Complex division, real part

Percentage Accurate: 60.8% → 90.1%

Time: 14.1s

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: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \mathsf{fma}\left(a, \frac{c}{d}, b\right)\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{t\_0 \cdot t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ d (hypot c d))) (t_1 (fma a (/ c d) b)))
   (if (<= d -1.3e-67)
     (/ (* t_0 t_1) (hypot c d))
     (if (<= d 1.95e-42)
       (/ (+ a (* b (/ d c))) c)
       (* t_0 (/ t_1 (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = d / hypot(c, d);
	double t_1 = fma(a, (c / d), b);
	double tmp;
	if (d <= -1.3e-67) {
		tmp = (t_0 * t_1) / hypot(c, d);
	} else if (d <= 1.95e-42) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = t_0 * (t_1 / hypot(c, d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(d / hypot(c, d))
	t_1 = fma(a, Float64(c / d), b)
	tmp = 0.0
	if (d <= -1.3e-67)
		tmp = Float64(Float64(t_0 * t_1) / hypot(c, d));
	elseif (d <= 1.95e-42)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision]}, If[LessEqual[d, -1.3e-67], N[(N[(t$95$0 * t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.95e-42], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2999999999999999e-67

   1. Initial program 56.1%

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

   3. Taylor expanded in d around inf 56.1%

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

      1. associate-/l* 56.0%

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

   5. Simplified 56.0%

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

      1. *-commutative 56.0%

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

      2. add-sqr-sqrt 56.0%

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

      3. hypot-undefine 56.0%

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

      4. hypot-undefine 56.0%

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

      5. times-frac 94.0%

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

      6. +-commutative 94.0%

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

      7. fma-define 94.0%

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

   7. Applied egg-rr 94.0%

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

      1. associate-*l/ 94.1%

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

      2. fma-undefine 94.1%

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

      3. associate-/l* 88.2%

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

      4. +-commutative 88.2%

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

      5. *-commutative 88.2%

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

      6. +-commutative 88.2%

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

      7. associate-/l* 94.1%

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

      8. fma-undefine 94.1%

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

   9. Applied egg-rr 94.1%

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

   if -1.2999999999999999e-67 < d < 1.9500000000000001e-42

   1. Initial program 68.9%

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

   3. Taylor expanded in c around inf 88.4%

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

      1. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

   if 1.9500000000000001e-42 < d

   1. Initial program 52.1%

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

   3. Taylor expanded in d around inf 52.1%

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

      1. associate-/l* 52.1%

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

   5. Simplified 52.1%

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

      1. *-commutative 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. hypot-undefine 52.1%

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

      4. hypot-undefine 52.1%

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

      5. times-frac 96.8%

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

      6. +-commutative 96.8%

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

      7. fma-define 96.8%

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

   7. Applied egg-rr 96.8%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-42}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* c a) (* d b)) (+ (* c c) (* d d))) INFINITY)
   (/ (/ (fma a c (* d b)) (hypot c d)) (hypot c d))
   (/ (+ b (* a (/ c d))) d)))

C

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

Julia

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

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(c * a), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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))) < +inf.0

   1. Initial program 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. associate-*r/ 75.5%

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

      3. fma-define 75.5%

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

      4. add-sqr-sqrt 75.5%

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

      5. times-frac 75.4%

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

      6. fma-define 75.4%

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

      7. hypot-define 75.4%

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

      8. fma-define 75.4%

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

      9. fma-define 75.5%

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

      10. hypot-define 94.5%

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

   4. Applied egg-rr 94.5%

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

      1. associate-*l/ 94.7%

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

      2. *-un-lft-identity 94.7%

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

   6. Applied egg-rr 94.7%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   3. Taylor expanded in d around inf 44.6%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

4. Final simplification 88.6%

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

Alternative 3: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-66} \lor \neg \left(d \leq 1.95 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.65e-66) (not (<= d 1.95e-41)))
   (* (/ d (hypot c d)) (/ (fma a (/ c d) b) (hypot c d)))
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.65e-66) || !(d <= 1.95e-41)) {
		tmp = (d / hypot(c, d)) * (fma(a, (c / d), b) / hypot(c, d));
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.65e-66) || !(d <= 1.95e-41))
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(fma(a, Float64(c / d), b) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.65e-66], N[Not[LessEqual[d, 1.95e-41]], $MachinePrecision]], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6499999999999999e-66 or 1.94999999999999995e-41 < d

   1. Initial program 54.3%

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

   3. Taylor expanded in d around inf 54.3%

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

      1. associate-/l* 54.3%

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

   5. Simplified 54.3%

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

      1. *-commutative 54.3%

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

      2. add-sqr-sqrt 54.3%

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

      3. hypot-undefine 54.3%

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

      4. hypot-undefine 54.3%

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

      5. times-frac 95.2%

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

      6. +-commutative 95.2%

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

      7. fma-define 95.2%

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

   7. Applied egg-rr 95.2%

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

   if -1.6499999999999999e-66 < d < 1.94999999999999995e-41

   1. Initial program 68.9%

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

   3. Taylor expanded in c around inf 88.4%

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

      1. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{-66} \lor \neg \left(d \leq 1.95 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * a) + (d * b);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * a) + (d * b)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= math.inf:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * a) + (d * b);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= Inf)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. associate-*r/ 75.5%

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

      3. fma-define 75.5%

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

      4. add-sqr-sqrt 75.5%

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

      5. times-frac 75.4%

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

      6. fma-define 75.4%

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

      7. hypot-define 75.4%

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

      8. fma-define 75.4%

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

      9. fma-define 75.5%

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

      10. hypot-define 94.5%

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

   4. Applied egg-rr 94.5%

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

      1. fma-define 94.5%

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

      2. +-commutative 94.5%

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

   6. Applied egg-rr 94.5%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   3. Taylor expanded in d around inf 44.6%

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

      1. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

4. Final simplification 88.4%

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

Alternative 5: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{-45}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

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 -1.3e+106)
     (/ t_1 d)
     (if (<= d -7.5e-67)
       t_0
       (if (<= d 1.06e-45)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 1.45e+146) 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 <= -1.3e+106) {
		tmp = t_1 / d;
	} else if (d <= -7.5e-67) {
		tmp = t_0;
	} else if (d <= 1.06e-45) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.45e+146) {
		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 <= -1.3e+106) {
		tmp = t_1 / d;
	} else if (d <= -7.5e-67) {
		tmp = t_0;
	} else if (d <= 1.06e-45) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 1.45e+146) {
		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 <= -1.3e+106:
		tmp = t_1 / d
	elif d <= -7.5e-67:
		tmp = t_0
	elif d <= 1.06e-45:
		tmp = (a + (b * (d / c))) / c
	elif d <= 1.45e+146:
		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 <= -1.3e+106)
		tmp = Float64(t_1 / d);
	elseif (d <= -7.5e-67)
		tmp = t_0;
	elseif (d <= 1.06e-45)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 1.45e+146)
		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 <= -1.3e+106)
		tmp = t_1 / d;
	elseif (d <= -7.5e-67)
		tmp = t_0;
	elseif (d <= 1.06e-45)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 1.45e+146)
		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, -1.3e+106], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -7.5e-67], t$95$0, If[LessEqual[d, 1.06e-45], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.45e+146], 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 < -1.3000000000000001e106

   1. Initial program 35.2%

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

   3. Taylor expanded in d around inf 82.7%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -1.3000000000000001e106 < d < -7.5000000000000005e-67 or 1.06000000000000004e-45 < d < 1.4499999999999999e146

   1. Initial program 79.1%

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

   if -7.5000000000000005e-67 < d < 1.06000000000000004e-45

   1. Initial program 68.6%

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

   3. Taylor expanded in c around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

   if 1.4499999999999999e146 < d

   1. Initial program 24.3%

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

      1. *-un-lft-identity 24.3%

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

      2. associate-*r/ 24.3%

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

      3. fma-define 24.3%

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

      4. add-sqr-sqrt 24.3%

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

      5. times-frac 24.3%

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

      6. fma-define 24.3%

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

      7. hypot-define 24.3%

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

      8. fma-define 24.3%

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

      9. fma-define 24.3%

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

      10. hypot-define 55.8%

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

   4. Applied egg-rr 55.8%

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

   5. Taylor expanded in c around 0 72.8%

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

      1. associate-/l* 85.9%

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

      2. *-commutative 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.06 \cdot 10^{-45}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* c a) (* d b)) (+ (* c c) (* d d)))))
   (if (<= d -9.5e+103)
     (/ (+ b (* a (/ c d))) d)
     (if (<= d -7.2e-67)
       t_0
       (if (<= d 1.05e-45)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 4.1e+148)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (* c (/ a d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -9.5e+103) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -7.2e-67) {
		tmp = t_0;
	} else if (d <= 1.05e-45) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 4.1e+148) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (c * (a / d)));
	}
	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 tmp;
	if (d <= -9.5e+103) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -7.2e-67) {
		tmp = t_0;
	} else if (d <= 1.05e-45) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 4.1e+148) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (c * (a / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -9.5e+103:
		tmp = (b + (a * (c / d))) / d
	elif d <= -7.2e-67:
		tmp = t_0
	elif d <= 1.05e-45:
		tmp = (a + (b * (d / c))) / c
	elif d <= 4.1e+148:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (c * (a / d)))
	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)))
	tmp = 0.0
	if (d <= -9.5e+103)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -7.2e-67)
		tmp = t_0;
	elseif (d <= 1.05e-45)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 4.1e+148)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * a) + (d * b)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -9.5e+103)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -7.2e-67)
		tmp = t_0;
	elseif (d <= 1.05e-45)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 4.1e+148)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (c * (a / d)));
	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]}, If[LessEqual[d, -9.5e+103], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -7.2e-67], t$95$0, If[LessEqual[d, 1.05e-45], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.1e+148], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.49999999999999922e103

   1. Initial program 35.2%

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

   3. Taylor expanded in d around inf 82.7%

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

      1. associate-/l* 91.4%

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

   5. Simplified 91.4%

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

   if -9.49999999999999922e103 < d < -7.19999999999999998e-67 or 1.04999999999999998e-45 < d < 4.0999999999999998e148

   1. Initial program 79.1%

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

   if -7.19999999999999998e-67 < d < 1.04999999999999998e-45

   1. Initial program 68.6%

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

   3. Taylor expanded in c around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

   if 4.0999999999999998e148 < d

   1. Initial program 24.3%

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

      1. *-un-lft-identity 24.3%

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

      2. associate-*r/ 24.3%

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

      3. fma-define 24.3%

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

      4. add-sqr-sqrt 24.3%

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

      5. times-frac 24.3%

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

      6. fma-define 24.3%

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

      7. hypot-define 24.3%

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

      8. fma-define 24.3%

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

      9. fma-define 24.3%

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

      10. hypot-define 55.8%

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

   4. Applied egg-rr 55.8%

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

   5. Taylor expanded in c around 0 72.8%

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

      1. *-commutative 72.8%

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

      2. *-un-lft-identity 72.8%

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

      3. times-frac 86.0%

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

   7. Applied egg-rr 86.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-45}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 9.7 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2.1e+105)
     t_1
     (if (<= d -9.6e-68)
       t_0
       (if (<= d 2.05e-44)
         (/ (+ a (* b (/ d c))) c)
         (if (<= d 9.7e+145) 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 <= -2.1e+105) {
		tmp = t_1;
	} else if (d <= -9.6e-68) {
		tmp = t_0;
	} else if (d <= 2.05e-44) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 9.7e+145) {
		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 <= (-2.1d+105)) then
        tmp = t_1
    else if (d <= (-9.6d-68)) then
        tmp = t_0
    else if (d <= 2.05d-44) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 9.7d+145) 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 <= -2.1e+105) {
		tmp = t_1;
	} else if (d <= -9.6e-68) {
		tmp = t_0;
	} else if (d <= 2.05e-44) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 9.7e+145) {
		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 <= -2.1e+105:
		tmp = t_1
	elif d <= -9.6e-68:
		tmp = t_0
	elif d <= 2.05e-44:
		tmp = (a + (b * (d / c))) / c
	elif d <= 9.7e+145:
		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 <= -2.1e+105)
		tmp = t_1;
	elseif (d <= -9.6e-68)
		tmp = t_0;
	elseif (d <= 2.05e-44)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 9.7e+145)
		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 <= -2.1e+105)
		tmp = t_1;
	elseif (d <= -9.6e-68)
		tmp = t_0;
	elseif (d <= 2.05e-44)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 9.7e+145)
		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, -2.1e+105], t$95$1, If[LessEqual[d, -9.6e-68], t$95$0, If[LessEqual[d, 2.05e-44], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 9.7e+145], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.1000000000000001e105 or 9.69999999999999987e145 < d

   1. Initial program 30.8%

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

   3. Taylor expanded in d around inf 78.7%

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

      1. associate-/l* 89.1%

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

   5. Simplified 89.1%

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

   if -2.1000000000000001e105 < d < -9.59999999999999965e-68 or 2.04999999999999996e-44 < d < 9.69999999999999987e145

   1. Initial program 79.1%

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

   if -9.59999999999999965e-68 < d < 2.04999999999999996e-44

   1. Initial program 68.6%

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

   3. Taylor expanded in c around inf 88.3%

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

      1. associate-/l* 88.4%

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

   5. Simplified 88.4%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -9.6 \cdot 10^{-68}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 9.7 \cdot 10^{+145}:\\ \;\;\;\;\frac{c \cdot a + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -3.85 \cdot 10^{+21}:\\ \;\;\;\;\frac{d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-21} \lor \neg \left(d \leq 9.5 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2e+85)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d -3.85e+21)
     (/ (* d b) (+ (* c c) (* d d)))
     (if (or (<= d -5e-21) (not (<= d 9.5e-37)))
       (/ (+ b (/ a (/ d c))) d)
       (/ (+ a (* b (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2e+85) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -3.85e+21) {
		tmp = (d * b) / ((c * c) + (d * d));
	} else if ((d <= -5e-21) || !(d <= 9.5e-37)) {
		tmp = (b + (a / (d / c))) / 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 <= (-2d+85)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-3.85d+21)) then
        tmp = (d * b) / ((c * c) + (d * d))
    else if ((d <= (-5d-21)) .or. (.not. (d <= 9.5d-37))) then
        tmp = (b + (a / (d / c))) / 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 <= -2e+85) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -3.85e+21) {
		tmp = (d * b) / ((c * c) + (d * d));
	} else if ((d <= -5e-21) || !(d <= 9.5e-37)) {
		tmp = (b + (a / (d / c))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2e+85:
		tmp = (b + (a * (c / d))) / d
	elif d <= -3.85e+21:
		tmp = (d * b) / ((c * c) + (d * d))
	elif (d <= -5e-21) or not (d <= 9.5e-37):
		tmp = (b + (a / (d / c))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2e+85)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -3.85e+21)
		tmp = Float64(Float64(d * b) / Float64(Float64(c * c) + Float64(d * d)));
	elseif ((d <= -5e-21) || !(d <= 9.5e-37))
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / 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 <= -2e+85)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -3.85e+21)
		tmp = (d * b) / ((c * c) + (d * d));
	elseif ((d <= -5e-21) || ~((d <= 9.5e-37)))
		tmp = (b + (a / (d / c))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2e+85], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3.85e+21], N[(N[(d * b), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[d, -5e-21], N[Not[LessEqual[d, 9.5e-37]], $MachinePrecision]], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2e85

   1. Initial program 36.6%

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

   3. Taylor expanded in d around inf 80.2%

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

      1. associate-/l* 88.2%

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

   5. Simplified 88.2%

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

   if -2e85 < d < -3.85e21

   1. Initial program 88.7%

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

   3. Taylor expanded in a around 0 83.1%

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

   if -3.85e21 < d < -4.99999999999999973e-21 or 9.49999999999999927e-37 < d

   1. Initial program 56.5%

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

   3. Taylor expanded in d around inf 64.9%

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

      1. associate-/l* 70.2%

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

   5. Simplified 70.2%

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

      1. clear-num 70.2%

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

      2. un-div-inv 70.2%

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

   7. Applied egg-rr 70.2%

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

   if -4.99999999999999973e-21 < d < 9.49999999999999927e-37

   1. Initial program 69.1%

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

   3. Taylor expanded in c around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+85}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -3.85 \cdot 10^{+21}:\\ \;\;\;\;\frac{d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-21} \lor \neg \left(d \leq 9.5 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+30} \lor \neg \left(d \leq 4.8 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.8e+30) (not (<= d 4.8e+74)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+30) || !(d <= 4.8e+74)) {
		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 <= (-4.8d+30)) .or. (.not. (d <= 4.8d+74))) 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 <= -4.8e+30) || !(d <= 4.8e+74)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.8e+30) or not (d <= 4.8e+74):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.8e+30) || !(d <= 4.8e+74))
		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 <= -4.8e+30) || ~((d <= 4.8e+74)))
		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, -4.8e+30], N[Not[LessEqual[d, 4.8e+74]], $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 < -4.7999999999999999e30 or 4.80000000000000017e74 < d

   1. Initial program 44.8%

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

   3. Taylor expanded in c around 0 76.1%

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

   if -4.7999999999999999e30 < d < 4.80000000000000017e74

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 76.3%

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

      1. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+30} \lor \neg \left(d \leq 4.8 \cdot 10^{+74}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-25} \lor \neg \left(d \leq 9.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.8e-25) (not (<= d 9.5e-31)))
   (/ (+ 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 <= -5.8e-25) || !(d <= 9.5e-31)) {
		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 <= (-5.8d-25)) .or. (.not. (d <= 9.5d-31))) 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 <= -5.8e-25) || !(d <= 9.5e-31)) {
		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 <= -5.8e-25) or not (d <= 9.5e-31):
		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 <= -5.8e-25) || !(d <= 9.5e-31))
		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 <= -5.8e-25) || ~((d <= 9.5e-31)))
		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, -5.8e-25], N[Not[LessEqual[d, 9.5e-31]], $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 < -5.8000000000000001e-25 or 9.5000000000000008e-31 < d

   1. Initial program 53.5%

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

   3. Taylor expanded in d around inf 69.4%

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

      1. associate-/l* 75.0%

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

   5. Simplified 75.0%

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

   if -5.8000000000000001e-25 < d < 9.5000000000000008e-31

   1. Initial program 69.1%

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

   3. Taylor expanded in c around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{-25} \lor \neg \left(d \leq 9.5 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1e-20)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 1.15e-30) (/ (+ a (* b (/ d c))) c) (/ (+ b (/ a (/ d c))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1e-20) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.15e-30) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / 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 (d <= (-1d-20)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 1.15d-30) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1e-20) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.15e-30) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1e-20:
		tmp = (b + (a * (c / d))) / d
	elif d <= 1.15e-30:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1e-20)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 1.15e-30)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1e-20)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 1.15e-30)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1e-20], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.15e-30], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.99999999999999945e-21

   1. Initial program 55.3%

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

   3. Taylor expanded in d around inf 73.5%

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

      1. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   if -9.99999999999999945e-21 < d < 1.14999999999999992e-30

   1. Initial program 69.1%

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

   3. Taylor expanded in c around inf 86.1%

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

      1. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if 1.14999999999999992e-30 < d

   1. Initial program 51.4%

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

   3. Taylor expanded in d around inf 64.4%

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

      1. associate-/l* 70.7%

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

   5. Simplified 70.7%

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

      1. clear-num 70.7%

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

      2. un-div-inv 70.7%

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

   7. Applied egg-rr 70.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-20}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-30}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{+14} \lor \neg \left(d \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.3e+14) (not (<= d 1.3e-37))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.3e+14) || !(d <= 1.3e-37)) {
		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 <= (-4.3d+14)) .or. (.not. (d <= 1.3d-37))) 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 <= -4.3e+14) || !(d <= 1.3e-37)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.3e+14) or not (d <= 1.3e-37):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.3e+14) || !(d <= 1.3e-37))
		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 <= -4.3e+14) || ~((d <= 1.3e-37)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.3e+14], N[Not[LessEqual[d, 1.3e-37]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.3e14 or 1.2999999999999999e-37 < d

   1. Initial program 51.4%

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

   3. Taylor expanded in c around 0 68.3%

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

   if -4.3e14 < d < 1.2999999999999999e-37

   1. Initial program 70.0%

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

   3. Taylor expanded in c around inf 65.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.3 \cdot 10^{+14} \lor \neg \left(d \leq 1.3 \cdot 10^{-37}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% 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 60.4%

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

3. Taylor expanded in c around inf 43.0%

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

4. Final simplification 43.0%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(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))))