Complex division, real part

Percentage Accurate: 61.9% → 85.0%

Time: 6.3s

Alternatives: 9

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: 61.9% 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: 85.0% accurate, 0.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b / d) + ((a / (d / c)) / 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))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a / Float64(d / c)) / 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], Infinity], 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 / d), $MachinePrecision] + N[(N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision] / 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))) < +inf.0

   1. Initial program 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. add-sqr-sqrt 74.1%

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

      3. times-frac 74.1%

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

      4. hypot-def 74.1%

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

      5. fma-def 74.1%

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

      6. hypot-def 95.1%

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

   3. Applied egg-rr 95.1%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   2. Taylor expanded in c around 0 43.2%

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

      1. unpow2 43.2%

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

      2. times-frac 54.4%

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

   4. Simplified 54.4%

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

      1. associate-*r/ 54.4%

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

   6. Applied egg-rr 54.4%

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

      1. *-commutative 54.4%

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

      2. clear-num 54.4%

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

      3. un-div-inv 54.4%

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

   8. Applied egg-rr 54.4%

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

4. Final simplification 87.8%

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

Alternative 2: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.86:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.86)
   (/ b d)
   (if (<= d 3.8e-29)
     (+ (/ a c) (/ (* d (/ b c)) c))
     (if (<= d 6.4e+62)
       (/ b d)
       (if (<= d 3.5e+127) (+ (/ a c) (* (/ d c) (/ b c))) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.86) {
		tmp = b / d;
	} else if (d <= 3.8e-29) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= 6.4e+62) {
		tmp = b / d;
	} else if (d <= 3.5e+127) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-0.86d0)) then
        tmp = b / d
    else if (d <= 3.8d-29) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else if (d <= 6.4d+62) then
        tmp = b / d
    else if (d <= 3.5d+127) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.86) {
		tmp = b / d;
	} else if (d <= 3.8e-29) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= 6.4e+62) {
		tmp = b / d;
	} else if (d <= 3.5e+127) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -0.86:
		tmp = b / d
	elif d <= 3.8e-29:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif d <= 6.4e+62:
		tmp = b / d
	elif d <= 3.5e+127:
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.86)
		tmp = Float64(b / d);
	elseif (d <= 3.8e-29)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (d <= 6.4e+62)
		tmp = Float64(b / d);
	elseif (d <= 3.5e+127)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -0.86)
		tmp = b / d;
	elseif (d <= 3.8e-29)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (d <= 6.4e+62)
		tmp = b / d;
	elseif (d <= 3.5e+127)
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.86], N[(b / d), $MachinePrecision], If[LessEqual[d, 3.8e-29], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.4e+62], N[(b / d), $MachinePrecision], If[LessEqual[d, 3.5e+127], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -0.859999999999999987 or 3.79999999999999976e-29 < d < 6.39999999999999968e62 or 3.49999999999999978e127 < d

   1. Initial program 51.4%

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

   2. Taylor expanded in c around 0 70.6%

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

   if -0.859999999999999987 < d < 3.79999999999999976e-29

   1. Initial program 70.3%

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

   2. Taylor expanded in c around inf 74.5%

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

      1. unpow2 74.5%

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

      2. times-frac 77.2%

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

   4. Simplified 77.2%

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

      1. associate-*l/ 78.9%

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

   6. Applied egg-rr 78.9%

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

   if 6.39999999999999968e62 < d < 3.49999999999999978e127

   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 67.8%

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

      1. unpow2 67.8%

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

      2. times-frac 74.2%

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

   4. Simplified 74.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.86:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+127}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 3: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.8e+107)
   (+ (/ a c) (* (/ d c) (/ b c)))
   (if (<= c -2.35e-142)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= c 6.8e-33)
       (+ (/ b d) (/ (* a (/ c d)) d))
       (+ (/ a c) (/ (* d (/ b c)) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.8e+107) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= -2.35e-142) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 6.8e-33) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-7.8d+107)) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (c <= (-2.35d-142)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (c <= 6.8d-33) then
        tmp = (b / d) + ((a * (c / d)) / d)
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.8e+107) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= -2.35e-142) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 6.8e-33) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.8e+107:
		tmp = (a / c) + ((d / c) * (b / c))
	elif c <= -2.35e-142:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif c <= 6.8e-33:
		tmp = (b / d) + ((a * (c / d)) / d)
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.8e+107)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (c <= -2.35e-142)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 6.8e-33)
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7.8e+107)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (c <= -2.35e-142)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (c <= 6.8e-33)
		tmp = (b / d) + ((a * (c / d)) / d);
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.8e+107], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.35e-142], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.8e-33], N[(N[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.7999999999999997e107

   1. Initial program 35.8%

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

   2. Taylor expanded in c around inf 78.9%

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

      1. unpow2 78.9%

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

      2. times-frac 89.1%

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

   4. Simplified 89.1%

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

   if -7.7999999999999997e107 < c < -2.34999999999999995e-142

   1. Initial program 77.9%

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

   if -2.34999999999999995e-142 < c < 6.8000000000000001e-33

   1. Initial program 69.1%

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

   2. Taylor expanded in c around 0 80.1%

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

      1. unpow2 80.1%

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

      2. times-frac 85.4%

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

   4. Simplified 85.4%

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

      1. associate-*r/ 89.4%

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

   6. Applied egg-rr 89.4%

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

   if 6.8000000000000001e-33 < c

   1. Initial program 53.8%

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

   2. Taylor expanded in c around inf 76.4%

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

      1. unpow2 76.4%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 78.2%

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

   6. Applied egg-rr 78.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{+107}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 4: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.86 \lor \neg \left(d \leq 1.6 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.86) (not (<= d 1.6e+127)))
   (/ b d)
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.86) || !(d <= 1.6e+127)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / 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 <= (-0.86d0)) .or. (.not. (d <= 1.6d+127))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -0.86) || !(d <= 1.6e+127)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -0.86) or not (d <= 1.6e+127):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -0.86) || !(d <= 1.6e+127))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -0.86) || ~((d <= 1.6e+127)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -0.86], N[Not[LessEqual[d, 1.6e+127]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -0.859999999999999987 or 1.59999999999999988e127 < d

   1. Initial program 44.4%

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

   2. Taylor expanded in c around 0 74.8%

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

   if -0.859999999999999987 < d < 1.59999999999999988e127

   1. Initial program 71.5%

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

   2. Taylor expanded in c around inf 67.4%

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

      1. unpow2 67.4%

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

      2. times-frac 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.86 \lor \neg \left(d \leq 1.6 \cdot 10^{+127}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9.6e+75)
   (+ (/ a c) (* (/ d c) (/ b c)))
   (if (<= c 7e-37)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.6e+75) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 7e-37) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-9.6d+75)) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (c <= 7d-37) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.6e+75) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 7e-37) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -9.6e+75:
		tmp = (a / c) + ((d / c) * (b / c))
	elif c <= 7e-37:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9.6e+75)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (c <= 7e-37)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -9.6e+75)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (c <= 7e-37)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9.6e+75], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7e-37], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.5999999999999999e75

   1. Initial program 39.2%

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

   2. Taylor expanded in c around inf 76.7%

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

      1. unpow2 76.7%

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

      2. times-frac 86.0%

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

   4. Simplified 86.0%

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

   if -9.5999999999999999e75 < c < 7.0000000000000003e-37

   1. Initial program 72.1%

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

   2. Taylor expanded in c around 0 72.7%

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

      1. unpow2 72.7%

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

      2. times-frac 77.0%

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

   4. Simplified 77.0%

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

   if 7.0000000000000003e-37 < c

   1. Initial program 53.8%

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

   2. Taylor expanded in c around inf 76.4%

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

      1. unpow2 76.4%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 78.2%

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

   6. Applied egg-rr 78.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-37}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 6: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.5e+75)
   (+ (/ a c) (* (/ d c) (/ b c)))
   (if (<= c 1.95e-38)
     (+ (/ b d) (/ a (* d (/ d c))))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.5e+75) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 1.95e-38) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-8.5d+75)) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (c <= 1.95d-38) then
        tmp = (b / d) + (a / (d * (d / c)))
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.5e+75) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 1.95e-38) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8.5e+75:
		tmp = (a / c) + ((d / c) * (b / c))
	elif c <= 1.95e-38:
		tmp = (b / d) + (a / (d * (d / c)))
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.5e+75)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (c <= 1.95e-38)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -8.5e+75)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (c <= 1.95e-38)
		tmp = (b / d) + (a / (d * (d / c)));
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.5e+75], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.95e-38], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.4999999999999993e75

   1. Initial program 39.2%

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

   2. Taylor expanded in c around inf 76.7%

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

      1. unpow2 76.7%

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

      2. times-frac 86.0%

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

   4. Simplified 86.0%

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

   if -8.4999999999999993e75 < c < 1.95e-38

   1. Initial program 72.1%

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

   2. Taylor expanded in c around 0 72.7%

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

      1. unpow2 72.7%

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

      2. times-frac 77.0%

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

   4. Simplified 77.0%

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

      1. clear-num 77.0%

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

      2. frac-times 78.9%

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

      3. *-un-lft-identity 78.9%

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

   6. Applied egg-rr 78.9%

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

   if 1.95e-38 < c

   1. Initial program 53.8%

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

   2. Taylor expanded in c around inf 76.4%

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

      1. unpow2 76.4%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 78.2%

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

   6. Applied egg-rr 78.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-38}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 7: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e+79)
   (+ (/ a c) (* (/ d c) (/ b c)))
   (if (<= c 7e-35)
     (+ (/ b d) (/ (* a (/ c d)) d))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e+79) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 7e-35) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.35d+79)) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (c <= 7d-35) then
        tmp = (b / d) + ((a * (c / d)) / d)
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e+79) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= 7e-35) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e+79:
		tmp = (a / c) + ((d / c) * (b / c))
	elif c <= 7e-35:
		tmp = (b / d) + ((a * (c / d)) / d)
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e+79)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (c <= 7e-35)
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.35e+79)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (c <= 7e-35)
		tmp = (b / d) + ((a * (c / d)) / d);
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e+79], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7e-35], N[(N[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.35e79

   1. Initial program 39.2%

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

   2. Taylor expanded in c around inf 76.7%

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

      1. unpow2 76.7%

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

      2. times-frac 86.0%

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

   4. Simplified 86.0%

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

   if -1.35e79 < c < 6.99999999999999992e-35

   1. Initial program 72.1%

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

   2. Taylor expanded in c around 0 72.7%

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

      1. unpow2 72.7%

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

      2. times-frac 77.0%

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

   4. Simplified 77.0%

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

      1. associate-*r/ 79.8%

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

   6. Applied egg-rr 79.8%

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

   if 6.99999999999999992e-35 < c

   1. Initial program 53.8%

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

   2. Taylor expanded in c around inf 76.4%

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

      1. unpow2 76.4%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 78.2%

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

   6. Applied egg-rr 78.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{+79}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 8: 63.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9e+75) (/ a c) (if (<= c 3.4e-34) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9e+75) {
		tmp = a / c;
	} else if (c <= 3.4e-34) {
		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 <= (-9d+75)) then
        tmp = a / c
    else if (c <= 3.4d-34) 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 <= -9e+75) {
		tmp = a / c;
	} else if (c <= 3.4e-34) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -9e+75:
		tmp = a / c
	elif c <= 3.4e-34:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9e+75)
		tmp = Float64(a / c);
	elseif (c <= 3.4e-34)
		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 <= -9e+75)
		tmp = a / c;
	elseif (c <= 3.4e-34)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9e+75], N[(a / c), $MachinePrecision], If[LessEqual[c, 3.4e-34], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -9.0000000000000007e75 or 3.4000000000000001e-34 < c

   1. Initial program 48.0%

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

   2. Taylor expanded in c around inf 69.4%

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

   if -9.0000000000000007e75 < c < 3.4000000000000001e-34

   1. Initial program 72.1%

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

   2. Taylor expanded in c around 0 64.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+75}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 9: 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 60.8%

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

2. Taylor expanded in c around inf 41.6%

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

3. Final simplification 41.6%

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

Developer target: 99.3% 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 2023171 
(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))))