Complex division, real part

Percentage Accurate: 61.5% → 83.4%

Time: 10.1s

Alternatives: 16

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% 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: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -6.1 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.62 \cdot 10^{-105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a c (* b d)) (fma c c (* d d))))
        (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -6.1e+88)
     t_1
     (if (<= d -1e-89)
       t_0
       (if (<= d 1.62e-105)
         (/ (fma b (/ d c) a) c)
         (if (<= d 3.4e+38) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, c, (b * d)) / fma(c, c, (d * d));
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -6.1e+88) {
		tmp = t_1;
	} else if (d <= -1e-89) {
		tmp = t_0;
	} else if (d <= 1.62e-105) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 3.4e+38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -6.1e+88)
		tmp = t_1;
	elseif (d <= -1e-89)
		tmp = t_0;
	elseif (d <= 1.62e-105)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 3.4e+38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + 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, -6.1e+88], t$95$1, If[LessEqual[d, -1e-89], t$95$0, If[LessEqual[d, 1.62e-105], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.4e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.0999999999999998e88 or 3.39999999999999996e38 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in d around inf 80.8%

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

      1. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if -6.0999999999999998e88 < d < -1.00000000000000004e-89 or 1.62e-105 < d < 3.39999999999999996e38

   1. Initial program 91.6%

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

   if -1.00000000000000004e-89 < d < 1.62e-105

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf 92.4%

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

      1. +-commutative 92.4%

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

      2. associate-/l* 93.6%

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

      3. fma-define 93.6%

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

   5. Simplified 93.6%

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

Alternative 2: 86.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+306)
   (* (/ 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+306) {
		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+306)
		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+306], 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))) < 1.00000000000000002e306

   1. Initial program 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. associate-*r/ 81.6%

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

      3. fma-define 81.6%

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

      4. add-sqr-sqrt 81.6%

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

      5. times-frac 81.5%

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

      6. fma-define 81.5%

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

      7. hypot-define 81.6%

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

      8. fma-define 81.6%

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

      9. fma-define 81.6%

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

      10. hypot-define 95.8%

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

   4. Applied egg-rr 95.8%

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

   if 1.00000000000000002e306 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.7%

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

   3. Taylor expanded in d around inf 60.1%

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

      1. associate-/l* 65.2%

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

   5. Simplified 65.2%

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

Alternative 3: 86.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+306)
     (* (/ 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+306) {
		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+306) {
		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+306:
		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+306)
		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+306)
		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+306], 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))) < 1.00000000000000002e306

   1. Initial program 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. associate-*r/ 81.6%

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

      3. fma-define 81.6%

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

      4. add-sqr-sqrt 81.6%

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

      5. times-frac 81.5%

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

      6. fma-define 81.5%

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

      7. hypot-define 81.6%

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

      8. fma-define 81.6%

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

      9. fma-define 81.6%

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

      10. hypot-define 95.8%

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

   4. Applied egg-rr 95.8%

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

      1. fma-define 95.8%

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

      2. +-commutative 95.8%

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

   6. Applied egg-rr 95.8%

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

   if 1.00000000000000002e306 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 11.7%

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

   3. Taylor expanded in d around inf 60.1%

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

      1. associate-/l* 65.2%

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

   5. Simplified 65.2%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a \cdot c + b \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-106}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1.32e+94)
     t_1
     (if (<= d -1.1e-82)
       t_0
       (if (<= d 1.25e-106)
         (/ (fma b (/ d c) a) c)
         (if (<= d 3.4e+38) 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 <= -1.32e+94) {
		tmp = t_1;
	} else if (d <= -1.1e-82) {
		tmp = t_0;
	} else if (d <= 1.25e-106) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 3.4e+38) {
		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 <= -1.32e+94)
		tmp = t_1;
	elseif (d <= -1.1e-82)
		tmp = t_0;
	elseif (d <= 1.25e-106)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 3.4e+38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return 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, -1.32e+94], t$95$1, If[LessEqual[d, -1.1e-82], t$95$0, If[LessEqual[d, 1.25e-106], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.4e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.32000000000000003e94 or 3.39999999999999996e38 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in d around inf 80.8%

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

      1. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if -1.32000000000000003e94 < d < -1.09999999999999993e-82 or 1.24999999999999996e-106 < d < 3.39999999999999996e38

   1. Initial program 91.6%

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

   if -1.09999999999999993e-82 < d < 1.24999999999999996e-106

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf 92.4%

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

      1. +-commutative 92.4%

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

      2. associate-/l* 93.6%

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

      3. fma-define 93.6%

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

   5. Simplified 93.6%

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

Alternative 5: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2.8e+92)
     t_1
     (if (<= d -9.2e-81)
       t_0
       (if (<= d 8.8e-107)
         (/ (+ a (/ b (/ c d))) c)
         (if (<= d 3.4e+38) 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 <= -2.8e+92) {
		tmp = t_1;
	} else if (d <= -9.2e-81) {
		tmp = t_0;
	} else if (d <= 8.8e-107) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 3.4e+38) {
		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 <= (-2.8d+92)) then
        tmp = t_1
    else if (d <= (-9.2d-81)) then
        tmp = t_0
    else if (d <= 8.8d-107) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= 3.4d+38) 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 <= -2.8e+92) {
		tmp = t_1;
	} else if (d <= -9.2e-81) {
		tmp = t_0;
	} else if (d <= 8.8e-107) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 3.4e+38) {
		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 <= -2.8e+92:
		tmp = t_1
	elif d <= -9.2e-81:
		tmp = t_0
	elif d <= 8.8e-107:
		tmp = (a + (b / (c / d))) / c
	elif d <= 3.4e+38:
		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 <= -2.8e+92)
		tmp = t_1;
	elseif (d <= -9.2e-81)
		tmp = t_0;
	elseif (d <= 8.8e-107)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 3.4e+38)
		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 <= -2.8e+92)
		tmp = t_1;
	elseif (d <= -9.2e-81)
		tmp = t_0;
	elseif (d <= 8.8e-107)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= 3.4e+38)
		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, -2.8e+92], t$95$1, If[LessEqual[d, -9.2e-81], t$95$0, If[LessEqual[d, 8.8e-107], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.4e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.80000000000000001e92 or 3.39999999999999996e38 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in d around inf 80.8%

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

      1. associate-/l* 84.9%

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

   5. Simplified 84.9%

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

   if -2.80000000000000001e92 < d < -9.19999999999999965e-81 or 8.8000000000000005e-107 < d < 3.39999999999999996e38

   1. Initial program 91.6%

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

   if -9.19999999999999965e-81 < d < 8.8000000000000005e-107

   1. Initial program 70.8%

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

   3. Taylor expanded in c around inf 92.4%

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

      1. +-commutative 92.4%

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

      2. associate-/l* 93.6%

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

      3. fma-define 93.6%

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

   5. Simplified 93.6%

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

      1. fma-undefine 93.6%

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

   7. Applied egg-rr 93.6%

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

      1. clear-num 93.6%

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

      2. un-div-inv 93.6%

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

   9. Applied egg-rr 93.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-81}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{d \cdot \left(\frac{a}{d} + \frac{b}{c}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -1.35e+24)
     t_0
     (if (<= d 4.8e-106)
       (/ (+ a (/ b (/ c d))) c)
       (if (<= d 1.9e+17)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 1.45e+36) (/ (* d (+ (/ a d) (/ b c))) 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 <= -1.35e+24) {
		tmp = t_0;
	} else if (d <= 4.8e-106) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.9e+17) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 1.45e+36) {
		tmp = (d * ((a / d) + (b / c))) / 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 <= (-1.35d+24)) then
        tmp = t_0
    else if (d <= 4.8d-106) then
        tmp = (a + (b / (c / d))) / c
    else if (d <= 1.9d+17) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 1.45d+36) then
        tmp = (d * ((a / d) + (b / c))) / 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 <= -1.35e+24) {
		tmp = t_0;
	} else if (d <= 4.8e-106) {
		tmp = (a + (b / (c / d))) / c;
	} else if (d <= 1.9e+17) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 1.45e+36) {
		tmp = (d * ((a / d) + (b / c))) / 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 <= -1.35e+24:
		tmp = t_0
	elif d <= 4.8e-106:
		tmp = (a + (b / (c / d))) / c
	elif d <= 1.9e+17:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 1.45e+36:
		tmp = (d * ((a / d) + (b / c))) / 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 <= -1.35e+24)
		tmp = t_0;
	elseif (d <= 4.8e-106)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	elseif (d <= 1.9e+17)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.45e+36)
		tmp = Float64(Float64(d * Float64(Float64(a / d) + Float64(b / c))) / 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 <= -1.35e+24)
		tmp = t_0;
	elseif (d <= 4.8e-106)
		tmp = (a + (b / (c / d))) / c;
	elseif (d <= 1.9e+17)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 1.45e+36)
		tmp = (d * ((a / d) + (b / c))) / 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, -1.35e+24], t$95$0, If[LessEqual[d, 4.8e-106], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.9e+17], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.45e+36], N[(N[(d * N[(N[(a / d), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.35e24 or 1.45e36 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -1.35e24 < d < 4.7999999999999995e-106

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. +-commutative 86.0%

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

      2. associate-/l* 86.9%

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

      3. fma-define 86.9%

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

   5. Simplified 86.9%

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

      1. fma-undefine 86.9%

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

   7. Applied egg-rr 86.9%

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

      1. clear-num 86.9%

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

      2. un-div-inv 86.9%

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

   9. Applied egg-rr 86.9%

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

   if 4.7999999999999995e-106 < d < 1.9e17

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0 70.4%

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

   if 1.9e17 < d < 1.45e36

   1. Initial program 72.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. +-commutative 86.1%

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

      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. Taylor expanded in d around inf 86.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+24}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+36}:\\ \;\;\;\;\frac{d \cdot \left(\frac{a}{d} + \frac{b}{c}\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+26} \lor \neg \left(d \leq 1.35 \cdot 10^{-76}\right) \land \left(d \leq 3.8 \cdot 10^{-23} \lor \neg \left(d \leq 3.8 \cdot 10^{+36}\right)\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.85e+26)
         (and (not (<= d 1.35e-76)) (or (<= d 3.8e-23) (not (<= d 3.8e+36)))))
   (/ (+ 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 <= -1.85e+26) || (!(d <= 1.35e-76) && ((d <= 3.8e-23) || !(d <= 3.8e+36)))) {
		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 <= (-1.85d+26)) .or. (.not. (d <= 1.35d-76)) .and. (d <= 3.8d-23) .or. (.not. (d <= 3.8d+36))) 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 <= -1.85e+26) || (!(d <= 1.35e-76) && ((d <= 3.8e-23) || !(d <= 3.8e+36)))) {
		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 <= -1.85e+26) or (not (d <= 1.35e-76) and ((d <= 3.8e-23) or not (d <= 3.8e+36))):
		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 <= -1.85e+26) || (!(d <= 1.35e-76) && ((d <= 3.8e-23) || !(d <= 3.8e+36))))
		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 <= -1.85e+26) || (~((d <= 1.35e-76)) && ((d <= 3.8e-23) || ~((d <= 3.8e+36)))))
		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, -1.85e+26], And[N[Not[LessEqual[d, 1.35e-76]], $MachinePrecision], Or[LessEqual[d, 3.8e-23], N[Not[LessEqual[d, 3.8e+36]], $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 < -1.84999999999999994e26 or 1.35e-76 < d < 3.80000000000000011e-23 or 3.80000000000000025e36 < d

   1. Initial program 53.3%

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

   3. Taylor expanded in d around inf 80.2%

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

      1. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

   if -1.84999999999999994e26 < d < 1.35e-76 or 3.80000000000000011e-23 < d < 3.80000000000000025e36

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. associate-/l* 83.3%

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

   5. Simplified 83.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.85 \cdot 10^{+26} \lor \neg \left(d \leq 1.35 \cdot 10^{-76}\right) \land \left(d \leq 3.8 \cdot 10^{-23} \lor \neg \left(d \leq 3.8 \cdot 10^{+36}\right)\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{+36} \lor \neg \left(d \leq 1.35 \cdot 10^{-76} \lor \neg \left(d \leq 9.5 \cdot 10^{-26}\right) \land d \leq 1.35 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.6e+36)
         (not (or (<= d 1.35e-76) (and (not (<= d 9.5e-26)) (<= d 1.35e+37)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.6e+36) || !((d <= 1.35e-76) || (!(d <= 9.5e-26) && (d <= 1.35e+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 <= (-9.6d+36)) .or. (.not. (d <= 1.35d-76) .or. (.not. (d <= 9.5d-26)) .and. (d <= 1.35d+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 <= -9.6e+36) || !((d <= 1.35e-76) || (!(d <= 9.5e-26) && (d <= 1.35e+37)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9.6e+36) or not ((d <= 1.35e-76) or (not (d <= 9.5e-26) and (d <= 1.35e+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 <= -9.6e+36) || !((d <= 1.35e-76) || (!(d <= 9.5e-26) && (d <= 1.35e+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 <= -9.6e+36) || ~(((d <= 1.35e-76) || (~((d <= 9.5e-26)) && (d <= 1.35e+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, -9.6e+36], N[Not[Or[LessEqual[d, 1.35e-76], And[N[Not[LessEqual[d, 9.5e-26]], $MachinePrecision], LessEqual[d, 1.35e+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 < -9.5999999999999997e36 or 1.35e-76 < d < 9.4999999999999995e-26 or 1.34999999999999993e37 < d

   1. Initial program 53.3%

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

   3. Taylor expanded in c around 0 72.7%

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

   if -9.5999999999999997e36 < d < 1.35e-76 or 9.4999999999999995e-26 < d < 1.34999999999999993e37

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. associate-/l* 83.3%

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

   5. Simplified 83.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.6 \cdot 10^{+36} \lor \neg \left(d \leq 1.35 \cdot 10^{-76} \lor \neg \left(d \leq 9.5 \cdot 10^{-26}\right) \land d \leq 1.35 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + \frac{b}{\frac{c}{d}}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -2.4 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (/ b (/ c d))) c)) (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -2.4e+23)
     t_1
     (if (<= d 3.2e-105)
       t_0
       (if (<= d 5.6e+17)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 8.5e+37) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (b / (c / d))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.4e+23) {
		tmp = t_1;
	} else if (d <= 3.2e-105) {
		tmp = t_0;
	} else if (d <= 5.6e+17) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 8.5e+37) {
		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 + (b / (c / d))) / c
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-2.4d+23)) then
        tmp = t_1
    else if (d <= 3.2d-105) then
        tmp = t_0
    else if (d <= 5.6d+17) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 8.5d+37) 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 + (b / (c / d))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -2.4e+23) {
		tmp = t_1;
	} else if (d <= 3.2e-105) {
		tmp = t_0;
	} else if (d <= 5.6e+17) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 8.5e+37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (b / (c / d))) / c
	t_1 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -2.4e+23:
		tmp = t_1
	elif d <= 3.2e-105:
		tmp = t_0
	elif d <= 5.6e+17:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 8.5e+37:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(b / Float64(c / d))) / c)
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -2.4e+23)
		tmp = t_1;
	elseif (d <= 3.2e-105)
		tmp = t_0;
	elseif (d <= 5.6e+17)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 8.5e+37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (b / (c / d))) / c;
	t_1 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -2.4e+23)
		tmp = t_1;
	elseif (d <= 3.2e-105)
		tmp = t_0;
	elseif (d <= 5.6e+17)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 8.5e+37)
		tmp = t_0;
	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[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.4e+23], t$95$1, If[LessEqual[d, 3.2e-105], t$95$0, If[LessEqual[d, 5.6e+17], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.5e+37], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.4e23 or 8.4999999999999999e37 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -2.4e23 < d < 3.19999999999999981e-105 or 5.6e17 < d < 8.4999999999999999e37

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf 86.0%

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

      1. +-commutative 86.0%

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

      2. associate-/l* 86.9%

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

      3. fma-define 86.9%

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

   5. Simplified 86.9%

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

      1. fma-undefine 86.9%

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

   7. Applied egg-rr 86.9%

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

      1. clear-num 86.9%

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

      2. un-div-inv 86.9%

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

   9. Applied egg-rr 86.9%

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

   if 3.19999999999999981e-105 < d < 5.6e17

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0 70.4%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+23}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+17}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + \frac{b}{\frac{c}{d}}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (/ b (/ c d))) c)) (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -1.3e+34)
     t_1
     (if (<= d 1.35e-76)
       t_0
       (if (<= d 1.15e-25)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 2e+36) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (b / (c / d))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.3e+34) {
		tmp = t_1;
	} else if (d <= 1.35e-76) {
		tmp = t_0;
	} else if (d <= 1.15e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2e+36) {
		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 + (b / (c / d))) / c
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-1.3d+34)) then
        tmp = t_1
    else if (d <= 1.35d-76) then
        tmp = t_0
    else if (d <= 1.15d-25) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 2d+36) 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 + (b / (c / d))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.3e+34) {
		tmp = t_1;
	} else if (d <= 1.35e-76) {
		tmp = t_0;
	} else if (d <= 1.15e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2e+36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (b / (c / d))) / c
	t_1 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -1.3e+34:
		tmp = t_1
	elif d <= 1.35e-76:
		tmp = t_0
	elif d <= 1.15e-25:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 2e+36:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(b / Float64(c / d))) / c)
	t_1 = Float64(Float64(b + Float64(a * Float64(c / d))) / d)
	tmp = 0.0
	if (d <= -1.3e+34)
		tmp = t_1;
	elseif (d <= 1.35e-76)
		tmp = t_0;
	elseif (d <= 1.15e-25)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 2e+36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (b / (c / d))) / c;
	t_1 = (b + (a * (c / d))) / d;
	tmp = 0.0;
	if (d <= -1.3e+34)
		tmp = t_1;
	elseif (d <= 1.35e-76)
		tmp = t_0;
	elseif (d <= 1.15e-25)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 2e+36)
		tmp = t_0;
	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[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.3e+34], t$95$1, If[LessEqual[d, 1.35e-76], t$95$0, If[LessEqual[d, 1.15e-25], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2e+36], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.29999999999999999e34 or 2.00000000000000008e36 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -1.29999999999999999e34 < d < 1.35e-76 or 1.15e-25 < d < 2.00000000000000008e36

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-/l* 83.3%

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

      3. fma-define 83.4%

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

   5. Simplified 83.4%

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

      1. fma-undefine 83.3%

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

   7. Applied egg-rr 83.3%

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

      1. clear-num 83.3%

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

      2. un-div-inv 83.4%

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

   9. Applied egg-rr 83.4%

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

   if 1.35e-76 < d < 1.15e-25

   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 d around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 78.0%

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

      1. *-commutative 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+34}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+36}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -8.5e+23)
     t_0
     (if (<= d 1.12e-76)
       (/ (+ a (* b (/ d c))) c)
       (if (<= d 4.5e-25)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 1.9e+36) (/ (+ a (* d (/ b c))) 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 <= -8.5e+23) {
		tmp = t_0;
	} else if (d <= 1.12e-76) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 4.5e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 1.9e+36) {
		tmp = (a + (d * (b / c))) / 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 <= (-8.5d+23)) then
        tmp = t_0
    else if (d <= 1.12d-76) then
        tmp = (a + (b * (d / c))) / c
    else if (d <= 4.5d-25) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 1.9d+36) then
        tmp = (a + (d * (b / c))) / 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 <= -8.5e+23) {
		tmp = t_0;
	} else if (d <= 1.12e-76) {
		tmp = (a + (b * (d / c))) / c;
	} else if (d <= 4.5e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 1.9e+36) {
		tmp = (a + (d * (b / c))) / 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 <= -8.5e+23:
		tmp = t_0
	elif d <= 1.12e-76:
		tmp = (a + (b * (d / c))) / c
	elif d <= 4.5e-25:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 1.9e+36:
		tmp = (a + (d * (b / c))) / 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 <= -8.5e+23)
		tmp = t_0;
	elseif (d <= 1.12e-76)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (d <= 4.5e-25)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 1.9e+36)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / 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 <= -8.5e+23)
		tmp = t_0;
	elseif (d <= 1.12e-76)
		tmp = (a + (b * (d / c))) / c;
	elseif (d <= 4.5e-25)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 1.9e+36)
		tmp = (a + (d * (b / c))) / 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, -8.5e+23], t$95$0, If[LessEqual[d, 1.12e-76], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e-25], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.9e+36], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.5000000000000001e23 or 1.90000000000000012e36 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -8.5000000000000001e23 < d < 1.12e-76

   1. Initial program 76.3%

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

   3. Taylor expanded in c around inf 83.7%

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

      1. +-commutative 83.7%

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

      2. associate-/l* 84.5%

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

      3. fma-define 84.5%

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

   5. Simplified 84.5%

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

      1. fma-undefine 84.5%

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

   7. Applied egg-rr 84.5%

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

   if 1.12e-76 < d < 4.5000000000000001e-25

   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 d around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 78.0%

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

      1. *-commutative 78.0%

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

   8. Simplified 78.0%

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

   if 4.5000000000000001e-25 < d < 1.90000000000000012e36

   1. Initial program 87.4%

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

   3. Taylor expanded in c around inf 74.1%

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

      1. +-commutative 74.1%

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

      2. associate-/l* 74.1%

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

      3. fma-define 74.1%

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

   5. Simplified 74.1%

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

      1. fma-undefine 74.1%

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

   7. Applied egg-rr 74.1%

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

      1. clear-num 74.1%

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

      2. un-div-inv 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. associate-/r/ 74.1%

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

   11. Simplified 74.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.12 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + b \cdot \frac{d}{c}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* b (/ d c))) c)) (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -3.1e+26)
     t_1
     (if (<= d 1.05e-76)
       t_0
       (if (<= d 1.2e-23)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 4.2e+37) t_0 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 tmp;
	if (d <= -3.1e+26) {
		tmp = t_1;
	} else if (d <= 1.05e-76) {
		tmp = t_0;
	} else if (d <= 1.2e-23) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 4.2e+37) {
		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 + (b * (d / c))) / c
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-3.1d+26)) then
        tmp = t_1
    else if (d <= 1.05d-76) then
        tmp = t_0
    else if (d <= 1.2d-23) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 4.2d+37) 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 + (b * (d / c))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -3.1e+26) {
		tmp = t_1;
	} else if (d <= 1.05e-76) {
		tmp = t_0;
	} else if (d <= 1.2e-23) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 4.2e+37) {
		tmp = t_0;
	} 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
	tmp = 0
	if d <= -3.1e+26:
		tmp = t_1
	elif d <= 1.05e-76:
		tmp = t_0
	elif d <= 1.2e-23:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 4.2e+37:
		tmp = t_0
	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)
	tmp = 0.0
	if (d <= -3.1e+26)
		tmp = t_1;
	elseif (d <= 1.05e-76)
		tmp = t_0;
	elseif (d <= 1.2e-23)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 4.2e+37)
		tmp = t_0;
	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;
	tmp = 0.0;
	if (d <= -3.1e+26)
		tmp = t_1;
	elseif (d <= 1.05e-76)
		tmp = t_0;
	elseif (d <= 1.2e-23)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 4.2e+37)
		tmp = t_0;
	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]}, If[LessEqual[d, -3.1e+26], t$95$1, If[LessEqual[d, 1.05e-76], t$95$0, If[LessEqual[d, 1.2e-23], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 4.2e+37], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.1e26 or 4.2000000000000002e37 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -3.1e26 < d < 1.04999999999999996e-76 or 1.19999999999999998e-23 < d < 4.2000000000000002e37

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-/l* 83.3%

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

      3. fma-define 83.4%

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

   5. Simplified 83.4%

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

      1. fma-undefine 83.3%

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

   7. Applied egg-rr 83.3%

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

   if 1.04999999999999996e-76 < d < 1.19999999999999998e-23

   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 d around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 78.0%

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

      1. *-commutative 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+37}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + b \cdot \frac{d}{c}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* b (/ d c))) c)) (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -1.4e+25)
     t_1
     (if (<= d 1.35e-76)
       t_0
       (if (<= d 7.7e-25)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 2.65e+37) t_0 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 tmp;
	if (d <= -1.4e+25) {
		tmp = t_1;
	} else if (d <= 1.35e-76) {
		tmp = t_0;
	} else if (d <= 7.7e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2.65e+37) {
		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 + (b * (d / c))) / c
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-1.4d+25)) then
        tmp = t_1
    else if (d <= 1.35d-76) then
        tmp = t_0
    else if (d <= 7.7d-25) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 2.65d+37) 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 + (b * (d / c))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -1.4e+25) {
		tmp = t_1;
	} else if (d <= 1.35e-76) {
		tmp = t_0;
	} else if (d <= 7.7e-25) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2.65e+37) {
		tmp = t_0;
	} 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
	tmp = 0
	if d <= -1.4e+25:
		tmp = t_1
	elif d <= 1.35e-76:
		tmp = t_0
	elif d <= 7.7e-25:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 2.65e+37:
		tmp = t_0
	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)
	tmp = 0.0
	if (d <= -1.4e+25)
		tmp = t_1;
	elseif (d <= 1.35e-76)
		tmp = t_0;
	elseif (d <= 7.7e-25)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 2.65e+37)
		tmp = t_0;
	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;
	tmp = 0.0;
	if (d <= -1.4e+25)
		tmp = t_1;
	elseif (d <= 1.35e-76)
		tmp = t_0;
	elseif (d <= 7.7e-25)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 2.65e+37)
		tmp = t_0;
	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]}, If[LessEqual[d, -1.4e+25], t$95$1, If[LessEqual[d, 1.35e-76], t$95$0, If[LessEqual[d, 7.7e-25], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.65e+37], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.4000000000000001e25 or 2.6500000000000001e37 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -1.4000000000000001e25 < d < 1.35e-76 or 7.7000000000000002e-25 < d < 2.6500000000000001e37

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. associate-/l* 83.3%

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

   5. Simplified 83.3%

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

   if 1.35e-76 < d < 7.7000000000000002e-25

   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 d around inf 78.0%

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

      1. associate-/l* 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in a around 0 78.0%

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

      1. *-commutative 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{+37}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + b \cdot \frac{d}{c}}{c}\\ t_1 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;\frac{b + \frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* b (/ d c))) c)) (t_1 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -9e+30)
     t_1
     (if (<= d 7.5e-77)
       t_0
       (if (<= d 2.45e-23)
         (/ (+ b (/ (* a c) d)) d)
         (if (<= d 2.6e+36) t_0 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 tmp;
	if (d <= -9e+30) {
		tmp = t_1;
	} else if (d <= 7.5e-77) {
		tmp = t_0;
	} else if (d <= 2.45e-23) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2.6e+36) {
		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 + (b * (d / c))) / c
    t_1 = (b + (a * (c / d))) / d
    if (d <= (-9d+30)) then
        tmp = t_1
    else if (d <= 7.5d-77) then
        tmp = t_0
    else if (d <= 2.45d-23) then
        tmp = (b + ((a * c) / d)) / d
    else if (d <= 2.6d+36) 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 + (b * (d / c))) / c;
	double t_1 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -9e+30) {
		tmp = t_1;
	} else if (d <= 7.5e-77) {
		tmp = t_0;
	} else if (d <= 2.45e-23) {
		tmp = (b + ((a * c) / d)) / d;
	} else if (d <= 2.6e+36) {
		tmp = t_0;
	} 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
	tmp = 0
	if d <= -9e+30:
		tmp = t_1
	elif d <= 7.5e-77:
		tmp = t_0
	elif d <= 2.45e-23:
		tmp = (b + ((a * c) / d)) / d
	elif d <= 2.6e+36:
		tmp = t_0
	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)
	tmp = 0.0
	if (d <= -9e+30)
		tmp = t_1;
	elseif (d <= 7.5e-77)
		tmp = t_0;
	elseif (d <= 2.45e-23)
		tmp = Float64(Float64(b + Float64(Float64(a * c) / d)) / d);
	elseif (d <= 2.6e+36)
		tmp = t_0;
	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;
	tmp = 0.0;
	if (d <= -9e+30)
		tmp = t_1;
	elseif (d <= 7.5e-77)
		tmp = t_0;
	elseif (d <= 2.45e-23)
		tmp = (b + ((a * c) / d)) / d;
	elseif (d <= 2.6e+36)
		tmp = t_0;
	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]}, If[LessEqual[d, -9e+30], t$95$1, If[LessEqual[d, 7.5e-77], t$95$0, If[LessEqual[d, 2.45e-23], N[(N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.6e+36], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.9999999999999999e30 or 2.6000000000000001e36 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around inf 80.5%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

   if -8.9999999999999999e30 < d < 7.5000000000000006e-77 or 2.4499999999999999e-23 < d < 2.6000000000000001e36

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf 82.6%

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

      1. associate-/l* 83.3%

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

   5. Simplified 83.3%

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

   if 7.5000000000000006e-77 < d < 2.4499999999999999e-23

   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 d around inf 78.0%

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

Alternative 15: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.1 \cdot 10^{+110} \lor \neg \left(c \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.1e+110) (not (<= c 2e-8))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.1e+110) || !(c <= 2e-8)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.1d+110)) .or. (.not. (c <= 2d-8))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.1e+110) || !(c <= 2e-8)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.1e+110) or not (c <= 2e-8):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.1e+110) || !(c <= 2e-8))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.1e+110) || ~((c <= 2e-8)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.1e+110], N[Not[LessEqual[c, 2e-8]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.1000000000000002e110 or 2e-8 < c

   1. Initial program 59.0%

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

   3. Taylor expanded in c around inf 75.7%

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

   if -5.1000000000000002e110 < c < 2e-8

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0 61.1%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.1 \cdot 10^{+110} \lor \neg \left(c \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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))))