Complex division, real part

Percentage Accurate: 61.5% → 82.8%

Time: 8.9s

Alternatives: 12

Speedup: 0.8×

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-107}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -6.1e+88)
     t_0
     (if (<= d -1.95e-55)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= d 1.95e-107)
         (/ (+ a (/ b (/ c d))) c)
         (if (<= d 3e+28) (/ (fma a c (* b d)) (fma c c (* d d))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -6.1e+88) {
		tmp = t_0;
	} else if (d <= -1.95e-55) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 1.95e-107) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 3e+28) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -6.1e+88)
		tmp = t_0;
	elseif (d <= -1.95e-55)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.95e-107)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 3e+28)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6.1e+88], t$95$0, If[LessEqual[d, -1.95e-55], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.95e-107], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+28], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.0999999999999998e88 or 3.0000000000000001e28 < d

   1. Initial program 44.2%

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

   3. Taylor expanded in d around inf 79.5%

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

      1. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -6.0999999999999998e88 < d < -1.95e-55

   1. Initial program 89.4%

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

   if -1.95e-55 < d < 1.95e-107

   1. Initial program 72.4%

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

   3. Taylor expanded in c around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. associate-/l* 90.8%

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

      3. fma-define 90.9%

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

   5. Simplified 90.9%

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

      1. fma-undefine 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. clear-num 90.8%

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

      2. un-div-inv 90.9%

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

   9. Applied egg-rr 90.9%

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

   if 1.95e-107 < d < 3.0000000000000001e28

   1. Initial program 91.5%

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

      1. fma-define 91.6%

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

      2. fma-define 91.6%

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

   3. Simplified 91.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-55}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.95 \cdot 10^{-107}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+189)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1e+189) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * 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(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+189)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * 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[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+189], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(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))) < 1e189

   1. Initial program 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. associate-*r/ 80.5%

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

      3. fma-define 80.5%

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

      4. add-sqr-sqrt 80.5%

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

      5. times-frac 80.5%

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

      6. fma-define 80.5%

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

      7. hypot-define 80.5%

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

      8. fma-define 80.5%

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

      9. fma-define 80.5%

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

      10. hypot-define 95.5%

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

   4. Applied egg-rr 95.5%

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

   if 1e189 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 25.7%

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

   3. Taylor expanded in d around inf 61.1%

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

      1. associate-/l* 65.3%

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

   5. Simplified 65.3%

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

Alternative 3: 84.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+189)
     (* (/ 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 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+189) {
		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 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+189) {
		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 = (a * c) + (b * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+189:
		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(a * c) + Float64(b * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+189)
		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 = (a * c) + (b * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+189)
		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[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+189], 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))) < 1e189

   1. Initial program 80.5%

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

      1. *-un-lft-identity 80.5%

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

      2. associate-*r/ 80.5%

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

      3. fma-define 80.5%

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

      4. add-sqr-sqrt 80.5%

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

      5. times-frac 80.5%

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

      6. fma-define 80.5%

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

      7. hypot-define 80.5%

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

      8. fma-define 80.5%

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

      9. fma-define 80.5%

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

      10. hypot-define 95.5%

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

   4. Applied egg-rr 95.5%

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

      1. fma-define 95.5%

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

      2. +-commutative 95.5%

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

   6. Applied egg-rr 95.5%

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

   if 1e189 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 25.7%

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

   3. Taylor expanded in d around inf 61.1%

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

      1. associate-/l* 65.3%

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

   5. Simplified 65.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+189}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a \cdot c + b \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + b \cdot \frac{d}{c}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ t_2 := \frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+121}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* b (/ d c))) c))
        (t_1 (/ (+ b (* a (/ c d))) d))
        (t_2 (/ (+ b (/ (* a c) d)) d)))
   (if (<= d -2.1e+44)
     t_1
     (if (<= d 3.2e-105)
       t_0
       (if (<= d 1.25e-47)
         t_2
         (if (<= d 3.7e+37)
           t_0
           (if (<= d 1.85e+121) t_2 (if (<= d 1.9e+121) (/ a c) t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (b * (d / c))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double t_2 = (b + ((a * c) / d)) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_1;
	} else if (d <= 3.2e-105) {
		tmp = t_0;
	} else if (d <= 1.25e-47) {
		tmp = t_2;
	} else if (d <= 3.7e+37) {
		tmp = t_0;
	} else if (d <= 1.85e+121) {
		tmp = t_2;
	} else if (d <= 1.9e+121) {
		tmp = a / c;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = (a + (b * (d / c))) / c
    t_1 = (b + (a * (c / d))) / d
    t_2 = (b + ((a * c) / d)) / d
    if (d <= (-2.1d+44)) then
        tmp = t_1
    else if (d <= 3.2d-105) then
        tmp = t_0
    else if (d <= 1.25d-47) then
        tmp = t_2
    else if (d <= 3.7d+37) then
        tmp = t_0
    else if (d <= 1.85d+121) then
        tmp = t_2
    else if (d <= 1.9d+121) then
        tmp = a / c
    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 = (a + (b * (d / c))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double t_2 = (b + ((a * c) / d)) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_1;
	} else if (d <= 3.2e-105) {
		tmp = t_0;
	} else if (d <= 1.25e-47) {
		tmp = t_2;
	} else if (d <= 3.7e+37) {
		tmp = t_0;
	} else if (d <= 1.85e+121) {
		tmp = t_2;
	} else if (d <= 1.9e+121) {
		tmp = a / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (b * (d / c))) / c
	t_1 = (b + (a * (c / d))) / d
	t_2 = (b + ((a * c) / d)) / d
	tmp = 0
	if d <= -2.1e+44:
		tmp = t_1
	elif d <= 3.2e-105:
		tmp = t_0
	elif d <= 1.25e-47:
		tmp = t_2
	elif d <= 3.7e+37:
		tmp = t_0
	elif d <= 1.85e+121:
		tmp = t_2
	elif d <= 1.9e+121:
		tmp = a / c
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(b * Float64(d / c))) / c)
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	t_2 = Float64(Float64(b + Float64(Float64(a * c) / d)) / d)
	tmp = 0.0
	if (d <= -2.1e+44)
		tmp = t_1;
	elseif (d <= 3.2e-105)
		tmp = t_0;
	elseif (d <= 1.25e-47)
		tmp = t_2;
	elseif (d <= 3.7e+37)
		tmp = t_0;
	elseif (d <= 1.85e+121)
		tmp = t_2;
	elseif (d <= 1.9e+121)
		tmp = Float64(a / c);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (b * (d / c))) / c;
	t_1 = (b + (a * (c / d))) / d;
	t_2 = (b + ((a * c) / d)) / d;
	tmp = 0.0;
	if (d <= -2.1e+44)
		tmp = t_1;
	elseif (d <= 3.2e-105)
		tmp = t_0;
	elseif (d <= 1.25e-47)
		tmp = t_2;
	elseif (d <= 3.7e+37)
		tmp = t_0;
	elseif (d <= 1.85e+121)
		tmp = t_2;
	elseif (d <= 1.9e+121)
		tmp = a / c;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.1e+44], t$95$1, If[LessEqual[d, 3.2e-105], t$95$0, If[LessEqual[d, 1.25e-47], t$95$2, If[LessEqual[d, 3.7e+37], t$95$0, If[LessEqual[d, 1.85e+121], t$95$2, If[LessEqual[d, 1.9e+121], N[(a / c), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.09999999999999987e44 or 1.9e121 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in d around inf 80.3%

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

      1. associate-/l* 85.5%

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

   5. Simplified 85.5%

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

   if -2.09999999999999987e44 < d < 3.19999999999999981e-105 or 1.25000000000000003e-47 < d < 3.6999999999999999e37

   1. Initial program 77.4%

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

   3. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 82.5%

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

   5. Simplified 82.5%

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

   if 3.19999999999999981e-105 < d < 1.25000000000000003e-47 or 3.6999999999999999e37 < d < 1.85000000000000006e121

   1. Initial program 79.8%

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

   3. Taylor expanded in d around inf 76.3%

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

   if 1.85000000000000006e121 < d < 1.9e121

   1. Initial program 5.3%

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

   3. Taylor expanded in c around inf 100.0%

      \[\leadsto \color{blue}{\frac{a}{c}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9.8 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.02 \cdot 10^{-109}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -5.2e+92)
     t_1
     (if (<= d -9.8e-53)
       t_0
       (if (<= d 2.02e-109)
         (/ (+ a (/ b (/ c d))) c)
         (if (<= d 3e+28) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -5.2e+92) {
		tmp = t_1;
	} else if (d <= -9.8e-53) {
		tmp = t_0;
	} else if (d <= 2.02e-109) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 3e+28) {
		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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-5.2d+92)) then
        tmp = t_1
    else if (d <= (-9.8d-53)) then
        tmp = t_0
    else if (d <= 2.02d-109) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= 3d+28) 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 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -5.2e+92) {
		tmp = t_1;
	} else if (d <= -9.8e-53) {
		tmp = t_0;
	} else if (d <= 2.02e-109) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 3e+28) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -5.2e+92:
		tmp = t_1
	elif d <= -9.8e-53:
		tmp = t_0
	elif d <= 2.02e-109:
		tmp = (a + (b / (c / d))) / c
	elif d <= 3e+28:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -5.2e+92)
		tmp = t_1;
	elseif (d <= -9.8e-53)
		tmp = t_0;
	elseif (d <= 2.02e-109)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 3e+28)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -5.2e+92)
		tmp = t_1;
	elseif (d <= -9.8e-53)
		tmp = t_0;
	elseif (d <= 2.02e-109)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= 3e+28)
		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[(a * c), $MachinePrecision] + N[(b * d), $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, -5.2e+92], t$95$1, If[LessEqual[d, -9.8e-53], t$95$0, If[LessEqual[d, 2.02e-109], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+28], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.1999999999999998e92 or 3.0000000000000001e28 < d

   1. Initial program 44.2%

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

   3. Taylor expanded in d around inf 79.5%

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

      1. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -5.1999999999999998e92 < d < -9.79999999999999926e-53 or 2.02e-109 < d < 3.0000000000000001e28

   1. Initial program 90.6%

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

   if -9.79999999999999926e-53 < d < 2.02e-109

   1. Initial program 72.4%

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

   3. Taylor expanded in c around inf 89.8%

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

      1. +-commutative 89.8%

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

      2. associate-/l* 90.8%

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

      3. fma-define 90.9%

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

   5. Simplified 90.9%

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

      1. fma-undefine 90.8%

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

   7. Applied egg-rr 90.8%

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

      1. clear-num 90.8%

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

      2. un-div-inv 90.9%

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

   9. Applied egg-rr 90.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -9.8 \cdot 10^{-53}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.02 \cdot 10^{-109}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+28}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+110} \lor \neg \left(c \leq -1.15 \cdot 10^{+40}\right) \land \left(c \leq -4.5 \cdot 10^{+38} \lor \neg \left(c \leq 2.05 \cdot 10^{-8} \lor \neg \left(c \leq 8 \cdot 10^{+133}\right) \land c \leq 2.6 \cdot 10^{+151}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5e+110)
         (and (not (<= c -1.15e+40))
              (or (<= c -4.5e+38)
                  (not
                   (or (<= c 2.05e-8)
                       (and (not (<= c 8e+133)) (<= c 2.6e+151)))))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5e+110) || (!(c <= -1.15e+40) && ((c <= -4.5e+38) || !((c <= 2.05e-8) || (!(c <= 8e+133) && (c <= 2.6e+151)))))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5d+110)) .or. (.not. (c <= (-1.15d+40))) .and. (c <= (-4.5d+38)) .or. (.not. (c <= 2.05d-8) .or. (.not. (c <= 8d+133)) .and. (c <= 2.6d+151))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5e+110) || (!(c <= -1.15e+40) && ((c <= -4.5e+38) || !((c <= 2.05e-8) || (!(c <= 8e+133) && (c <= 2.6e+151)))))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5e+110) or (not (c <= -1.15e+40) and ((c <= -4.5e+38) or not ((c <= 2.05e-8) or (not (c <= 8e+133) and (c <= 2.6e+151))))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5e+110) || (!(c <= -1.15e+40) && ((c <= -4.5e+38) || !((c <= 2.05e-8) || (!(c <= 8e+133) && (c <= 2.6e+151))))))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5e+110) || (~((c <= -1.15e+40)) && ((c <= -4.5e+38) || ~(((c <= 2.05e-8) || (~((c <= 8e+133)) && (c <= 2.6e+151)))))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5e+110], And[N[Not[LessEqual[c, -1.15e+40]], $MachinePrecision], Or[LessEqual[c, -4.5e+38], N[Not[Or[LessEqual[c, 2.05e-8], And[N[Not[LessEqual[c, 8e+133]], $MachinePrecision], LessEqual[c, 2.6e+151]]]], $MachinePrecision]]]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.99999999999999978e110 or -1.14999999999999997e40 < c < -4.4999999999999998e38 or 2.05000000000000016e-8 < c < 8.0000000000000002e133 or 2.60000000000000013e151 < c

   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 78.7%

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

   if -4.99999999999999978e110 < c < -1.14999999999999997e40 or -4.4999999999999998e38 < c < 2.05000000000000016e-8 or 8.0000000000000002e133 < c < 2.60000000000000013e151

   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 0 61.9%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+110} \lor \neg \left(c \leq -1.15 \cdot 10^{+40}\right) \land \left(c \leq -4.5 \cdot 10^{+38} \lor \neg \left(c \leq 2.05 \cdot 10^{-8} \lor \neg \left(c \leq 8 \cdot 10^{+133}\right) \land c \leq 2.6 \cdot 10^{+151}\right)\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+31} \lor \neg \left(d \leq 1.35 \cdot 10^{-76}\right) \land \left(d \leq 1.95 \cdot 10^{-44} \lor \neg \left(d \leq 2.3 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.2e+31)
         (and (not (<= d 1.35e-76)) (or (<= d 1.95e-44) (not (<= d 2.3e+37)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.2e+31) || (!(d <= 1.35e-76) && ((d <= 1.95e-44) || !(d <= 2.3e+37)))) {
		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 <= (-1.2d+31)) .or. (.not. (d <= 1.35d-76)) .and. (d <= 1.95d-44) .or. (.not. (d <= 2.3d+37))) 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 <= -1.2e+31) || (!(d <= 1.35e-76) && ((d <= 1.95e-44) || !(d <= 2.3e+37)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.2e+31) or (not (d <= 1.35e-76) and ((d <= 1.95e-44) or not (d <= 2.3e+37))):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.2e+31) || (!(d <= 1.35e-76) && ((d <= 1.95e-44) || !(d <= 2.3e+37))))
		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 <= -1.2e+31) || (~((d <= 1.35e-76)) && ((d <= 1.95e-44) || ~((d <= 2.3e+37)))))
		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, -1.2e+31], And[N[Not[LessEqual[d, 1.35e-76]], $MachinePrecision], Or[LessEqual[d, 1.95e-44], N[Not[LessEqual[d, 2.3e+37]], $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 < -1.19999999999999991e31 or 1.35e-76 < d < 1.9500000000000001e-44 or 2.30000000000000002e37 < 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 c around 0 74.4%

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

   if -1.19999999999999991e31 < d < 1.35e-76 or 1.9500000000000001e-44 < d < 2.30000000000000002e37

   1. Initial program 78.6%

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

   3. Taylor expanded in c around inf 80.6%

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

      1. associate-/l* 81.3%

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

   5. Simplified 81.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+31} \lor \neg \left(d \leq 1.35 \cdot 10^{-76}\right) \land \left(d \leq 1.95 \cdot 10^{-44} \lor \neg \left(d \leq 2.3 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-43} \lor \neg \left(d \leq 1.38 \cdot 10^{-40}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -2.1e+44)
     t_0
     (if (<= d 3.2e-105)
       (/ (+ a (* b (/ d c))) c)
       (if (or (<= d 2.5e-43) (not (<= d 1.38e-40))) t_0 (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_0;
	} else if (d <= 3.2e-105) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((d <= 2.5e-43) || !(d <= 1.38e-40)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    if (d <= (-2.1d+44)) then
        tmp = t_0
    else if (d <= 3.2d-105) then
        tmp = (a + (b * (d / c))) / c
    else if ((d <= 2.5d-43) .or. (.not. (d <= 1.38d-40))) then
        tmp = t_0
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_0;
	} else if (d <= 3.2e-105) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((d <= 2.5e-43) || !(d <= 1.38e-40)) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -2.1e+44:
		tmp = t_0
	elif d <= 3.2e-105:
		tmp = (a + (b * (d / c))) / c
	elif (d <= 2.5e-43) or not (d <= 1.38e-40):
		tmp = t_0
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -2.1e+44)
		tmp = t_0;
	elseif (d <= 3.2e-105)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif ((d <= 2.5e-43) || !(d <= 1.38e-40))
		tmp = t_0;
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -2.1e+44)
		tmp = t_0;
	elseif (d <= 3.2e-105)
		tmp = (a + (b * (d / c))) / c;
	elseif ((d <= 2.5e-43) || ~((d <= 1.38e-40)))
		tmp = t_0;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.1e+44], t$95$0, If[LessEqual[d, 3.2e-105], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 2.5e-43], N[Not[LessEqual[d, 1.38e-40]], $MachinePrecision]], t$95$0, N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.09999999999999987e44 or 3.19999999999999981e-105 < d < 2.50000000000000009e-43 or 1.37999999999999996e-40 < d

   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 d around inf 74.4%

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

      1. associate-/l* 77.2%

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

   5. Simplified 77.2%

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

   if -2.09999999999999987e44 < d < 3.19999999999999981e-105

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf 85.2%

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 2.50000000000000009e-43 < d < 1.37999999999999996e-40

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{-43} \lor \neg \left(d \leq 1.38 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -2.3e+44)
     t_0
     (if (<= d 2.9e-105)
       (/ (+ a (/ b (/ c d))) c)
       (if (<= d 1.35e-46)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 1.18e-39) (/ a c) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.3e+44) {
		tmp = t_0;
	} else if (d <= 2.9e-105) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.35e-46) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 1.18e-39) {
		tmp = a / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    if (d <= (-2.3d+44)) then
        tmp = t_0
    else if (d <= 2.9d-105) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= 1.35d-46) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 1.18d-39) then
        tmp = a / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.3e+44) {
		tmp = t_0;
	} else if (d <= 2.9e-105) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.35e-46) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 1.18e-39) {
		tmp = a / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -2.3e+44:
		tmp = t_0
	elif d <= 2.9e-105:
		tmp = (a + (b / (c / d))) / c
	elif d <= 1.35e-46:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 1.18e-39:
		tmp = a / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -2.3e+44)
		tmp = t_0;
	elseif (d <= 2.9e-105)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 1.35e-46)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 1.18e-39)
		tmp = Float64(a / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -2.3e+44)
		tmp = t_0;
	elseif (d <= 2.9e-105)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= 1.35e-46)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 1.18e-39)
		tmp = a / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.3e+44], t$95$0, If[LessEqual[d, 2.9e-105], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.35e-46], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.18e-39], N[(a / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.30000000000000004e44 or 1.17999999999999993e-39 < d

   1. Initial program 53.7%

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

   3. Taylor expanded in d around inf 75.1%

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

      1. associate-/l* 78.3%

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

   5. Simplified 78.3%

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

   if -2.30000000000000004e44 < d < 2.90000000000000003e-105

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf 85.2%

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

      1. +-commutative 85.2%

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

      2. associate-/l* 86.1%

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

      3. fma-define 86.1%

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

   5. Simplified 86.1%

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

      1. fma-undefine 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. clear-num 86.1%

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

      2. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 2.90000000000000003e-105 < d < 1.35e-46

   1. Initial program 99.6%

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

   3. Taylor expanded in d around inf 67.9%

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

   if 1.35e-46 < d < 1.17999999999999993e-39

   1. Initial program 100.0%

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

   3. Taylor expanded in c around inf 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-46}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -2.1e+44)
     t_0
     (if (<= d 3.2e-105)
       (/ (+ a (/ b (/ c d))) c)
       (if (<= d 1.15e-50) (/ (* b d) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_0;
	} else if (d <= 3.2e-105) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.15e-50) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    if (d <= (-2.1d+44)) then
        tmp = t_0
    else if (d <= 3.2d-105) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= 1.15d-50) then
        tmp = (b * d) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.1e+44) {
		tmp = t_0;
	} else if (d <= 3.2e-105) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.15e-50) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -2.1e+44:
		tmp = t_0
	elif d <= 3.2e-105:
		tmp = (a + (b / (c / d))) / c
	elif d <= 1.15e-50:
		tmp = (b * d) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -2.1e+44)
		tmp = t_0;
	elseif (d <= 3.2e-105)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 1.15e-50)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -2.1e+44)
		tmp = t_0;
	elseif (d <= 3.2e-105)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= 1.15e-50)
		tmp = (b * d) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.1e+44], t$95$0, If[LessEqual[d, 3.2e-105], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.15e-50], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.09999999999999987e44 or 1.1500000000000001e-50 < d

   1. Initial program 55.1%

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

   3. Taylor expanded in d around inf 74.5%

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

      1. associate-/l* 77.5%

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

   5. Simplified 77.5%

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

   if -2.09999999999999987e44 < d < 3.19999999999999981e-105

   1. Initial program 74.8%

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

   3. Taylor expanded in c around inf 85.2%

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

      1. +-commutative 85.2%

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

      2. associate-/l* 86.1%

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

      3. fma-define 86.1%

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

   5. Simplified 86.1%

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

      1. fma-undefine 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. clear-num 86.1%

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

      2. un-div-inv 86.1%

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

   9. Applied egg-rr 86.1%

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

   if 3.19999999999999981e-105 < d < 1.1500000000000001e-50

   1. Initial program 99.5%

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

   3. Taylor expanded in a around 0 77.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-50}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+125}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.16e+125)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 7e-67) (/ (+ b (* a (/ c d))) d) (/ (+ a (* d (/ b c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.16e+125) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 7e-67) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.16d+125)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 7d-67) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.16e+125) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 7e-67) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.16e+125:
		tmp = (a + (b * (d / c))) / c
	elif c <= 7e-67:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.16e+125)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 7e-67)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.16e+125)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 7e-67)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.16e+125], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7e-67], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.16000000000000009e125

   1. Initial program 53.9%

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

   3. Taylor expanded in c around inf 79.2%

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

      1. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -1.16000000000000009e125 < c < 7.0000000000000001e-67

   1. Initial program 69.2%

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

   3. Taylor expanded in d around inf 74.9%

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

      1. associate-/l* 76.2%

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

   5. Simplified 76.2%

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

   if 7.0000000000000001e-67 < c

   1. Initial program 64.0%

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

   3. Taylor expanded in c around inf 71.2%

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

      1. +-commutative 71.2%

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

      2. associate-/l* 71.4%

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

      3. fma-define 71.4%

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

   5. Simplified 71.4%

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

      1. fma-undefine 71.4%

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

   7. Applied egg-rr 71.4%

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

      1. clear-num 71.4%

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

      2. un-div-inv 71.4%

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

   9. Applied egg-rr 71.4%

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

       1. associate-/r/ 74.9%

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

   11. Simplified 74.9%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+125}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-67}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.1% 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 65.7%

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

3. Taylor expanded in c around inf 42.8%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
4. Add Preprocessing

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(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))))