Complex division, real part

Percentage Accurate: 60.6% → 79.3%

Time: 6.3s

Alternatives: 8

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.6% 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: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 2.85 \cdot 10^{+20}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e-20)
   (+ (/ a c) (/ (* d (/ b c)) c))
   (if (<= c 3.8e-126)
     (+ (/ b d) (/ a (* d (/ d c))))
     (if (<= c 2.85e+20)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (+ (/ a c) (/ (/ b c) (/ c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-20) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 3.8e-126) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 2.85e+20) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + ((b / 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 (c <= (-7.5d-20)) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else if (c <= 3.8d-126) then
        tmp = (b / d) + (a / (d * (d / c)))
    else if (c <= 2.85d+20) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = (a / c) + ((b / 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 (c <= -7.5e-20) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 3.8e-126) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else if (c <= 2.85e+20) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + ((b / c) / (c / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.5e-20:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif c <= 3.8e-126:
		tmp = (b / d) + (a / (d * (d / c)))
	elif c <= 2.85e+20:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (a / c) + ((b / c) / (c / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e-20)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (c <= 3.8e-126)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	elseif (c <= 2.85e+20)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.5e-20)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (c <= 3.8e-126)
		tmp = (b / d) + (a / (d * (d / c)));
	elseif (c <= 2.85e+20)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (a / c) + ((b / c) / (c / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e-20], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e-126], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.85e+20], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.49999999999999981e-20

   1. Initial program 52.3%

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

   2. Taylor expanded in c around inf 74.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 74.6%

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

      3. times-frac 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -7.49999999999999981e-20 < c < 3.7999999999999999e-126

   1. Initial program 68.8%

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

   2. Taylor expanded in c around 0 86.4%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. associate-/l* 86.5%

         \[\leadsto \frac{b}{d} + \color{blue}{\frac{a}{\frac{{d}^{2}}{c}}} \]

      2. unpow2 86.5%

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

   4. Simplified 86.5%

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

   5. Taylor expanded in d around 0 86.5%

      \[\leadsto \frac{b}{d} + \frac{a}{\color{blue}{\frac{{d}^{2}}{c}}} \]
   6. Step-by-step derivation

      1. unpow2 86.5%

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

      2. associate-*r/ 88.3%

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

   7. Simplified 88.3%

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

   if 3.7999999999999999e-126 < c < 2.85e20

   1. Initial program 86.1%

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

   if 2.85e20 < c

   1. Initial program 44.7%

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

   2. Taylor expanded in c around inf 81.2%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 81.2%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 81.2%

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

      3. times-frac 86.4%

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

   4. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. clear-num 86.4%

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

      3. un-div-inv 86.4%

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

   6. Applied egg-rr 86.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{elif}\;c \leq 2.85 \cdot 10^{+20}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+286)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (+ (* (/ c d) (/ a d)) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+286) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = ((c / d) * (a / d)) + (b / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+286)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(a / d)) + Float64(b / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+286], N[(N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $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))) < 5.0000000000000004e286

   1. Initial program 77.9%

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

      1. *-un-lft-identity 77.9%

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

      2. add-sqr-sqrt 77.9%

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

      3. times-frac 77.8%

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

      4. hypot-def 77.8%

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

      5. fma-def 77.8%

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

      6. hypot-def 93.3%

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

   3. Applied egg-rr 93.3%

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

      1. associate-*l/ 93.6%

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

      2. *-un-lft-identity 93.6%

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

   5. Applied egg-rr 93.6%

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

   if 5.0000000000000004e286 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 5.8%

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

   2. Taylor expanded in c around 0 47.6%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 47.6%

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. *-commutative 47.6%

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{d}^{2}} + \frac{b}{d} \]

      3. unpow2 47.6%

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

      4. times-frac 55.2%

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

      5. fma-def 55.2%

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

   4. Simplified 55.2%

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

      1. fma-udef 55.2%

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

   6. Applied egg-rr 55.2%

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

4. Final simplification 84.9%

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

Alternative 3: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-21} \lor \neg \left(c \leq 1.75 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.9e-21) (not (<= c 1.75e+47)))
   (+ (/ a c) (* (/ b c) (/ d c)))
   (+ (* (/ c d) (/ a d)) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-21) || !(c <= 1.75e+47)) {
		tmp = (a / c) + ((b / c) * (d / c));
	} else {
		tmp = ((c / d) * (a / d)) + (b / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.9d-21)) .or. (.not. (c <= 1.75d+47))) then
        tmp = (a / c) + ((b / c) * (d / c))
    else
        tmp = ((c / d) * (a / d)) + (b / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-21) || !(c <= 1.75e+47)) {
		tmp = (a / c) + ((b / c) * (d / c));
	} else {
		tmp = ((c / d) * (a / d)) + (b / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.9e-21) or not (c <= 1.75e+47):
		tmp = (a / c) + ((b / c) * (d / c))
	else:
		tmp = ((c / d) * (a / d)) + (b / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.9e-21) || !(c <= 1.75e+47))
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(a / d)) + Float64(b / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.9e-21) || ~((c <= 1.75e+47)))
		tmp = (a / c) + ((b / c) * (d / c));
	else
		tmp = ((c / d) * (a / d)) + (b / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.9e-21], N[Not[LessEqual[c, 1.75e+47]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.8999999999999999e-21 or 1.75000000000000008e47 < c

   1. Initial program 48.6%

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

   2. Taylor expanded in c around inf 78.7%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 78.7%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 78.7%

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

      3. times-frac 83.8%

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

   4. Simplified 83.8%

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

   if -1.8999999999999999e-21 < c < 1.75000000000000008e47

   1. Initial program 72.0%

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

   2. Taylor expanded in c around 0 79.8%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{d}^{2}} + \frac{b}{d} \]

      3. unpow2 79.8%

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

      4. times-frac 81.7%

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

      5. fma-def 81.7%

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

   4. Simplified 81.7%

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

      1. fma-udef 81.7%

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

   6. Applied egg-rr 81.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-21} \lor \neg \left(c \leq 1.75 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \end{array} \]

Alternative 4: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.8e-16)
   (/ a c)
   (if (<= c 4.5e+48) (+ (* (/ c d) (/ a d)) (/ b d)) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.8e-16) {
		tmp = a / c;
	} else if (c <= 4.5e+48) {
		tmp = ((c / d) * (a / d)) + (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 (c <= (-4.8d-16)) then
        tmp = a / c
    else if (c <= 4.5d+48) then
        tmp = ((c / d) * (a / d)) + (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 (c <= -4.8e-16) {
		tmp = a / c;
	} else if (c <= 4.5e+48) {
		tmp = ((c / d) * (a / d)) + (b / d);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.8e-16:
		tmp = a / c
	elif c <= 4.5e+48:
		tmp = ((c / d) * (a / d)) + (b / d)
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.8e-16)
		tmp = Float64(a / c);
	elseif (c <= 4.5e+48)
		tmp = Float64(Float64(Float64(c / d) * Float64(a / d)) + 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 (c <= -4.8e-16)
		tmp = a / c;
	elseif (c <= 4.5e+48)
		tmp = ((c / d) * (a / d)) + (b / d);
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.8e-16], N[(a / c), $MachinePrecision], If[LessEqual[c, 4.5e+48], N[(N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.8000000000000001e-16 or 4.49999999999999995e48 < c

   1. Initial program 48.1%

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

   2. Taylor expanded in c around inf 71.1%

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

   if -4.8000000000000001e-16 < c < 4.49999999999999995e48

   1. Initial program 72.2%

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

   2. Taylor expanded in c around 0 79.3%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 79.3%

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. *-commutative 79.3%

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{d}^{2}} + \frac{b}{d} \]

      3. unpow2 79.3%

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

      4. times-frac 81.2%

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

      5. fma-def 81.2%

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

   4. Simplified 81.2%

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

      1. fma-udef 81.2%

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

   6. Applied egg-rr 81.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 5: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e-19)
   (+ (/ a c) (/ (* d (/ b c)) c))
   (if (<= c 2.6e+48)
     (+ (* (/ c d) (/ a d)) (/ b d))
     (+ (/ a c) (* (/ b c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e-19) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 2.6e+48) {
		tmp = ((c / d) * (a / d)) + (b / d);
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-5.5d-19)) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else if (c <= 2.6d+48) then
        tmp = ((c / d) * (a / d)) + (b / d)
    else
        tmp = (a / c) + ((b / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e-19) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 2.6e+48) {
		tmp = ((c / d) * (a / d)) + (b / d);
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e-19:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif c <= 2.6e+48:
		tmp = ((c / d) * (a / d)) + (b / d)
	else:
		tmp = (a / c) + ((b / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e-19)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (c <= 2.6e+48)
		tmp = Float64(Float64(Float64(c / d) * Float64(a / d)) + Float64(b / d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.5e-19)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (c <= 2.6e+48)
		tmp = ((c / d) * (a / d)) + (b / d);
	else
		tmp = (a / c) + ((b / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e-19], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+48], N[(N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.4999999999999996e-19

   1. Initial program 52.3%

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

   2. Taylor expanded in c around inf 74.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 74.6%

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

      3. times-frac 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -5.4999999999999996e-19 < c < 2.59999999999999995e48

   1. Initial program 72.0%

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

   2. Taylor expanded in c around 0 79.8%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{d}^{2}} + \frac{b}{d} \]

      3. unpow2 79.8%

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

      4. times-frac 81.7%

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

      5. fma-def 81.7%

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

   4. Simplified 81.7%

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

      1. fma-udef 81.7%

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

   6. Applied egg-rr 81.7%

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

   if 2.59999999999999995e48 < c

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 83.4%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 83.4%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 83.4%

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

      3. times-frac 89.1%

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

   4. Simplified 89.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 6: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.65e-18)
   (+ (/ a c) (/ (* d (/ b c)) c))
   (if (<= c 8.6e+51)
     (+ (* (/ c d) (/ a d)) (/ b d))
     (+ (/ a c) (/ (/ b c) (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.65e-18) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 8.6e+51) {
		tmp = ((c / d) * (a / d)) + (b / d);
	} else {
		tmp = (a / c) + ((b / 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 (c <= (-1.65d-18)) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else if (c <= 8.6d+51) then
        tmp = ((c / d) * (a / d)) + (b / d)
    else
        tmp = (a / c) + ((b / 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 (c <= -1.65e-18) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (c <= 8.6e+51) {
		tmp = ((c / d) * (a / d)) + (b / d);
	} else {
		tmp = (a / c) + ((b / c) / (c / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.65e-18:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif c <= 8.6e+51:
		tmp = ((c / d) * (a / d)) + (b / d)
	else:
		tmp = (a / c) + ((b / c) / (c / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.65e-18)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (c <= 8.6e+51)
		tmp = Float64(Float64(Float64(c / d) * Float64(a / d)) + Float64(b / d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.65e-18)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (c <= 8.6e+51)
		tmp = ((c / d) * (a / d)) + (b / d);
	else
		tmp = (a / c) + ((b / c) / (c / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.65e-18], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.6e+51], N[(N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.6500000000000001e-18

   1. Initial program 52.3%

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

   2. Taylor expanded in c around inf 74.6%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 74.6%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 74.6%

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

      3. times-frac 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 80.8%

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

   6. Applied egg-rr 80.8%

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

   if -1.6500000000000001e-18 < c < 8.5999999999999994e51

   1. Initial program 72.0%

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

   2. Taylor expanded in c around 0 79.8%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a \cdot c}{{d}^{2}}} \]
   3. Step-by-step derivation

      1. +-commutative 79.8%

         \[\leadsto \color{blue}{\frac{a \cdot c}{{d}^{2}} + \frac{b}{d}} \]

      2. *-commutative 79.8%

         \[\leadsto \frac{\color{blue}{c \cdot a}}{{d}^{2}} + \frac{b}{d} \]

      3. unpow2 79.8%

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

      4. times-frac 81.7%

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

      5. fma-def 81.7%

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

   4. Simplified 81.7%

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

      1. fma-udef 81.7%

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

   6. Applied egg-rr 81.7%

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

   if 8.5999999999999994e51 < c

   1. Initial program 44.2%

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

   2. Taylor expanded in c around inf 83.4%

      \[\leadsto \color{blue}{\frac{a}{c} + \frac{b \cdot d}{{c}^{2}}} \]
   3. Step-by-step derivation

      1. *-commutative 83.4%

         \[\leadsto \frac{a}{c} + \frac{\color{blue}{d \cdot b}}{{c}^{2}} \]

      2. unpow2 83.4%

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

      3. times-frac 89.1%

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

   4. Simplified 89.1%

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

      1. *-commutative 89.1%

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

      2. clear-num 89.1%

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

      3. un-div-inv 89.1%

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

   6. Applied egg-rr 89.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-18}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d} + \frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 7: 64.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 0.00024:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.45e-16) (/ a c) (if (<= c 0.00024) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.45e-16) {
		tmp = a / c;
	} else if (c <= 0.00024) {
		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 (c <= (-1.45d-16)) then
        tmp = a / c
    else if (c <= 0.00024d0) 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 (c <= -1.45e-16) {
		tmp = a / c;
	} else if (c <= 0.00024) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.45e-16:
		tmp = a / c
	elif c <= 0.00024:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.45e-16)
		tmp = Float64(a / c);
	elseif (c <= 0.00024)
		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 (c <= -1.45e-16)
		tmp = a / c;
	elseif (c <= 0.00024)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.45e-16], N[(a / c), $MachinePrecision], If[LessEqual[c, 0.00024], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.4499999999999999e-16 or 2.40000000000000006e-4 < c

   1. Initial program 49.3%

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

   2. Taylor expanded in c around inf 69.8%

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

   if -1.4499999999999999e-16 < c < 2.40000000000000006e-4

   1. Initial program 72.5%

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

   2. Taylor expanded in c around 0 74.4%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-16}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 0.00024:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8: 42.5% 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 61.6%

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

2. Taylor expanded in c around inf 41.1%

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

3. Final simplification 41.1%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (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))))