Complex division, real part

Percentage Accurate: 61.7% → 85.4%

Time: 8.4s

Alternatives: 8

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: 61.7% 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: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+264}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 2e+264)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ a (/ b (/ c d))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 2e+264) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	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))) <= 2e+264)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	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], 2e+264], 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[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $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))) < 2.00000000000000009e264

   1. Initial program 76.6%

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

      1. *-un-lft-identity 76.6%

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

      2. associate-*r/ 76.6%

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

      3. fma-define 76.6%

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

      4. add-sqr-sqrt 76.6%

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

      5. times-frac 76.5%

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

      6. fma-define 76.5%

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

      7. hypot-define 76.5%

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

      8. fma-define 76.5%

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

      9. fma-define 76.6%

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

      10. hypot-define 96.5%

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

   4. Applied egg-rr 96.5%

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

   if 2.00000000000000009e264 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.9%

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

   3. Taylor expanded in c around inf 57.7%

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

      1. associate-/l* 67.9%

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

   5. Simplified 67.9%

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

   6. Taylor expanded in b around 0 57.7%

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

      1. *-rgt-identity 57.7%

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

      2. times-frac 64.8%

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

      3. /-rgt-identity 64.8%

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

      4. associate-/r/ 68.0%

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

   8. Simplified 68.0%

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

Alternative 2: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;c \leq -2.9 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-150}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ a (* d (/ b c))) c)))
   (if (<= c -2.9e-14)
     t_0
     (if (<= c 1.12e-150)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c 5e+83) (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a + (d * (b / c))) / c;
	double tmp;
	if (c <= -2.9e-14) {
		tmp = t_0;
	} else if (c <= 1.12e-150) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5e+83) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a + (d * (b / c))) / c
    if (c <= (-2.9d-14)) then
        tmp = t_0
    else if (c <= 1.12d-150) then
        tmp = (b + (a * (c / d))) / d
    else if (c <= 5d+83) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a + (d * (b / c))) / c;
	double tmp;
	if (c <= -2.9e-14) {
		tmp = t_0;
	} else if (c <= 1.12e-150) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5e+83) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a + (d * (b / c))) / c
	tmp = 0
	if c <= -2.9e-14:
		tmp = t_0
	elif c <= 1.12e-150:
		tmp = (b + (a * (c / d))) / d
	elif c <= 5e+83:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a + Float64(d * Float64(b / c))) / c)
	tmp = 0.0
	if (c <= -2.9e-14)
		tmp = t_0;
	elseif (c <= 1.12e-150)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 5e+83)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a + (d * (b / c))) / c;
	tmp = 0.0;
	if (c <= -2.9e-14)
		tmp = t_0;
	elseif (c <= 1.12e-150)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 5e+83)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.9e-14], t$95$0, If[LessEqual[c, 1.12e-150], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5e+83], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.9000000000000003e-14 or 5.00000000000000029e83 < c

   1. Initial program 47.1%

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

   3. Taylor expanded in c around inf 76.6%

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

      1. associate-/l* 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in b around 0 76.6%

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

      1. *-rgt-identity 76.6%

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

      2. times-frac 83.2%

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

      3. /-rgt-identity 83.2%

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

      4. associate-/r/ 82.2%

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

   8. Simplified 82.2%

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

      1. associate-/r/ 83.2%

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

   10. Applied egg-rr 83.2%

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

   if -2.9000000000000003e-14 < c < 1.12e-150

   1. Initial program 69.6%

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

   3. Taylor expanded in d around inf 90.0%

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

      1. associate-/l* 90.0%

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

   5. Simplified 90.0%

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

   if 1.12e-150 < c < 5.00000000000000029e83

   1. Initial program 86.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{-14}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-150}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2500000 \lor \neg \left(d \leq 2 \cdot 10^{-61} \lor \neg \left(d \leq 7.8 \cdot 10^{+112}\right) \land d \leq 1.02 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2500000.0)
         (not (or (<= d 2e-61) (and (not (<= d 7.8e+112)) (<= d 1.02e+133)))))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2500000.0) || !((d <= 2e-61) || (!(d <= 7.8e+112) && (d <= 1.02e+133)))) {
		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 <= (-2500000.0d0)) .or. (.not. (d <= 2d-61) .or. (.not. (d <= 7.8d+112)) .and. (d <= 1.02d+133))) 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 <= -2500000.0) || !((d <= 2e-61) || (!(d <= 7.8e+112) && (d <= 1.02e+133)))) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2500000.0) or not ((d <= 2e-61) or (not (d <= 7.8e+112) and (d <= 1.02e+133))):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2500000.0) || !((d <= 2e-61) || (!(d <= 7.8e+112) && (d <= 1.02e+133))))
		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 <= -2500000.0) || ~(((d <= 2e-61) || (~((d <= 7.8e+112)) && (d <= 1.02e+133)))))
		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, -2500000.0], N[Not[Or[LessEqual[d, 2e-61], And[N[Not[LessEqual[d, 7.8e+112]], $MachinePrecision], LessEqual[d, 1.02e+133]]]], $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 < -2.5e6 or 2.0000000000000001e-61 < d < 7.79999999999999937e112 or 1.02e133 < d

   1. Initial program 59.4%

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

   3. Taylor expanded in c around 0 69.9%

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

   if -2.5e6 < d < 2.0000000000000001e-61 or 7.79999999999999937e112 < d < 1.02e133

   1. Initial program 67.9%

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

   3. Taylor expanded in c around inf 82.8%

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

      1. associate-/l* 84.1%

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

   5. Simplified 84.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2500000 \lor \neg \left(d \leq 2 \cdot 10^{-61} \lor \neg \left(d \leq 7.8 \cdot 10^{+112}\right) \land d \leq 1.02 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+115} \lor \neg \left(d \leq 5.4 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3000000.0)
   (/ b d)
   (if (<= d 2e-61)
     (/ (+ a (/ (* b d) c)) c)
     (if (or (<= d 5e+115) (not (<= d 5.4e+135)))
       (/ b d)
       (/ (+ a (* d (/ b c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3000000.0) {
		tmp = b / d;
	} else if (d <= 2e-61) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 5e+115) || !(d <= 5.4e+135)) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3000000.0d0)) then
        tmp = b / d
    else if (d <= 2d-61) then
        tmp = (a + ((b * d) / c)) / c
    else if ((d <= 5d+115) .or. (.not. (d <= 5.4d+135))) then
        tmp = b / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3000000.0) {
		tmp = b / d;
	} else if (d <= 2e-61) {
		tmp = (a + ((b * d) / c)) / c;
	} else if ((d <= 5e+115) || !(d <= 5.4e+135)) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3000000.0:
		tmp = b / d
	elif d <= 2e-61:
		tmp = (a + ((b * d) / c)) / c
	elif (d <= 5e+115) or not (d <= 5.4e+135):
		tmp = b / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3000000.0)
		tmp = Float64(b / d);
	elseif (d <= 2e-61)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif ((d <= 5e+115) || !(d <= 5.4e+135))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3000000.0)
		tmp = b / d;
	elseif (d <= 2e-61)
		tmp = (a + ((b * d) / c)) / c;
	elseif ((d <= 5e+115) || ~((d <= 5.4e+135)))
		tmp = b / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3000000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 2e-61], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 5e+115], N[Not[LessEqual[d, 5.4e+135]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3e6 or 2.0000000000000001e-61 < d < 5.00000000000000008e115 or 5.3999999999999997e135 < d

   1. Initial program 59.4%

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

   3. Taylor expanded in c around 0 69.9%

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

   if -3e6 < d < 2.0000000000000001e-61

   1. Initial program 70.7%

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

   3. Taylor expanded in c around inf 85.8%

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

   if 5.00000000000000008e115 < d < 5.3999999999999997e135

   1. Initial program 34.7%

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

   3. Taylor expanded in c around inf 45.6%

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

      1. associate-/l* 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in b around 0 45.6%

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

      1. *-rgt-identity 45.6%

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

      2. times-frac 78.2%

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

      3. /-rgt-identity 78.2%

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

      4. associate-/r/ 78.2%

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

   8. Simplified 78.2%

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

      1. associate-/r/ 78.2%

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

   10. Applied egg-rr 78.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3000000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-61}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+115} \lor \neg \left(d \leq 5.4 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -90000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+114} \lor \neg \left(d \leq 1.05 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -90000.0)
   (/ b d)
   (if (<= d 1.9e-61)
     (/ (+ a (* b (/ d c))) c)
     (if (or (<= d 9e+114) (not (<= d 1.05e+133)))
       (/ b d)
       (/ (+ a (* d (/ b c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -90000.0) {
		tmp = b / d;
	} else if (d <= 1.9e-61) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((d <= 9e+114) || !(d <= 1.05e+133)) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-90000.0d0)) then
        tmp = b / d
    else if (d <= 1.9d-61) then
        tmp = (a + (b * (d / c))) / c
    else if ((d <= 9d+114) .or. (.not. (d <= 1.05d+133))) then
        tmp = b / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -90000.0) {
		tmp = b / d;
	} else if (d <= 1.9e-61) {
		tmp = (a + (b * (d / c))) / c;
	} else if ((d <= 9e+114) || !(d <= 1.05e+133)) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -90000.0:
		tmp = b / d
	elif d <= 1.9e-61:
		tmp = (a + (b * (d / c))) / c
	elif (d <= 9e+114) or not (d <= 1.05e+133):
		tmp = b / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -90000.0)
		tmp = Float64(b / d);
	elseif (d <= 1.9e-61)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif ((d <= 9e+114) || !(d <= 1.05e+133))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -90000.0)
		tmp = b / d;
	elseif (d <= 1.9e-61)
		tmp = (a + (b * (d / c))) / c;
	elseif ((d <= 9e+114) || ~((d <= 1.05e+133)))
		tmp = b / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -90000.0], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.9e-61], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 9e+114], N[Not[LessEqual[d, 1.05e+133]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9e4 or 1.8999999999999999e-61 < d < 9.0000000000000001e114 or 1.05e133 < d

   1. Initial program 59.4%

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

   3. Taylor expanded in c around 0 69.9%

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

   if -9e4 < d < 1.8999999999999999e-61

   1. Initial program 70.7%

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

   3. Taylor expanded in c around inf 85.8%

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

      1. associate-/l* 84.6%

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

   5. Simplified 84.6%

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

   if 9.0000000000000001e114 < d < 1.05e133

   1. Initial program 34.7%

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

   3. Taylor expanded in c around inf 45.6%

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

      1. associate-/l* 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in b around 0 45.6%

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

      1. *-rgt-identity 45.6%

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

      2. times-frac 78.2%

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

      3. /-rgt-identity 78.2%

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

      4. associate-/r/ 78.2%

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

   8. Simplified 78.2%

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

      1. associate-/r/ 78.2%

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

   10. Applied egg-rr 78.2%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -90000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+114} \lor \neg \left(d \leq 1.05 \cdot 10^{+133}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-14} \lor \neg \left(c \leq 1.02 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.2e-14) (not (<= c 1.02e-57)))
   (/ (+ a (* d (/ b c))) c)
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.2e-14) || !(c <= 1.02e-57)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / 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 <= (-2.2d-14)) .or. (.not. (c <= 1.02d-57))) then
        tmp = (a + (d * (b / c))) / c
    else
        tmp = (b + (a * (c / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.2e-14) || !(c <= 1.02e-57)) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.2e-14) or not (c <= 1.02e-57):
		tmp = (a + (d * (b / c))) / c
	else:
		tmp = (b + (a * (c / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.2e-14) || !(c <= 1.02e-57))
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.2e-14) || ~((c <= 1.02e-57)))
		tmp = (a + (d * (b / c))) / c;
	else
		tmp = (b + (a * (c / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.2e-14], N[Not[LessEqual[c, 1.02e-57]], $MachinePrecision]], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.2000000000000001e-14 or 1.02e-57 < c

   1. Initial program 55.9%

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

   3. Taylor expanded in c around inf 73.3%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in b around 0 73.3%

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

      1. *-rgt-identity 73.3%

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

      2. times-frac 78.5%

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

      3. /-rgt-identity 78.5%

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

      4. associate-/r/ 77.0%

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

   8. Simplified 77.0%

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

      1. associate-/r/ 78.5%

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

   10. Applied egg-rr 78.5%

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

   if -2.2000000000000001e-14 < c < 1.02e-57

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf 86.9%

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

      1. associate-/l* 86.9%

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

   5. Simplified 86.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{-14} \lor \neg \left(c \leq 1.02 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -46000 \lor \neg \left(d \leq 9.2 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -46000.0) (not (<= d 9.2e-62))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -46000.0) || !(d <= 9.2e-62)) {
		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 <= (-46000.0d0)) .or. (.not. (d <= 9.2d-62))) 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 <= -46000.0) || !(d <= 9.2e-62)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -46000.0) or not (d <= 9.2e-62):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -46000.0) || !(d <= 9.2e-62))
		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 <= -46000.0) || ~((d <= 9.2e-62)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -46000.0], N[Not[LessEqual[d, 9.2e-62]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -46000 or 9.20000000000000002e-62 < d

   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 0 65.9%

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

   if -46000 < d < 9.20000000000000002e-62

   1. Initial program 70.7%

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

   3. Taylor expanded in c around inf 66.8%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -46000 \lor \neg \left(d \leq 9.2 \cdot 10^{-62}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.3%

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

3. Taylor expanded in c around inf 41.1%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024118 
(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))))