Complex division, real part

Percentage Accurate: 61.6% → 85.7%

Time: 8.0s

Alternatives: 12

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: 61.6% 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: 85.7% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+306)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (+ (/ b d) (/ (* a (/ c d)) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+306) {
		tmp = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (b / d) + ((a * (c / d)) / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+306)
		tmp = Float64(Float64(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+306], 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[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $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))) < 4.99999999999999993e306

   1. Initial program 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. add-sqr-sqrt 76.8%

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

      3. times-frac 76.8%

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

      4. hypot-def 76.8%

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

      5. fma-def 76.8%

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

      6. hypot-def 96.5%

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

   3. Applied egg-rr 96.5%

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

      1. associate-*l/ 96.7%

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

      2. *-un-lft-identity 96.7%

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

   5. Applied egg-rr 96.7%

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

   if 4.99999999999999993e306 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 9.1%

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

   2. Taylor expanded in c around 0 38.9%

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

      1. unpow2 38.9%

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

      2. times-frac 57.0%

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

   4. Simplified 57.0%

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

      1. associate-*r/ 57.1%

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

   6. Applied egg-rr 57.1%

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

4. Final simplification 85.7%

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

Alternative 2: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\\ \mathbf{elif}\;c \leq -4.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.85 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.4e+89)
   (/ (/ c (hypot c d)) (/ (hypot c d) a))
   (if (<= c -4.4e-71)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= c 2.85e+24)
       (+ (/ 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 <= -5.4e+89) {
		tmp = (c / hypot(c, d)) / (hypot(c, d) / a);
	} else if (c <= -4.4e-71) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 2.85e+24) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.4e+89) {
		tmp = (c / Math.hypot(c, d)) / (Math.hypot(c, d) / a);
	} else if (c <= -4.4e-71) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 2.85e+24) {
		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 <= -5.4e+89:
		tmp = (c / math.hypot(c, d)) / (math.hypot(c, d) / a)
	elif c <= -4.4e-71:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif c <= 2.85e+24:
		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 <= -5.4e+89)
		tmp = Float64(Float64(c / hypot(c, d)) / Float64(hypot(c, d) / a));
	elseif (c <= -4.4e-71)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.85e+24)
		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 <= -5.4e+89)
		tmp = (c / hypot(c, d)) / (hypot(c, d) / a);
	elseif (c <= -4.4e-71)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (c <= 2.85e+24)
		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, -5.4e+89], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -4.4e-71], 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, 2.85e+24], 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 < -5.4e89

   1. Initial program 31.4%

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

      1. *-un-lft-identity 31.4%

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

      2. add-sqr-sqrt 31.4%

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

      3. times-frac 31.4%

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

      4. hypot-def 31.4%

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

      5. fma-def 31.4%

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

      6. hypot-def 54.2%

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

   3. Applied egg-rr 54.2%

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

   4. Taylor expanded in a around inf 50.7%

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

      1. *-un-lft-identity 50.7%

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

      2. times-frac 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. frac-times 50.7%

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

      2. *-un-lft-identity 50.7%

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

      3. associate-/l* 81.0%

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

      4. associate-*r/ 80.8%

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

      5. associate-*l/ 81.0%

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

      6. *-un-lft-identity 81.0%

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

   8. Applied egg-rr 81.0%

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

   if -5.4e89 < c < -4.39999999999999995e-71

   1. Initial program 84.9%

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

   if -4.39999999999999995e-71 < c < 2.8500000000000002e24

   1. Initial program 67.7%

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

   2. Taylor expanded in c around 0 80.5%

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

      1. unpow2 80.5%

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

      2. times-frac 86.0%

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

   4. Simplified 86.0%

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

      1. associate-*r/ 87.8%

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

   6. Applied egg-rr 87.8%

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

   if 2.8500000000000002e24 < c

   1. Initial program 48.9%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. associate-*l/ 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{c}{\mathsf{hypot}\left(c, d\right)}}{\frac{\mathsf{hypot}\left(c, d\right)}{a}}\\ \mathbf{elif}\;c \leq -4.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.85 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 3: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+95}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.35e+95)
   (+ (/ a c) (/ d (* c (/ c b))))
   (if (<= c -3.8e-71)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= c 4.5e+28)
       (+ (/ 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 <= -2.35e+95) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= -3.8e-71) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 4.5e+28) {
		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 <= (-2.35d+95)) then
        tmp = (a / c) + (d / (c * (c / b)))
    else if (c <= (-3.8d-71)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (c <= 4.5d+28) 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 <= -2.35e+95) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= -3.8e-71) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 4.5e+28) {
		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 <= -2.35e+95:
		tmp = (a / c) + (d / (c * (c / b)))
	elif c <= -3.8e-71:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif c <= 4.5e+28:
		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 <= -2.35e+95)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	elseif (c <= -3.8e-71)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 4.5e+28)
		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 <= -2.35e+95)
		tmp = (a / c) + (d / (c * (c / b)));
	elseif (c <= -3.8e-71)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (c <= 4.5e+28)
		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, -2.35e+95], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.8e-71], 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, 4.5e+28], 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 < -2.34999999999999986e95

   1. Initial program 31.2%

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

   2. Taylor expanded in c around inf 73.8%

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

      1. unpow2 73.8%

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

      2. times-frac 78.5%

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

   4. Simplified 78.5%

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

      1. *-commutative 78.5%

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

      2. clear-num 78.5%

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

      3. frac-times 82.3%

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

      4. *-un-lft-identity 82.3%

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

   6. Applied egg-rr 82.3%

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

   if -2.34999999999999986e95 < c < -3.79999999999999992e-71

   1. Initial program 80.8%

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

   if -3.79999999999999992e-71 < c < 4.4999999999999997e28

   1. Initial program 67.7%

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

   2. Taylor expanded in c around 0 80.5%

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

      1. unpow2 80.5%

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

      2. times-frac 86.0%

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

   4. Simplified 86.0%

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

      1. associate-*r/ 87.8%

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

   6. Applied egg-rr 87.8%

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

   if 4.4999999999999997e28 < c

   1. Initial program 48.9%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. associate-*l/ 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+95}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 4: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\frac{d}{c} \cdot \frac{d}{a}}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e+45)
   (/ a c)
   (if (<= c 2.75e-129)
     (/ b d)
     (if (<= c 1.65e-91)
       (/ 1.0 (* (/ d c) (/ d a)))
       (if (<= c 1.65e+26) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 2.75e-129) {
		tmp = b / d;
	} else if (c <= 1.65e-91) {
		tmp = 1.0 / ((d / c) * (d / a));
	} else if (c <= 1.65e+26) {
		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 <= (-5.5d+45)) then
        tmp = a / c
    else if (c <= 2.75d-129) then
        tmp = b / d
    else if (c <= 1.65d-91) then
        tmp = 1.0d0 / ((d / c) * (d / a))
    else if (c <= 1.65d+26) 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 <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 2.75e-129) {
		tmp = b / d;
	} else if (c <= 1.65e-91) {
		tmp = 1.0 / ((d / c) * (d / a));
	} else if (c <= 1.65e+26) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e+45:
		tmp = a / c
	elif c <= 2.75e-129:
		tmp = b / d
	elif c <= 1.65e-91:
		tmp = 1.0 / ((d / c) * (d / a))
	elif c <= 1.65e+26:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e+45)
		tmp = Float64(a / c);
	elseif (c <= 2.75e-129)
		tmp = Float64(b / d);
	elseif (c <= 1.65e-91)
		tmp = Float64(1.0 / Float64(Float64(d / c) * Float64(d / a)));
	elseif (c <= 1.65e+26)
		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 <= -5.5e+45)
		tmp = a / c;
	elseif (c <= 2.75e-129)
		tmp = b / d;
	elseif (c <= 1.65e-91)
		tmp = 1.0 / ((d / c) * (d / a));
	elseif (c <= 1.65e+26)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e+45], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.75e-129], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.65e-91], N[(1.0 / N[(N[(d / c), $MachinePrecision] * N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.65e+26], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.5000000000000001e45 or 1.64999999999999997e26 < c

   1. Initial program 45.4%

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

   2. Taylor expanded in c around inf 69.6%

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

   if -5.5000000000000001e45 < c < 2.75000000000000012e-129 or 1.65000000000000006e-91 < c < 1.64999999999999997e26

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 68.9%

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

   if 2.75000000000000012e-129 < c < 1.65000000000000006e-91

   1. Initial program 80.9%

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

   2. Taylor expanded in c around 0 61.1%

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

      1. unpow2 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in a around inf 61.1%

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

      1. clear-num 60.9%

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

      2. inv-pow 60.9%

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

      3. *-commutative 60.9%

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

      4. times-frac 71.4%

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

   7. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{{\left(\frac{d}{a} \cdot \frac{d}{c}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 71.4%

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

      2. *-commutative 71.4%

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

   9. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{d}{c} \cdot \frac{d}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-91}:\\ \;\;\;\;\frac{1}{\frac{d}{c} \cdot \frac{d}{a}}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 5: 68.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+52} \lor \neg \left(d \leq 1.85 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9e+52) (not (<= d 1.85e+168)))
   (/ b d)
   (+ (/ a c) (* (/ b c) (/ d c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e+52) || !(d <= 1.85e+168)) {
		tmp = 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 ((d <= (-9d+52)) .or. (.not. (d <= 1.85d+168))) then
        tmp = 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 ((d <= -9e+52) || !(d <= 1.85e+168)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9e+52) or not (d <= 1.85e+168):
		tmp = b / d
	else:
		tmp = (a / c) + ((b / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9e+52) || !(d <= 1.85e+168))
		tmp = 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 ((d <= -9e+52) || ~((d <= 1.85e+168)))
		tmp = b / d;
	else
		tmp = (a / c) + ((b / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9e+52], N[Not[LessEqual[d, 1.85e+168]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.9999999999999999e52 or 1.85000000000000005e168 < d

   1. Initial program 39.2%

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

   2. Taylor expanded in c around 0 74.3%

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

   if -8.9999999999999999e52 < d < 1.85000000000000005e168

   1. Initial program 67.4%

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

   2. Taylor expanded in c around inf 64.5%

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

      1. unpow2 64.5%

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

      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 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+52} \lor \neg \left(d \leq 1.85 \cdot 10^{+168}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+168}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.1e+53)
   (/ b d)
   (if (<= d 1.85e+168) (+ (/ a c) (/ (* d (/ b c)) c)) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.1e+53) {
		tmp = b / d;
	} else if (d <= 1.85e+168) {
		tmp = (a / c) + ((d * (b / c)) / 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 <= (-1.1d+53)) then
        tmp = b / d
    else if (d <= 1.85d+168) then
        tmp = (a / c) + ((d * (b / c)) / 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 <= -1.1e+53) {
		tmp = b / d;
	} else if (d <= 1.85e+168) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.1e+53:
		tmp = b / d
	elif d <= 1.85e+168:
		tmp = (a / c) + ((d * (b / c)) / c)
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.1e+53)
		tmp = Float64(b / d);
	elseif (d <= 1.85e+168)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.1e+53)
		tmp = b / d;
	elseif (d <= 1.85e+168)
		tmp = (a / c) + ((d * (b / c)) / c);
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.1e+53], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.85e+168], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.09999999999999999e53 or 1.85000000000000005e168 < d

   1. Initial program 39.2%

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

   2. Taylor expanded in c around 0 74.3%

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

   if -1.09999999999999999e53 < d < 1.85000000000000005e168

   1. Initial program 67.4%

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

   2. Taylor expanded in c around inf 64.5%

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

      1. unpow2 64.5%

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

      2. times-frac 70.7%

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

   4. Simplified 70.7%

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

      1. associate-*l/ 71.8%

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

   6. Applied egg-rr 71.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.1 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+168}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6e+45)
   (+ (/ a c) (/ d (* c (/ c b))))
   (if (<= c 1.85e+25)
     (+ (/ 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 <= -6e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 1.85e+25) {
		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 <= (-6d+45)) then
        tmp = (a / c) + (d / (c * (c / b)))
    else if (c <= 1.85d+25) 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 <= -6e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 1.85e+25) {
		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 <= -6e+45:
		tmp = (a / c) + (d / (c * (c / b)))
	elif c <= 1.85e+25:
		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 <= -6e+45)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	elseif (c <= 1.85e+25)
		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 <= -6e+45)
		tmp = (a / c) + (d / (c * (c / b)));
	elseif (c <= 1.85e+25)
		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, -6e+45], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.85e+25], 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 < -6.00000000000000021e45

   1. Initial program 41.9%

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

   2. Taylor expanded in c around inf 72.8%

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

      1. unpow2 72.8%

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

      2. times-frac 75.0%

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

   4. Simplified 75.0%

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

      1. *-commutative 75.0%

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

      2. clear-num 75.0%

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

      3. frac-times 79.4%

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

      4. *-un-lft-identity 79.4%

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

   6. Applied egg-rr 79.4%

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

   if -6.00000000000000021e45 < c < 1.8499999999999999e25

   1. Initial program 70.1%

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

   2. Taylor expanded in c around 0 77.2%

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

      1. unpow2 77.2%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

   if 1.8499999999999999e25 < c

   1. Initial program 48.9%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. associate-*l/ 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 8: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e+45)
   (+ (/ a c) (/ d (* c (/ c b))))
   (if (<= c 1e+27)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 1e+27) {
		tmp = (b / d) + ((c * (a / 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 <= (-5.5d+45)) then
        tmp = (a / c) + (d / (c * (c / b)))
    else if (c <= 1d+27) then
        tmp = (b / d) + ((c * (a / 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 <= -5.5e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 1e+27) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e+45:
		tmp = (a / c) + (d / (c * (c / b)))
	elif c <= 1e+27:
		tmp = (b / d) + ((c * (a / d)) / d)
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e+45)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	elseif (c <= 1e+27)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / 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 <= -5.5e+45)
		tmp = (a / c) + (d / (c * (c / b)));
	elseif (c <= 1e+27)
		tmp = (b / d) + ((c * (a / d)) / d);
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e+45], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e+27], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / 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 < -5.5000000000000001e45

   1. Initial program 41.9%

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

   2. Taylor expanded in c around inf 72.8%

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

      1. unpow2 72.8%

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

      2. times-frac 75.0%

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

   4. Simplified 75.0%

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

      1. *-commutative 75.0%

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

      2. clear-num 75.0%

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

      3. frac-times 79.4%

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

      4. *-un-lft-identity 79.4%

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

   6. Applied egg-rr 79.4%

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

   if -5.5000000000000001e45 < c < 1e27

   1. Initial program 70.1%

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

   2. Taylor expanded in c around 0 77.2%

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

      1. unpow2 77.2%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. associate-*l/ 83.6%

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

   6. Applied egg-rr 83.6%

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

   if 1e27 < c

   1. Initial program 48.9%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. associate-*l/ 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 10^{+27}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 9: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.2e+45)
   (+ (/ a c) (/ d (* c (/ c b))))
   (if (<= c 3.2e+26)
     (+ (/ 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 <= -6.2e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 3.2e+26) {
		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 <= (-6.2d+45)) then
        tmp = (a / c) + (d / (c * (c / b)))
    else if (c <= 3.2d+26) 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 <= -6.2e+45) {
		tmp = (a / c) + (d / (c * (c / b)));
	} else if (c <= 3.2e+26) {
		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 <= -6.2e+45:
		tmp = (a / c) + (d / (c * (c / b)))
	elif c <= 3.2e+26:
		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 <= -6.2e+45)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	elseif (c <= 3.2e+26)
		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 <= -6.2e+45)
		tmp = (a / c) + (d / (c * (c / b)));
	elseif (c <= 3.2e+26)
		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, -6.2e+45], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.2e+26], 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 < -6.19999999999999975e45

   1. Initial program 41.9%

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

   2. Taylor expanded in c around inf 72.8%

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

      1. unpow2 72.8%

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

      2. times-frac 75.0%

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

   4. Simplified 75.0%

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

      1. *-commutative 75.0%

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

      2. clear-num 75.0%

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

      3. frac-times 79.4%

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

      4. *-un-lft-identity 79.4%

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

   6. Applied egg-rr 79.4%

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

   if -6.19999999999999975e45 < c < 3.20000000000000029e26

   1. Initial program 70.1%

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

   2. Taylor expanded in c around 0 77.2%

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

      1. unpow2 77.2%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. associate-*r/ 84.7%

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

   6. Applied egg-rr 84.7%

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

   if 3.20000000000000029e26 < c

   1. Initial program 48.9%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. associate-*l/ 83.4%

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

   6. Applied egg-rr 83.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 10: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e+45)
   (/ a c)
   (if (<= c 9.5e-129)
     (/ b d)
     (if (<= c 6.5e-85)
       (* (/ c d) (/ a d))
       (if (<= c 7.6e+24) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 9.5e-129) {
		tmp = b / d;
	} else if (c <= 6.5e-85) {
		tmp = (c / d) * (a / d);
	} else if (c <= 7.6e+24) {
		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 <= (-5.5d+45)) then
        tmp = a / c
    else if (c <= 9.5d-129) then
        tmp = b / d
    else if (c <= 6.5d-85) then
        tmp = (c / d) * (a / d)
    else if (c <= 7.6d+24) 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 <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 9.5e-129) {
		tmp = b / d;
	} else if (c <= 6.5e-85) {
		tmp = (c / d) * (a / d);
	} else if (c <= 7.6e+24) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e+45:
		tmp = a / c
	elif c <= 9.5e-129:
		tmp = b / d
	elif c <= 6.5e-85:
		tmp = (c / d) * (a / d)
	elif c <= 7.6e+24:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e+45)
		tmp = Float64(a / c);
	elseif (c <= 9.5e-129)
		tmp = Float64(b / d);
	elseif (c <= 6.5e-85)
		tmp = Float64(Float64(c / d) * Float64(a / d));
	elseif (c <= 7.6e+24)
		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 <= -5.5e+45)
		tmp = a / c;
	elseif (c <= 9.5e-129)
		tmp = b / d;
	elseif (c <= 6.5e-85)
		tmp = (c / d) * (a / d);
	elseif (c <= 7.6e+24)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e+45], N[(a / c), $MachinePrecision], If[LessEqual[c, 9.5e-129], N[(b / d), $MachinePrecision], If[LessEqual[c, 6.5e-85], N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.6e+24], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.5000000000000001e45 or 7.6000000000000003e24 < c

   1. Initial program 45.4%

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

   2. Taylor expanded in c around inf 69.6%

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

   if -5.5000000000000001e45 < c < 9.5000000000000006e-129 or 6.5e-85 < c < 7.6000000000000003e24

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 68.9%

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

   if 9.5000000000000006e-129 < c < 6.5e-85

   1. Initial program 80.9%

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

   2. Taylor expanded in c around 0 68.8%

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

      1. unpow2 68.8%

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

      2. times-frac 78.8%

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

   4. Simplified 78.8%

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

      1. frac-times 68.8%

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

      2. *-commutative 68.8%

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

      3. frac-add 37.0%

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

   6. Applied egg-rr 37.0%

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

   7. Taylor expanded in b around 0 61.1%

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

      1. unpow2 61.1%

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

      2. times-frac 71.1%

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

   9. Simplified 71.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 11: 62.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e+45) (/ a c) (if (<= c 1.8e+24) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 1.8e+24) {
		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 <= (-5.5d+45)) then
        tmp = a / c
    else if (c <= 1.8d+24) 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 <= -5.5e+45) {
		tmp = a / c;
	} else if (c <= 1.8e+24) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e+45:
		tmp = a / c
	elif c <= 1.8e+24:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e+45)
		tmp = Float64(a / c);
	elseif (c <= 1.8e+24)
		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 <= -5.5e+45)
		tmp = a / c;
	elseif (c <= 1.8e+24)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e+45], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.8e+24], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.5000000000000001e45 or 1.79999999999999992e24 < c

   1. Initial program 45.4%

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

   2. Taylor expanded in c around inf 69.6%

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

   if -5.5000000000000001e45 < c < 1.79999999999999992e24

   1. Initial program 70.1%

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

   2. Taylor expanded in c around 0 66.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+24}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 12: 42.4% 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 58.0%

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

2. Taylor expanded in c around inf 42.3%

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

3. Final simplification 42.3%

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

Developer target: 99.2% 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 2023192 
(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))))