-72}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+67}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -4.6 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* c c) (* d d))))
        (t_1 (/ (+ a (* b (/ d c))) c)))
   (if (<= c -4.6e+95)
     t_1
     (if (<= c -4.8e-27)
       t_0
       (if (<= c 4.5e-74)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 5.8e+74) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = (a + (b * (d / c))) / c;
	double tmp;
	if (c <= -4.6e+95) {
		tmp = t_1;
	} else if (c <= -4.8e-27) {
		tmp = t_0;
	} else if (c <= 4.5e-74) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5.8e+74) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d))
    t_1 = (a + (b * (d / c))) / c
    if (c <= (-4.6d+95)) then
        tmp = t_1
    else if (c <= (-4.8d-27)) then
        tmp = t_0
    else if (c <= 4.5d-74) then
        tmp = (b + (a * (c / d))) / d
    else if (c <= 5.8d+74) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double t_1 = (a + (b * (d / c))) / c;
	double tmp;
	if (c <= -4.6e+95) {
		tmp = t_1;
	} else if (c <= -4.8e-27) {
		tmp = t_0;
	} else if (c <= 4.5e-74) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5.8e+74) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d))
	t_1 = (a + (b * (d / c))) / c
	tmp = 0
	if c <= -4.6e+95:
		tmp = t_1
	elif c <= -4.8e-27:
		tmp = t_0
	elif c <= 4.5e-74:
		tmp = (b + (a * (c / d))) / d
	elif c <= 5.8e+74:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(a + Float64(b * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -4.6e+95)
		tmp = t_1;
	elseif (c <= -4.8e-27)
		tmp = t_0;
	elseif (c <= 4.5e-74)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 5.8e+74)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	t_1 = (a + (b * (d / c))) / c;
	tmp = 0.0;
	if (c <= -4.6e+95)
		tmp = t_1;
	elseif (c <= -4.8e-27)
		tmp = t_0;
	elseif (c <= 4.5e-74)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 5.8e+74)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.6e+95], t$95$1, If[LessEqual[c, -4.8e-27], t$95$0, If[LessEqual[c, 4.5e-74], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.8e+74], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.59999999999999994e95 or 5.8000000000000005e74 < c

   1. Initial program 41.2%

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

   3. Taylor expanded in c around inf 77.5%

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

      1. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

   if -4.59999999999999994e95 < c < -4.80000000000000004e-27 or 4.4999999999999999e-74 < c < 5.8000000000000005e74

   1. Initial program 80.5%

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

   if -4.80000000000000004e-27 < c < 4.4999999999999999e-74

   1. Initial program 69.6%

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

   3. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 89.9%

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

   5. Simplified 89.9%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+95}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-74}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+101} \lor \neg \left(d \leq 1.1 \cdot 10^{+15}\right) \land \left(d \leq 1.7 \cdot 10^{+73} \lor \neg \left(d \leq 3.5 \cdot 10^{+170}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.2e+101)
         (and (not (<= d 1.1e+15)) (or (<= d 1.7e+73) (not (<= d 3.5e+170)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e+101) || (!(d <= 1.1e+15) && ((d <= 1.7e+73) || !(d <= 3.5e+170)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.2d+101)) .or. (.not. (d <= 1.1d+15)) .and. (d <= 1.7d+73) .or. (.not. (d <= 3.5d+170))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e+101) || (!(d <= 1.1e+15) && ((d <= 1.7e+73) || !(d <= 3.5e+170)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.2e+101) or (not (d <= 1.1e+15) and ((d <= 1.7e+73) or not (d <= 3.5e+170))):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.2e+101) || (!(d <= 1.1e+15) && ((d <= 1.7e+73) || !(d <= 3.5e+170))))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.2e+101) || (~((d <= 1.1e+15)) && ((d <= 1.7e+73) || ~((d <= 3.5e+170)))))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.2e+101], And[N[Not[LessEqual[d, 1.1e+15]], $MachinePrecision], Or[LessEqual[d, 1.7e+73], N[Not[LessEqual[d, 3.5e+170]], $MachinePrecision]]]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.1999999999999999e101 or 1.1e15 < d < 1.7000000000000001e73 or 3.50000000000000005e170 < d

   1. Initial program 51.5%

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

   3. Taylor expanded in c around 0 78.3%

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

   if -8.1999999999999999e101 < d < 1.1e15 or 1.7000000000000001e73 < d < 3.50000000000000005e170

   1. Initial program 67.4%

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

   3. Taylor expanded in c around inf 70.1%

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

      1. associate-/l* 72.0%

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

   5. Simplified 72.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+101} \lor \neg \left(d \leq 1.1 \cdot 10^{+15}\right) \land \left(d \leq 1.7 \cdot 10^{+73} \lor \neg \left(d \leq 3.5 \cdot 10^{+170}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 96000000000000:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+73} \lor \neg \left(d \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.7e+101)
   (/ b d)
   (if (<= d 96000000000000.0)
     (/ (+ a (/ (* d b) c)) c)
     (if (or (<= d 1.35e+73) (not (<= d 3.5e+170)))
       (/ b d)
       (/ (+ a (* b (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.7e+101) {
		tmp = b / d;
	} else if (d <= 96000000000000.0) {
		tmp = (a + ((d * b) / c)) / c;
	} else if ((d <= 1.35e+73) || !(d <= 3.5e+170)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-4.7d+101)) then
        tmp = b / d
    else if (d <= 96000000000000.0d0) then
        tmp = (a + ((d * b) / c)) / c
    else if ((d <= 1.35d+73) .or. (.not. (d <= 3.5d+170))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.7e+101) {
		tmp = b / d;
	} else if (d <= 96000000000000.0) {
		tmp = (a + ((d * b) / c)) / c;
	} else if ((d <= 1.35e+73) || !(d <= 3.5e+170)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.7e+101:
		tmp = b / d
	elif d <= 96000000000000.0:
		tmp = (a + ((d * b) / c)) / c
	elif (d <= 1.35e+73) or not (d <= 3.5e+170):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.7e+101)
		tmp = Float64(b / d);
	elseif (d <= 96000000000000.0)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif ((d <= 1.35e+73) || !(d <= 3.5e+170))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4.7e+101)
		tmp = b / d;
	elseif (d <= 96000000000000.0)
		tmp = (a + ((d * b) / c)) / c;
	elseif ((d <= 1.35e+73) || ~((d <= 3.5e+170)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.7e+101], N[(b / d), $MachinePrecision], If[LessEqual[d, 96000000000000.0], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 1.35e+73], N[Not[LessEqual[d, 3.5e+170]], $MachinePrecision]], 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 d < -4.69999999999999971e101 or 9.6e13 < d < 1.35e73 or 3.50000000000000005e170 < d

   1. Initial program 51.5%

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

   3. Taylor expanded in c around 0 78.3%

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

   if -4.69999999999999971e101 < d < 9.6e13

   1. Initial program 69.6%

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

   3. Taylor expanded in c around inf 73.8%

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

   if 1.35e73 < d < 3.50000000000000005e170

   1. Initial program 44.1%

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

   3. Taylor expanded in c around inf 31.2%

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

      1. associate-/l* 59.0%

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

   5. Simplified 59.0%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 96000000000000:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{+73} \lor \neg \left(d \leq 3.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{-23} \lor \neg \left(c \leq 9.8 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.95e-23) (not (<= c 9.8e+45)))
   (/ (+ a (* b (/ d c))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.95e-23) || !(c <= 9.8e+45)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.95d-23)) .or. (.not. (c <= 9.8d+45))) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.95e-23) || !(c <= 9.8e+45)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.95e-23) or not (c <= 9.8e+45):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.95e-23) || !(c <= 9.8e+45))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.95e-23) || ~((c <= 9.8e+45)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.95e-23], N[Not[LessEqual[c, 9.8e+45]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.95e-23 or 9.8000000000000004e45 < c

   1. Initial program 49.6%

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

   3. Taylor expanded in c around inf 73.5%

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

      1. associate-/l* 76.3%

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

   5. Simplified 76.3%

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

   if -1.95e-23 < c < 9.8000000000000004e45

   1. Initial program 71.3%

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

   3. Taylor expanded in d around inf 84.3%

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

      1. associate-/l* 86.0%

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

   5. Simplified 86.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{-23} \lor \neg \left(c \leq 9.8 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-39} \lor \neg \left(c \leq 2.75 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.8e-39) (not (<= c 2.75e+42))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.8e-39) || !(c <= 2.75e+42)) {
		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 <= (-5.8d-39)) .or. (.not. (c <= 2.75d+42))) 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 <= -5.8e-39) || !(c <= 2.75e+42)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.8e-39) or not (c <= 2.75e+42):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.8e-39) || !(c <= 2.75e+42))
		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 <= -5.8e-39) || ~((c <= 2.75e+42)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.8e-39], N[Not[LessEqual[c, 2.75e+42]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.79999999999999975e-39 or 2.75000000000000001e42 < c

   1. Initial program 49.7%

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

   3. Taylor expanded in c around inf 65.5%

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

   if -5.79999999999999975e-39 < c < 2.75000000000000001e42

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0 64.6%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-39} \lor \neg \left(c \leq 2.75 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.0% 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 61.6%

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

3. Taylor expanded in c around inf 39.7%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
4. 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 2024101 
(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))))