Complex division, real part

Percentage Accurate: 62.1% → 90.1%

Time: 8.9s

Alternatives: 11

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: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{-162} \lor \neg \left(c \leq 1.22 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.25e-162) (not (<= c 1.22e-54)))
   (* (/ (fma b (/ d c) a) (hypot c d)) (/ c (hypot c d)))
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.25e-162) || !(c <= 1.22e-54)) {
		tmp = (fma(b, (d / c), a) / hypot(c, d)) * (c / hypot(c, d));
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.25e-162) || !(c <= 1.22e-54))
		tmp = Float64(Float64(fma(b, Float64(d / c), a) / hypot(c, d)) * Float64(c / 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[Or[LessEqual[c, -2.25e-162], N[Not[LessEqual[c, 1.22e-54]], $MachinePrecision]], N[(N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / 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 c < -2.25000000000000011e-162 or 1.22e-54 < c

   1. Initial program 57.9%

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

   3. Taylor expanded in c around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. add-sqr-sqrt 57.3%

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

      3. hypot-undefine 57.3%

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

      4. hypot-undefine 57.3%

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

      5. times-frac 87.7%

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

      6. +-commutative 87.7%

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

      7. associate-/l* 93.7%

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

      8. fma-define 93.7%

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

   5. Applied egg-rr 93.7%

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

   if -2.25000000000000011e-162 < c < 1.22e-54

   1. Initial program 66.9%

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

   3. Taylor expanded in d around inf 90.5%

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

      1. associate-/l* 90.7%

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

   5. Simplified 90.7%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{-162} \lor \neg \left(c \leq 1.22 \cdot 10^{-54}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+286}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* c a) (* b d)) (+ (* c c) (* d d))) 1e+286)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ b (/ a (/ d c))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((c * a) + (b * d)) / ((c * c) + (d * d))) <= 1e+286) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+286)
		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(d / c))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+286], 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[(d / c), $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))) < 1.00000000000000003e286

   1. Initial program 75.5%

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

      1. *-un-lft-identity 75.5%

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

      2. associate-*r/ 75.5%

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

      3. fma-define 75.5%

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

      4. add-sqr-sqrt 75.5%

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

      5. times-frac 75.5%

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

      6. fma-define 75.5%

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

      7. hypot-define 75.5%

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

      8. fma-define 75.5%

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

      9. fma-define 75.5%

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

      10. hypot-define 94.6%

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

   4. Applied egg-rr 94.6%

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

   if 1.00000000000000003e286 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.2%

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

   3. Taylor expanded in d around inf 59.8%

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

      1. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

      1. clear-num 63.0%

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

      2. un-div-inv 63.0%

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

   7. Applied egg-rr 63.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+286}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+96}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7.0)
   (/ (+ b (/ (* c a) d)) d)
   (if (<= d 1.35e-83)
     (/ (+ a (/ (* b d) c)) c)
     (if (<= d 3e+96)
       (/ (+ (* c a) (* b d)) (+ (* c c) (* d d)))
       (/ (+ b (/ a (/ d c))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7.0) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 1.35e-83) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 3e+96) {
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-7.0d0)) then
        tmp = (b + ((c * a) / d)) / d
    else if (d <= 1.35d-83) then
        tmp = (a + ((b * d) / c)) / c
    else if (d <= 3d+96) then
        tmp = ((c * a) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7.0) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 1.35e-83) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 3e+96) {
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -7.0:
		tmp = (b + ((c * a) / d)) / d
	elif d <= 1.35e-83:
		tmp = (a + ((b * d) / c)) / c
	elif d <= 3e+96:
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7.0)
		tmp = Float64(Float64(b + Float64(Float64(c * a) / d)) / d);
	elseif (d <= 1.35e-83)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif (d <= 3e+96)
		tmp = Float64(Float64(Float64(c * a) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -7.0)
		tmp = (b + ((c * a) / d)) / d;
	elseif (d <= 1.35e-83)
		tmp = (a + ((b * d) / c)) / c;
	elseif (d <= 3e+96)
		tmp = ((c * a) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7.0], N[(N[(b + N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.35e-83], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+96], N[(N[(N[(c * a), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7

   1. Initial program 47.7%

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

   3. Taylor expanded in d around inf 85.6%

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

   if -7 < d < 1.34999999999999996e-83

   1. Initial program 73.8%

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

   3. Taylor expanded in c around inf 88.0%

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

   if 1.34999999999999996e-83 < d < 3e96

   1. Initial program 76.7%

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

   if 3e96 < d

   1. Initial program 36.8%

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

   3. Taylor expanded in d around inf 85.4%

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

      1. associate-/l* 89.6%

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

   5. Simplified 89.6%

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

      1. clear-num 89.6%

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

      2. un-div-inv 89.7%

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

   7. Applied egg-rr 89.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 1.35 \cdot 10^{-83}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+96}:\\ \;\;\;\;\frac{c \cdot a + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+33} \lor \neg \left(d \leq 6600000000000\right) \land \left(d \leq 1.15 \cdot 10^{+55} \lor \neg \left(d \leq 7.2 \cdot 10^{+108}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.5e+33)
         (and (not (<= d 6600000000000.0))
              (or (<= d 1.15e+55) (not (<= d 7.2e+108)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e+33) || (!(d <= 6600000000000.0) && ((d <= 1.15e+55) || !(d <= 7.2e+108)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.5d+33)) .or. (.not. (d <= 6600000000000.0d0)) .and. (d <= 1.15d+55) .or. (.not. (d <= 7.2d+108))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e+33) || (!(d <= 6600000000000.0) && ((d <= 1.15e+55) || !(d <= 7.2e+108)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.5e+33) or (not (d <= 6600000000000.0) and ((d <= 1.15e+55) or not (d <= 7.2e+108))):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.5e+33) || (!(d <= 6600000000000.0) && ((d <= 1.15e+55) || !(d <= 7.2e+108))))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.5e+33) || (~((d <= 6600000000000.0)) && ((d <= 1.15e+55) || ~((d <= 7.2e+108)))))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.5e+33], And[N[Not[LessEqual[d, 6600000000000.0]], $MachinePrecision], Or[LessEqual[d, 1.15e+55], N[Not[LessEqual[d, 7.2e+108]], $MachinePrecision]]]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.4999999999999998e33 or 6.6e12 < d < 1.14999999999999994e55 or 7.2e108 < d

   1. Initial program 46.9%

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

   3. Taylor expanded in c around 0 74.3%

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

   if -8.4999999999999998e33 < d < 6.6e12 or 1.14999999999999994e55 < d < 7.2e108

   1. Initial program 72.3%

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

   3. Taylor expanded in c around inf 82.5%

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

      1. associate-/l* 81.9%

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

   5. Simplified 81.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+33} \lor \neg \left(d \leq 6600000000000\right) \land \left(d \leq 1.15 \cdot 10^{+55} \lor \neg \left(d \leq 7.2 \cdot 10^{+108}\right)\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.025:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 18000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+56} \lor \neg \left(d \leq 2.45 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.025)
   (/ (+ b (/ (* c a) d)) d)
   (if (<= d 18000000000.0)
     (/ (+ a (/ (* b d) c)) c)
     (if (or (<= d 6.5e+56) (not (<= d 2.45e+100)))
       (/ (+ b (/ a (/ d c))) d)
       (/ (+ a (* b (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.025) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 18000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 6.5e+56) || !(d <= 2.45e+100)) {
		tmp = (b + (a / (d / c))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-0.025d0)) then
        tmp = (b + ((c * a) / d)) / d
    else if (d <= 18000000000.0d0) then
        tmp = (a + ((b * d) / c)) / c
    else if ((d <= 6.5d+56) .or. (.not. (d <= 2.45d+100))) then
        tmp = (b + (a / (d / c))) / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.025) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 18000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 6.5e+56) || !(d <= 2.45e+100)) {
		tmp = (b + (a / (d / c))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -0.025:
		tmp = (b + ((c * a) / d)) / d
	elif d <= 18000000000.0:
		tmp = (a + ((b * d) / c)) / c
	elif (d <= 6.5e+56) or not (d <= 2.45e+100):
		tmp = (b + (a / (d / c))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.025)
		tmp = Float64(Float64(b + Float64(Float64(c * a) / d)) / d);
	elseif (d <= 18000000000.0)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif ((d <= 6.5e+56) || !(d <= 2.45e+100))
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -0.025)
		tmp = (b + ((c * a) / d)) / d;
	elseif (d <= 18000000000.0)
		tmp = (a + ((b * d) / c)) / c;
	elseif ((d <= 6.5e+56) || ~((d <= 2.45e+100)))
		tmp = (b + (a / (d / c))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.025], N[(N[(b + N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 18000000000.0], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 6.5e+56], N[Not[LessEqual[d, 2.45e+100]], $MachinePrecision]], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -0.025000000000000001

   1. Initial program 47.7%

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

   3. Taylor expanded in d around inf 85.6%

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

   if -0.025000000000000001 < d < 1.8e10

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf 84.4%

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

   if 1.8e10 < d < 6.5000000000000001e56 or 2.44999999999999983e100 < 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 85.9%

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

      1. associate-/l* 89.5%

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

   5. Simplified 89.5%

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

      1. clear-num 89.5%

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

      2. un-div-inv 89.5%

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

   7. Applied egg-rr 89.5%

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

   if 6.5000000000000001e56 < d < 2.44999999999999983e100

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.025:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 18000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+56} \lor \neg \left(d \leq 2.45 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{+56} \lor \neg \left(d \leq 3 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.8e-5)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 2.1e+14)
     (/ (+ a (/ (* b d) c)) c)
     (if (or (<= d 1.22e+56) (not (<= d 3e+100)))
       (/ (+ b (/ a (/ d c))) d)
       (/ (+ a (* b (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.8e-5) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.1e+14) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 1.22e+56) || !(d <= 3e+100)) {
		tmp = (b + (a / (d / c))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3.8d-5)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 2.1d+14) then
        tmp = (a + ((b * d) / c)) / c
    else if ((d <= 1.22d+56) .or. (.not. (d <= 3d+100))) then
        tmp = (b + (a / (d / c))) / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.8e-5) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.1e+14) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 1.22e+56) || !(d <= 3e+100)) {
		tmp = (b + (a / (d / c))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.8e-5:
		tmp = (b + (a * (c / d))) / d
	elif d <= 2.1e+14:
		tmp = (a + ((b * d) / c)) / c
	elif (d <= 1.22e+56) or not (d <= 3e+100):
		tmp = (b + (a / (d / c))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.8e-5)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 2.1e+14)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif ((d <= 1.22e+56) || !(d <= 3e+100))
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.8e-5)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 2.1e+14)
		tmp = (a + ((b * d) / c)) / c;
	elseif ((d <= 1.22e+56) || ~((d <= 3e+100)))
		tmp = (b + (a / (d / c))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.8e-5], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.1e+14], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 1.22e+56], N[Not[LessEqual[d, 3e+100]], $MachinePrecision]], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.8000000000000002e-5

   1. Initial program 47.7%

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

   3. Taylor expanded in d around inf 85.6%

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

      1. associate-/l* 84.5%

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

   5. Simplified 84.5%

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

   if -3.8000000000000002e-5 < d < 2.1e14

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf 84.4%

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

   if 2.1e14 < d < 1.22e56 or 2.99999999999999985e100 < 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 85.9%

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

      1. associate-/l* 89.5%

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

   5. Simplified 89.5%

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

      1. clear-num 89.5%

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

      2. un-div-inv 89.5%

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

   7. Applied egg-rr 89.5%

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

   if 1.22e56 < d < 2.99999999999999985e100

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.22 \cdot 10^{+56} \lor \neg \left(d \leq 3 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{if}\;d \leq -0.043:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+56} \lor \neg \left(d \leq 1.22 \cdot 10^{+101}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* a (/ c d))) d)))
   (if (<= d -0.043)
     t_0
     (if (<= d 2.2e+15)
       (/ (+ a (/ (* b d) c)) c)
       (if (or (<= d 6.8e+56) (not (<= d 1.22e+101)))
         t_0
         (/ (+ a (* b (/ d c))) c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -0.043) {
		tmp = t_0;
	} else if (d <= 2.2e+15) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 6.8e+56) || !(d <= 1.22e+101)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (b + (a * (c / d))) / d
    if (d <= (-0.043d0)) then
        tmp = t_0
    else if (d <= 2.2d+15) then
        tmp = (a + ((b * d) / c)) / c
    else if ((d <= 6.8d+56) .or. (.not. (d <= 1.22d+101))) then
        tmp = t_0
    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 t_0 = (b + (a * (c / d))) / d;
	double tmp;
	if (d <= -0.043) {
		tmp = t_0;
	} else if (d <= 2.2e+15) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 6.8e+56) || !(d <= 1.22e+101)) {
		tmp = t_0;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (a * (c / d))) / d
	tmp = 0
	if d <= -0.043:
		tmp = t_0
	elif d <= 2.2e+15:
		tmp = (a + ((b * d) / c)) / c
	elif (d <= 6.8e+56) or not (d <= 1.22e+101):
		tmp = t_0
	else:
		tmp = (a + (b * (d / c))) / 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 <= -0.043)
		tmp = t_0;
	elseif (d <= 2.2e+15)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif ((d <= 6.8e+56) || !(d <= 1.22e+101))
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / 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 <= -0.043)
		tmp = t_0;
	elseif (d <= 2.2e+15)
		tmp = (a + ((b * d) / c)) / c;
	elseif ((d <= 6.8e+56) || ~((d <= 1.22e+101)))
		tmp = t_0;
	else
		tmp = (a + (b * (d / c))) / 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, -0.043], t$95$0, If[LessEqual[d, 2.2e+15], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 6.8e+56], N[Not[LessEqual[d, 1.22e+101]], $MachinePrecision]], t$95$0, N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -0.042999999999999997 or 2.2e15 < d < 6.80000000000000002e56 or 1.22e101 < d

   1. Initial program 47.8%

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

   3. Taylor expanded in d around inf 85.7%

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

      1. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

   if -0.042999999999999997 < d < 2.2e15

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf 84.4%

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

   if 6.80000000000000002e56 < d < 1.22e101

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.043:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+56} \lor \neg \left(d \leq 1.22 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3300000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.12 \cdot 10^{+55} \lor \neg \left(d \leq 4.5 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.6e+29)
   (/ b d)
   (if (<= d 3300000000000.0)
     (/ (+ a (/ (* b d) c)) c)
     (if (or (<= d 2.12e+55) (not (<= d 4.5e+100)))
       (/ b d)
       (/ (+ a (* b (/ d c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.6e+29) {
		tmp = b / d;
	} else if (d <= 3300000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 2.12e+55) || !(d <= 4.5e+100)) {
		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 <= (-2.6d+29)) then
        tmp = b / d
    else if (d <= 3300000000000.0d0) then
        tmp = (a + ((b * d) / c)) / c
    else if ((d <= 2.12d+55) .or. (.not. (d <= 4.5d+100))) 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 <= -2.6e+29) {
		tmp = b / d;
	} else if (d <= 3300000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 2.12e+55) || !(d <= 4.5e+100)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.6e+29:
		tmp = b / d
	elif d <= 3300000000000.0:
		tmp = (a + ((b * d) / c)) / c
	elif (d <= 2.12e+55) or not (d <= 4.5e+100):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.6e+29)
		tmp = Float64(b / d);
	elseif (d <= 3300000000000.0)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif ((d <= 2.12e+55) || !(d <= 4.5e+100))
		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 <= -2.6e+29)
		tmp = b / d;
	elseif (d <= 3300000000000.0)
		tmp = (a + ((b * d) / c)) / c;
	elseif ((d <= 2.12e+55) || ~((d <= 4.5e+100)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.6e+29], N[(b / d), $MachinePrecision], If[LessEqual[d, 3300000000000.0], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 2.12e+55], N[Not[LessEqual[d, 4.5e+100]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.6e29 or 3.3e12 < d < 2.12000000000000007e55 or 4.50000000000000036e100 < d

   1. Initial program 46.9%

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

   3. Taylor expanded in c around 0 74.3%

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

   if -2.6e29 < d < 3.3e12

   1. Initial program 73.6%

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

   3. Taylor expanded in c around inf 82.9%

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

   if 2.12000000000000007e55 < d < 4.50000000000000036e100

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3300000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.12 \cdot 10^{+55} \lor \neg \left(d \leq 4.5 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.00012:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 0.4:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.00012)
   (/ (+ b (/ (* c a) d)) d)
   (if (<= d 0.4)
     (/ (+ a (/ (* b d) c)) c)
     (if (<= d 2.6e+55)
       (/ (* b d) (+ (* c c) (* d d)))
       (if (<= d 2.5e+100)
         (/ (+ a (* b (/ d c))) c)
         (/ (+ b (/ a (/ d c))) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.00012) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 0.4) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 2.6e+55) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2.5e+100) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-0.00012d0)) then
        tmp = (b + ((c * a) / d)) / d
    else if (d <= 0.4d0) then
        tmp = (a + ((b * d) / c)) / c
    else if (d <= 2.6d+55) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 2.5d+100) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (a / (d / c))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.00012) {
		tmp = (b + ((c * a) / d)) / d;
	} else if (d <= 0.4) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 2.6e+55) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2.5e+100) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -0.00012:
		tmp = (b + ((c * a) / d)) / d
	elif d <= 0.4:
		tmp = (a + ((b * d) / c)) / c
	elif d <= 2.6e+55:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 2.5e+100:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (a / (d / c))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.00012)
		tmp = Float64(Float64(b + Float64(Float64(c * a) / d)) / d);
	elseif (d <= 0.4)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif (d <= 2.6e+55)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.5e+100)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -0.00012)
		tmp = (b + ((c * a) / d)) / d;
	elseif (d <= 0.4)
		tmp = (a + ((b * d) / c)) / c;
	elseif (d <= 2.6e+55)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 2.5e+100)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (a / (d / c))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.00012], N[(N[(b + N[(N[(c * a), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 0.4], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.6e+55], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e+100], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.20000000000000003e-4

   1. Initial program 47.7%

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

   3. Taylor expanded in d around inf 85.6%

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

   if -1.20000000000000003e-4 < d < 0.40000000000000002

   1. Initial program 73.1%

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

   3. Taylor expanded in c around inf 84.9%

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

   if 0.40000000000000002 < d < 2.6e55

   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 91.1%

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

   if 2.6e55 < d < 2.4999999999999999e100

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 87.7%

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

   5. Simplified 87.7%

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

   if 2.4999999999999999e100 < d

   1. Initial program 37.6%

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

   3. Taylor expanded in d around inf 87.2%

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

      1. associate-/l* 91.5%

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

   5. Simplified 91.5%

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

      1. clear-num 91.5%

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

      2. un-div-inv 91.6%

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

   7. Applied egg-rr 91.6%

      \[\leadsto \frac{b + \color{blue}{\frac{a}{\frac{d}{c}}}}{d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.00012:\\ \;\;\;\;\frac{b + \frac{c \cdot a}{d}}{d}\\ \mathbf{elif}\;d \leq 0.4:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-6} \lor \neg \left(d \leq 4200000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.5e-6) (not (<= d 4200000000.0))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.5e-6) || !(d <= 4200000000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4.5d-6)) .or. (.not. (d <= 4200000000.0d0))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.5e-6) || !(d <= 4200000000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.5e-6) or not (d <= 4200000000.0):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.5e-6) || !(d <= 4200000000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.5e-6) || ~((d <= 4200000000.0)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.5e-6], N[Not[LessEqual[d, 4200000000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.50000000000000011e-6 or 4.2e9 < 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 c around 0 69.4%

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

   if -4.50000000000000011e-6 < d < 4.2e9

   1. Initial program 73.5%

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

   3. Taylor expanded in c around inf 67.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-6} \lor \neg \left(d \leq 4200000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{a}{c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.7%

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

3. Taylor expanded in c around inf 39.8%

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

Developer target: 99.5% 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 2024092 
(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))))