Complex division, real part

Percentage Accurate: 61.1% → 88.5%

Time: 12.0s

Alternatives: 12

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: 61.1% 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: 88.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;t\_0 \cdot \left(c \cdot \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))))
   (if (<= d -5.7e+147)
     (* (/ d (hypot c d)) (/ b (hypot c d)))
     (if (<= d 7.5e+110)
       (* t_0 (* c (/ (fma b (/ d c) a) (hypot c d))))
       (* t_0 (+ b (* a (/ c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double tmp;
	if (d <= -5.7e+147) {
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	} else if (d <= 7.5e+110) {
		tmp = t_0 * (c * (fma(b, (d / c), a) / hypot(c, d)));
	} else {
		tmp = t_0 * (b + (a * (c / d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	tmp = 0.0
	if (d <= -5.7e+147)
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	elseif (d <= 7.5e+110)
		tmp = Float64(t_0 * Float64(c * Float64(fma(b, Float64(d / c), a) / hypot(c, d))));
	else
		tmp = Float64(t_0 * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.7e+147], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+110], N[(t$95$0 * N[(c * N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.69999999999999991e147

   1. Initial program 25.1%

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

   3. Taylor expanded in a around 0 25.1%

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

      1. *-commutative 25.1%

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

      2. add-sqr-sqrt 25.1%

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

      3. hypot-undefine 25.1%

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

      4. hypot-undefine 25.1%

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

      5. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

   if -5.69999999999999991e147 < d < 7.5e110

   1. Initial program 67.8%

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

   3. Taylor expanded in c around inf 64.7%

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

      1. associate-/l* 64.2%

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

   5. Simplified 64.2%

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

      1. *-un-lft-identity 64.2%

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

      2. add-sqr-sqrt 64.2%

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

      3. hypot-undefine 64.2%

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

      4. hypot-undefine 64.2%

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

      5. times-frac 78.2%

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

      6. +-commutative 78.2%

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

      7. fma-define 78.2%

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

   7. Applied egg-rr 78.2%

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

      1. associate-/l* 95.0%

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

   9. Simplified 95.0%

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

   if 7.5e110 < d

   1. Initial program 45.1%

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

      1. *-un-lft-identity 45.1%

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

      2. associate-*r/ 45.1%

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

      3. fma-define 45.1%

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

      4. add-sqr-sqrt 45.1%

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

      5. times-frac 45.0%

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

      6. fma-define 45.0%

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

      7. hypot-define 45.0%

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

      8. fma-define 45.0%

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

      9. fma-define 45.0%

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

      10. hypot-define 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in c around 0 79.8%

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

      1. associate-/l* 91.5%

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

   7. Simplified 91.5%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+147}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(c \cdot \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d} \leq 10^{+241}:\\ \;\;\;\;t\_0 \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(c \cdot \frac{b \cdot \left(\frac{d}{c} + \frac{a}{b}\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))))
   (if (<= (/ (+ (* c a) (* d b)) (+ (* c c) (* d d))) 1e+241)
     (* t_0 (/ (fma a c (* d b)) (hypot c d)))
     (* t_0 (* c (/ (* b (+ (/ d c) (/ a b))) (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double tmp;
	if ((((c * a) + (d * b)) / ((c * c) + (d * d))) <= 1e+241) {
		tmp = t_0 * (fma(a, c, (d * b)) / hypot(c, d));
	} else {
		tmp = t_0 * (c * ((b * ((d / c) + (a / b))) / hypot(c, d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	tmp = 0.0
	if (Float64(Float64(Float64(c * a) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+241)
		tmp = Float64(t_0 * Float64(fma(a, c, Float64(d * b)) / hypot(c, d)));
	else
		tmp = Float64(t_0 * Float64(c * Float64(Float64(b * Float64(Float64(d / c) + Float64(a / b))) / hypot(c, d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+241], N[(t$95$0 * N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(c * N[(N[(b * N[(N[(d / c), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.0000000000000001e241

   1. Initial program 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. associate-*r/ 74.6%

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

      3. fma-define 74.6%

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

      4. add-sqr-sqrt 74.5%

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

      5. times-frac 74.5%

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

      6. fma-define 74.5%

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

      7. hypot-define 74.5%

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

      8. fma-define 74.5%

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

      9. fma-define 74.5%

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

      10. hypot-define 96.4%

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

   4. Applied egg-rr 96.4%

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

   if 1.0000000000000001e241 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.2%

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

   3. Taylor expanded in c around inf 14.2%

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

      1. associate-/l* 14.2%

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

   5. Simplified 14.2%

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

      1. *-un-lft-identity 14.2%

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

      2. add-sqr-sqrt 14.2%

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

      3. hypot-undefine 14.2%

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

      4. hypot-undefine 14.2%

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

      5. times-frac 20.4%

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

      6. +-commutative 20.4%

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

      7. fma-define 20.4%

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

   7. Applied egg-rr 20.4%

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

      1. associate-/l* 76.6%

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

   9. Simplified 76.6%

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

   10. Taylor expanded in b around inf 59.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d} \leq 10^{+241}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(c \cdot \frac{b \cdot \left(\frac{d}{c} + \frac{a}{b}\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -6.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+110}:\\ \;\;\;\;t\_0 \cdot \left(c \cdot \frac{b \cdot \left(\frac{d}{c} + \frac{a}{b}\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))) (t_1 (+ b (* a (/ c d)))))
   (if (<= d -6.6e+121)
     (/ t_1 d)
     (if (<= d -2.3e-77)
       (/ (+ (* c a) (* d b)) (fma c c (* d d)))
       (if (<= d -9.2e-275)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 5.8e+110)
           (* t_0 (* c (/ (* b (+ (/ d c) (/ a b))) (hypot c d))))
           (* t_0 t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -6.6e+121) {
		tmp = t_1 / d;
	} else if (d <= -2.3e-77) {
		tmp = ((c * a) + (d * b)) / fma(c, c, (d * d));
	} else if (d <= -9.2e-275) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 5.8e+110) {
		tmp = t_0 * (c * ((b * ((d / c) + (a / b))) / hypot(c, d)));
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -6.6e+121)
		tmp = Float64(t_1 / d);
	elseif (d <= -2.3e-77)
		tmp = Float64(Float64(Float64(c * a) + Float64(d * b)) / fma(c, c, Float64(d * d)));
	elseif (d <= -9.2e-275)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 5.8e+110)
		tmp = Float64(t_0 * Float64(c * Float64(Float64(b * Float64(Float64(d / c) + Float64(a / b))) / hypot(c, d))));
	else
		tmp = Float64(t_0 * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.6e+121], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -2.3e-77], N[(N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -9.2e-275], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.8e+110], N[(t$95$0 * N[(c * N[(N[(b * N[(N[(d / c), $MachinePrecision] + N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.59999999999999958e121

   1. Initial program 27.1%

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

   3. Taylor expanded in d around inf 90.6%

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

      1. associate-/l* 92.9%

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

   5. Simplified 92.9%

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

   if -6.59999999999999958e121 < d < -2.29999999999999999e-77

   1. Initial program 86.0%

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

      1. fma-define 86.0%

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

      2. fma-define 86.1%

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

   3. Simplified 86.1%

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

      1. fma-define 86.1%

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

      2. +-commutative 86.1%

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

   6. Applied egg-rr 86.1%

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

   if -2.29999999999999999e-77 < d < -9.19999999999999959e-275

   1. Initial program 63.3%

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

   3. Taylor expanded in c around inf 90.4%

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

      1. associate-/l* 90.5%

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

   5. Simplified 90.5%

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

   if -9.19999999999999959e-275 < d < 5.7999999999999999e110

   1. Initial program 63.1%

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

   3. Taylor expanded in c around inf 61.0%

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

      1. associate-/l* 60.0%

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

   5. Simplified 60.0%

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

      1. *-un-lft-identity 60.0%

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

      2. add-sqr-sqrt 60.0%

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

      3. hypot-undefine 60.0%

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

      4. hypot-undefine 60.0%

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

      5. times-frac 76.0%

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

      6. +-commutative 76.0%

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

      7. fma-define 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. associate-/l* 94.3%

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

   9. Simplified 94.3%

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

   10. Taylor expanded in b around inf 87.8%

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

   if 5.7999999999999999e110 < d

   1. Initial program 45.1%

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

      1. *-un-lft-identity 45.1%

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

      2. associate-*r/ 45.1%

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

      3. fma-define 45.1%

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

      4. add-sqr-sqrt 45.1%

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

      5. times-frac 45.0%

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

      6. fma-define 45.0%

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

      7. hypot-define 45.0%

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

      8. fma-define 45.0%

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

      9. fma-define 45.0%

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

      10. hypot-define 59.8%

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

   4. Applied egg-rr 59.8%

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

   5. Taylor expanded in c around 0 79.8%

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

      1. associate-/l* 91.5%

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

   7. Simplified 91.5%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + a \cdot \frac{c}{d}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-275}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(c \cdot \frac{b \cdot \left(\frac{d}{c} + \frac{a}{b}\right)}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -1.1 \cdot 10^{+123}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-118}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* c a) (* d b)) (+ (* c c) (* d d))))
        (t_1 (+ b (* a (/ c d)))))
   (if (<= d -1.1e+123)
     (/ t_1 d)
     (if (<= d -3.3e-80)
       t_0
       (if (<= d 9e-118)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 1.5e+79) t_0 (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -1.1e+123) {
		tmp = t_1 / d;
	} else if (d <= -3.3e-80) {
		tmp = t_0;
	} else if (d <= 9e-118) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.5e+79) {
		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 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -1.1e+123) {
		tmp = t_1 / d;
	} else if (d <= -3.3e-80) {
		tmp = t_0;
	} else if (d <= 9e-118) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.5e+79) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d))
	t_1 = b + (a * (c / d))
	tmp = 0
	if d <= -1.1e+123:
		tmp = t_1 / d
	elif d <= -3.3e-80:
		tmp = t_0
	elif d <= 9e-118:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1.5e+79:
		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(c * a) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -1.1e+123)
		tmp = Float64(t_1 / d);
	elseif (d <= -3.3e-80)
		tmp = t_0;
	elseif (d <= 9e-118)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1.5e+79)
		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 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	t_1 = b + (a * (c / d));
	tmp = 0.0;
	if (d <= -1.1e+123)
		tmp = t_1 / d;
	elseif (d <= -3.3e-80)
		tmp = t_0;
	elseif (d <= 9e-118)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1.5e+79)
		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[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.1e+123], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -3.3e-80], t$95$0, If[LessEqual[d, 9e-118], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.5e+79], 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 d < -1.09999999999999996e123

   1. Initial program 27.1%

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

   3. Taylor expanded in d around inf 90.6%

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

      1. associate-/l* 92.9%

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

   5. Simplified 92.9%

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

   if -1.09999999999999996e123 < d < -3.3e-80 or 9.0000000000000001e-118 < d < 1.49999999999999987e79

   1. Initial program 83.9%

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

   if -3.3e-80 < d < 9.0000000000000001e-118

   1. Initial program 58.4%

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

   3. Taylor expanded in c around inf 88.8%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if 1.49999999999999987e79 < d

   1. Initial program 47.2%

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

      1. *-un-lft-identity 47.2%

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

      2. associate-*r/ 47.2%

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

      3. fma-define 47.2%

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

      4. add-sqr-sqrt 47.2%

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

      5. times-frac 47.2%

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

      6. fma-define 47.2%

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

      7. hypot-define 47.2%

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

      8. fma-define 47.2%

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

      9. fma-define 47.2%

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

      10. hypot-define 59.6%

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

   4. Applied egg-rr 59.6%

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

   5. Taylor expanded in c around 0 76.4%

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

      1. associate-/l* 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+123}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-80}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-118}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+79}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot a + d \cdot b\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c a) (* d b))) (t_1 (+ b (* a (/ c d)))))
   (if (<= d -1.3e+122)
     (/ t_1 d)
     (if (<= d -2.9e-72)
       (/ t_0 (fma c c (* d d)))
       (if (<= d 3.5e-117)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 5e+81)
           (/ t_0 (+ (* c c) (* d d)))
           (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * a) + (d * b);
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -1.3e+122) {
		tmp = t_1 / d;
	} else if (d <= -2.9e-72) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (d <= 3.5e-117) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 5e+81) {
		tmp = t_0 / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / hypot(c, d)) * t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * a) + Float64(d * b))
	t_1 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -1.3e+122)
		tmp = Float64(t_1 / d);
	elseif (d <= -2.9e-72)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (d <= 3.5e-117)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 5e+81)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e+122], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -2.9e-72], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.5e-117], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+81], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.30000000000000004e122

   1. Initial program 27.1%

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

   3. Taylor expanded in d around inf 90.6%

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

      1. associate-/l* 92.9%

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

   5. Simplified 92.9%

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

   if -1.30000000000000004e122 < d < -2.89999999999999998e-72

   1. Initial program 86.0%

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

      1. fma-define 86.0%

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

      2. fma-define 86.1%

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

   3. Simplified 86.1%

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

      1. fma-define 86.1%

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

      2. +-commutative 86.1%

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

   6. Applied egg-rr 86.1%

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

   if -2.89999999999999998e-72 < d < 3.4999999999999998e-117

   1. Initial program 58.4%

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

   3. Taylor expanded in c around inf 88.8%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if 3.4999999999999998e-117 < d < 4.9999999999999998e81

   1. Initial program 80.8%

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

   if 4.9999999999999998e81 < d

   1. Initial program 47.2%

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

      1. *-un-lft-identity 47.2%

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

      2. associate-*r/ 47.2%

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

      3. fma-define 47.2%

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

      4. add-sqr-sqrt 47.2%

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

      5. times-frac 47.2%

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

      6. fma-define 47.2%

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

      7. hypot-define 47.2%

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

      8. fma-define 47.2%

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

      9. fma-define 47.2%

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

      10. hypot-define 59.6%

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

   4. Applied egg-rr 59.6%

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

   5. Taylor expanded in c around 0 76.4%

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

      1. associate-/l* 86.1%

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

   7. Simplified 86.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + a \cdot \frac{c}{d}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{-72}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+81}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.85 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+74}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* c a) (* d b)) (+ (* c c) (* d d))))
        (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -1.85e+124)
     t_1
     (if (<= d -1.9e-75)
       t_0
       (if (<= d 2.4e-117)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 2.9e+74) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.85e+124) {
		tmp = t_1;
	} else if (d <= -1.9e-75) {
		tmp = t_0;
	} else if (d <= 2.4e-117) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 2.9e+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 = ((c * a) + (d * b)) / ((c * c) + (d * d))
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-1.85d+124)) then
        tmp = t_1
    else if (d <= (-1.9d-75)) then
        tmp = t_0
    else if (d <= 2.4d-117) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 2.9d+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 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.85e+124) {
		tmp = t_1;
	} else if (d <= -1.9e-75) {
		tmp = t_0;
	} else if (d <= 2.4e-117) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 2.9e+74) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d))
	t_1 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -1.85e+124:
		tmp = t_1
	elif d <= -1.9e-75:
		tmp = t_0
	elif d <= 2.4e-117:
		tmp = (a + (b * (d / c))) / c
	elif d <= 2.9e+74:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * a) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -1.85e+124)
		tmp = t_1;
	elseif (d <= -1.9e-75)
		tmp = t_0;
	elseif (d <= 2.4e-117)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 2.9e+74)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	t_1 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -1.85e+124)
		tmp = t_1;
	elseif (d <= -1.9e-75)
		tmp = t_0;
	elseif (d <= 2.4e-117)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 2.9e+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[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.85e+124], t$95$1, If[LessEqual[d, -1.9e-75], t$95$0, If[LessEqual[d, 2.4e-117], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.9e+74], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.85000000000000004e124 or 2.9000000000000002e74 < d

   1. Initial program 37.1%

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

   3. Taylor expanded in d around inf 83.5%

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

      1. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

   if -1.85000000000000004e124 < d < -1.89999999999999997e-75 or 2.40000000000000014e-117 < d < 2.9000000000000002e74

   1. Initial program 83.9%

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

   if -1.89999999999999997e-75 < d < 2.40000000000000014e-117

   1. Initial program 58.4%

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

   3. Taylor expanded in c around inf 88.8%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+124}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-75}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{+74}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.7e-52)
   (/ b d)
   (if (<= d -2.1e-308)
     (/ a c)
     (if (<= d 4.4e-274)
       (/ (* b (/ d c)) c)
       (if (<= d 1.25e-9) (/ a c) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.7e-52) {
		tmp = b / d;
	} else if (d <= -2.1e-308) {
		tmp = a / c;
	} else if (d <= 4.4e-274) {
		tmp = (b * (d / c)) / c;
	} else if (d <= 1.25e-9) {
		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 (d <= (-5.7d-52)) then
        tmp = b / d
    else if (d <= (-2.1d-308)) then
        tmp = a / c
    else if (d <= 4.4d-274) then
        tmp = (b * (d / c)) / c
    else if (d <= 1.25d-9) 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 (d <= -5.7e-52) {
		tmp = b / d;
	} else if (d <= -2.1e-308) {
		tmp = a / c;
	} else if (d <= 4.4e-274) {
		tmp = (b * (d / c)) / c;
	} else if (d <= 1.25e-9) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.7e-52:
		tmp = b / d
	elif d <= -2.1e-308:
		tmp = a / c
	elif d <= 4.4e-274:
		tmp = (b * (d / c)) / c
	elif d <= 1.25e-9:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.7e-52)
		tmp = Float64(b / d);
	elseif (d <= -2.1e-308)
		tmp = Float64(a / c);
	elseif (d <= 4.4e-274)
		tmp = Float64(Float64(b * Float64(d / c)) / c);
	elseif (d <= 1.25e-9)
		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 (d <= -5.7e-52)
		tmp = b / d;
	elseif (d <= -2.1e-308)
		tmp = a / c;
	elseif (d <= 4.4e-274)
		tmp = (b * (d / c)) / c;
	elseif (d <= 1.25e-9)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.7e-52], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.1e-308], N[(a / c), $MachinePrecision], If[LessEqual[d, 4.4e-274], N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.25e-9], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.6999999999999997e-52 or 1.25e-9 < d

   1. Initial program 54.4%

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

   3. Taylor expanded in c around 0 62.1%

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

   if -5.6999999999999997e-52 < d < -2.1e-308 or 4.3999999999999999e-274 < d < 1.25e-9

   1. Initial program 63.4%

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

   3. Taylor expanded in c around inf 71.2%

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

   if -2.1e-308 < d < 4.3999999999999999e-274

   1. Initial program 59.7%

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

   3. Taylor expanded in c around inf 61.0%

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

   4. Taylor expanded in a around 0 60.4%

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

      1. associate-*r/ 82.4%

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

   6. Simplified 82.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-308}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-274}:\\ \;\;\;\;\frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+63} \lor \neg \left(d \leq 1.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5e+63) (not (<= d 1.1e+18)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5e+63) || !(d <= 1.1e+18)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5d+63)) .or. (.not. (d <= 1.1d+18))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5e+63) || !(d <= 1.1e+18)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5e+63) or not (d <= 1.1e+18):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5e+63) || !(d <= 1.1e+18))
		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 <= -5e+63) || ~((d <= 1.1e+18)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5e+63], N[Not[LessEqual[d, 1.1e+18]], $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 < -5.00000000000000011e63 or 1.1e18 < d

   1. Initial program 47.1%

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

   3. Taylor expanded in c around 0 69.7%

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

   if -5.00000000000000011e63 < d < 1.1e18

   1. Initial program 66.6%

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

   3. Taylor expanded in c around inf 78.9%

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

      1. associate-/l* 79.1%

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

   5. Simplified 79.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+63} \lor \neg \left(d \leq 1.1 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-17} \lor \neg \left(d \leq 9.5 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.5e-17) (not (<= d 9.5e+17)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.5e-17) || !(d <= 9.5e+17)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.5d-17)) .or. (.not. (d <= 9.5d+17))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.5e-17) || !(d <= 9.5e+17)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.5e-17) or not (d <= 9.5e+17):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.5e-17) || !(d <= 9.5e+17))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.5e-17) || ~((d <= 9.5e+17)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.5e-17], N[Not[LessEqual[d, 9.5e+17]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.50000000000000001e-17 or 9.5e17 < d

   1. Initial program 51.0%

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

   3. Taylor expanded in d around inf 76.9%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   if -5.50000000000000001e-17 < d < 9.5e17

   1. Initial program 65.5%

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

   3. Taylor expanded in c around inf 82.9%

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

      1. associate-/l* 83.2%

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

   5. Simplified 83.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{-17} \lor \neg \left(d \leq 9.5 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.92 \cdot 10^{+152} \lor \neg \left(d \leq 3 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.92e+152) (not (<= d 3e+213))) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.92e+152) || !(d <= 3e+213)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.92d+152)) .or. (.not. (d <= 3d+213))) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.92e+152) || !(d <= 3e+213)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.92e+152) or not (d <= 3e+213):
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.92e+152) || !(d <= 3e+213))
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.92e+152) || ~((d <= 3e+213)))
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.92e+152], N[Not[LessEqual[d, 3e+213]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.92000000000000008e152 or 3.0000000000000001e213 < d

   1. Initial program 34.9%

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

      1. *-un-lft-identity 34.9%

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

      2. associate-*r/ 34.9%

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

      3. fma-define 34.9%

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

      4. add-sqr-sqrt 34.9%

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

      5. times-frac 34.9%

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

      6. fma-define 34.9%

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

      7. hypot-define 34.9%

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

      8. fma-define 34.9%

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

      9. fma-define 34.9%

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

      10. hypot-define 66.2%

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

   4. Applied egg-rr 66.2%

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

   5. Taylor expanded in c around 0 42.5%

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

      1. associate-/l* 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in c around inf 21.9%

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

   if -1.92000000000000008e152 < d < 3.0000000000000001e213

   1. Initial program 64.8%

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

   3. Taylor expanded in c around inf 49.6%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.92 \cdot 10^{+152} \lor \neg \left(d \leq 3 \cdot 10^{+213}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{-52} \lor \neg \left(d \leq 1.25 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.7e-52) (not (<= d 1.25e-10))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e-52) || !(d <= 1.25e-10)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.7d-52)) .or. (.not. (d <= 1.25d-10))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e-52) || !(d <= 1.25e-10)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.7e-52) or not (d <= 1.25e-10):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.7e-52) || !(d <= 1.25e-10))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.7e-52) || ~((d <= 1.25e-10)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.7e-52], N[Not[LessEqual[d, 1.25e-10]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.6999999999999997e-52 or 1.25000000000000008e-10 < d

   1. Initial program 54.4%

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

   3. Taylor expanded in c around 0 62.1%

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

   if -5.6999999999999997e-52 < d < 1.25000000000000008e-10

   1. Initial program 63.3%

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

   3. Taylor expanded in c around inf 68.5%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{-52} \lor \neg \left(d \leq 1.25 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 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 58.8%

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

3. Taylor expanded in c around inf 40.6%

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

4. Final simplification 40.6%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024067 
(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))))