Complex division, real part

Percentage Accurate: 62.0% → 84.9%

Time: 9.7s

Alternatives: 14

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: 62.0% 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: 84.9% accurate, 0.0× speedup

Math

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

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 (* (/ 1.0 (hypot d c)) (/ d (hypot d c))))))

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 * ((1.0 / hypot(d, c)) * (d / hypot(d, c)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(b * Float64(Float64(1.0 / hypot(d, c)) * Float64(d / hypot(d, c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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[(b * N[(N[(1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $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 75.3%

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

      1. *-un-lft-identity 75.3%

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

      2. add-sqr-sqrt 75.3%

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

      3. times-frac 75.4%

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

      4. hypot-def 75.4%

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

      5. fma-def 75.4%

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

      6. hypot-def 92.4%

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

   3. Applied egg-rr 92.4%

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

   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 a around 0 1.2%

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

      1. associate-/l* 3.6%

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

      2. associate-/r/ 3.6%

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

      3. unpow2 3.6%

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

      4. unpow2 3.6%

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

      5. fma-udef 3.6%

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

   4. Simplified 3.6%

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

      1. *-un-lft-identity 3.6%

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

      2. add-sqr-sqrt 3.6%

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

      3. times-frac 3.6%

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

      4. fma-udef 3.6%

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

      5. hypot-def 3.6%

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

      6. fma-udef 3.6%

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

      7. hypot-def 62.8%

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

   6. Applied egg-rr 62.8%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{d}{\mathsf{hypot}\left(d, c\right)}\right)} \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

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

Alternative 2: 84.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{b \cdot d}{t_0} + \frac{a}{\frac{t_0}{c}}\\ t_2 := \frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (+ (/ (* b d) t_0) (/ a (/ t_0 c))))
        (t_2 (+ (/ b d) (/ (* c (/ a d)) d))))
   (if (<= d -5e+65)
     t_2
     (if (<= d -9e-160)
       t_1
       (if (<= d 2.7e-162)
         (+ (/ a c) (/ 1.0 (/ c (/ b (/ c d)))))
         (if (<= d 4.9e+61) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = ((b * d) / t_0) + (a / (t_0 / c));
	double t_2 = (b / d) + ((c * (a / d)) / d);
	double tmp;
	if (d <= -5e+65) {
		tmp = t_2;
	} else if (d <= -9e-160) {
		tmp = t_1;
	} else if (d <= 2.7e-162) {
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	} else if (d <= 4.9e+61) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(Float64(b * d) / t_0) + Float64(a / Float64(t_0 / c)))
	t_2 = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d))
	tmp = 0.0
	if (d <= -5e+65)
		tmp = t_2;
	elseif (d <= -9e-160)
		tmp = t_1;
	elseif (d <= 2.7e-162)
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(c / Float64(b / Float64(c / d)))));
	elseif (d <= 4.9e+61)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * d), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(a / N[(t$95$0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e+65], t$95$2, If[LessEqual[d, -9e-160], t$95$1, If[LessEqual[d, 2.7e-162], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c / N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.9e+61], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.99999999999999973e65 or 4.90000000000000025e61 < d

   1. Initial program 35.3%

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

   2. Taylor expanded in c around 0 67.8%

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

      1. unpow2 67.8%

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

      2. times-frac 80.8%

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

   4. Simplified 80.8%

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

      1. associate-*l/ 81.9%

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

   6. Applied egg-rr 81.9%

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

   if -4.99999999999999973e65 < d < -9.00000000000000053e-160 or 2.69999999999999984e-162 < d < 4.90000000000000025e61

   1. Initial program 76.1%

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

   2. Taylor expanded in a around 0 76.1%

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

      1. unpow2 76.1%

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

      2. unpow2 76.1%

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

      3. fma-udef 76.1%

         \[\leadsto \frac{d \cdot b}{\color{blue}{\mathsf{fma}\left(d, d, c \cdot c\right)}} + \frac{c \cdot a}{{d}^{2} + {c}^{2}} \]

      4. *-commutative 76.1%

         \[\leadsto \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)} + \frac{\color{blue}{a \cdot c}}{{d}^{2} + {c}^{2}} \]

      5. associate-/l* 79.7%

         \[\leadsto \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)} + \color{blue}{\frac{a}{\frac{{d}^{2} + {c}^{2}}{c}}} \]

      6. unpow2 79.7%

         \[\leadsto \frac{d \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)} + \frac{a}{\frac{\color{blue}{d \cdot d} + {c}^{2}}{c}} \]

      7. unpow2 79.7%

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

      8. fma-udef 79.7%

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

   4. Simplified 79.7%

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

   if -9.00000000000000053e-160 < d < 2.69999999999999984e-162

   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 72.4%

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

      1. unpow2 72.4%

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

      2. associate-/l* 69.2%

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

   4. Simplified 69.2%

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

      1. clear-num 69.1%

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

      2. inv-pow 69.1%

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

   6. Applied egg-rr 69.1%

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

      1. unpow-1 69.1%

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

      2. unpow2 69.1%

         \[\leadsto \frac{a}{c} + \frac{1}{\frac{\frac{\color{blue}{{c}^{2}}}{b}}{d}} \]

      3. associate-/r* 72.4%

         \[\leadsto \frac{a}{c} + \frac{1}{\color{blue}{\frac{{c}^{2}}{b \cdot d}}} \]

      4. unpow2 72.4%

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

      5. times-frac 84.9%

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

   8. Simplified 84.9%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{\frac{c}{b} \cdot \frac{c}{d}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 86.6%

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

      2. associate-/l* 86.8%

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

   10. Applied egg-rr 86.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+65}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} + \frac{a}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{c}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-162}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;\frac{b \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)} + \frac{a}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-138}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 8e-138)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (if (<= c 1.15e+57)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (* (/ 1.0 (hypot c d)) (+ a (/ d (/ c b))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 8e-138) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= 1.15e+57) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (a + (d / (c / b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 8e-138) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= 1.15e+57) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (a + (d / (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 8e-138:
		tmp = (b / d) + ((c * (a / d)) / d)
	elif c <= 1.15e+57:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (1.0 / math.hypot(c, d)) * (a + (d / (c / b)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 8e-138)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	elseif (c <= 1.15e+57)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(a + Float64(d / Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 8e-138)
		tmp = (b / d) + ((c * (a / d)) / d);
	elseif (c <= 1.15e+57)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (1.0 / hypot(c, d)) * (a + (d / (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e-138], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.15e+57], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 8.00000000000000054e-138

   1. Initial program 66.8%

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

   2. Taylor expanded in c around 0 70.9%

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

      1. unpow2 70.9%

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

      2. times-frac 77.8%

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

   4. Simplified 77.8%

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

      1. associate-*l/ 78.7%

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

   6. Applied egg-rr 78.7%

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

   if 8.00000000000000054e-138 < c < 1.1499999999999999e57

   1. Initial program 80.4%

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

   if 1.1499999999999999e57 < c

   1. Initial program 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. hypot-def 37.8%

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

      5. fma-def 37.8%

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

      6. hypot-def 54.9%

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

   3. Applied egg-rr 54.9%

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

   4. Taylor expanded in c around inf 73.9%

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

      1. associate-/l* 82.9%

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

   6. Simplified 82.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(a + \frac{d}{\frac{c}{b}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-138}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \end{array} \]

Alternative 4: 63.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.3 \cdot 10^{+121}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{a \cdot c}{d \cdot d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-145}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* b (/ d (+ (* c c) (* d d))))))
   (if (<= d -2.3e+121)
     (/ b d)
     (if (<= d -9.5e+29)
       (/ (* a c) (* d d))
       (if (<= d -2.1e-81)
         t_0
         (if (<= d 1.25e-145) (/ a c) (if (<= d 1.7e+87) t_0 (/ b d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b * (d / ((c * c) + (d * d)));
	double tmp;
	if (d <= -2.3e+121) {
		tmp = b / d;
	} else if (d <= -9.5e+29) {
		tmp = (a * c) / (d * d);
	} else if (d <= -2.1e-81) {
		tmp = t_0;
	} else if (d <= 1.25e-145) {
		tmp = a / c;
	} else if (d <= 1.7e+87) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = b * (d / ((c * c) + (d * d)))
    if (d <= (-2.3d+121)) then
        tmp = b / d
    else if (d <= (-9.5d+29)) then
        tmp = (a * c) / (d * d)
    else if (d <= (-2.1d-81)) then
        tmp = t_0
    else if (d <= 1.25d-145) then
        tmp = a / c
    else if (d <= 1.7d+87) then
        tmp = t_0
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = b * (d / ((c * c) + (d * d)));
	double tmp;
	if (d <= -2.3e+121) {
		tmp = b / d;
	} else if (d <= -9.5e+29) {
		tmp = (a * c) / (d * d);
	} else if (d <= -2.1e-81) {
		tmp = t_0;
	} else if (d <= 1.25e-145) {
		tmp = a / c;
	} else if (d <= 1.7e+87) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = b * (d / ((c * c) + (d * d)))
	tmp = 0
	if d <= -2.3e+121:
		tmp = b / d
	elif d <= -9.5e+29:
		tmp = (a * c) / (d * d)
	elif d <= -2.1e-81:
		tmp = t_0
	elif d <= 1.25e-145:
		tmp = a / c
	elif d <= 1.7e+87:
		tmp = t_0
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(b * Float64(d / Float64(Float64(c * c) + Float64(d * d))))
	tmp = 0.0
	if (d <= -2.3e+121)
		tmp = Float64(b / d);
	elseif (d <= -9.5e+29)
		tmp = Float64(Float64(a * c) / Float64(d * d));
	elseif (d <= -2.1e-81)
		tmp = t_0;
	elseif (d <= 1.25e-145)
		tmp = Float64(a / c);
	elseif (d <= 1.7e+87)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = b * (d / ((c * c) + (d * d)));
	tmp = 0.0;
	if (d <= -2.3e+121)
		tmp = b / d;
	elseif (d <= -9.5e+29)
		tmp = (a * c) / (d * d);
	elseif (d <= -2.1e-81)
		tmp = t_0;
	elseif (d <= 1.25e-145)
		tmp = a / c;
	elseif (d <= 1.7e+87)
		tmp = t_0;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b * N[(d / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.3e+121], N[(b / d), $MachinePrecision], If[LessEqual[d, -9.5e+29], N[(N[(a * c), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.1e-81], t$95$0, If[LessEqual[d, 1.25e-145], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.7e+87], t$95$0, N[(b / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.2999999999999999e121 or 1.7000000000000001e87 < d

   1. Initial program 30.5%

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

   2. Taylor expanded in c around 0 67.8%

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

   if -2.2999999999999999e121 < d < -9.5000000000000003e29

   1. Initial program 58.5%

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

   2. Taylor expanded in c around 0 58.5%

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

      1. unpow2 58.5%

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

   4. Simplified 58.5%

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

   5. Taylor expanded in a around inf 51.3%

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

      1. unpow2 51.3%

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

   7. Simplified 51.3%

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

   if -9.5000000000000003e29 < d < -2.0999999999999999e-81 or 1.2499999999999999e-145 < d < 1.7000000000000001e87

   1. Initial program 78.9%

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

   2. Taylor expanded in a around 0 56.2%

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

      1. associate-/l* 58.5%

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

      2. associate-/r/ 60.6%

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

      3. unpow2 60.6%

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

      4. unpow2 60.6%

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

      5. fma-udef 60.6%

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

   4. Simplified 60.6%

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

      1. fma-udef 60.6%

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

   6. Applied egg-rr 60.6%

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

   if -2.0999999999999999e-81 < d < 1.2499999999999999e-145

   1. Initial program 67.3%

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

   2. Taylor expanded in c around inf 64.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+121}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{a \cdot c}{d \cdot d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-81}:\\ \;\;\;\;b \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{-145}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;b \cdot \frac{d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-136}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 10^{+57}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.2e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 1.3e-136)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (if (<= c 1e+57)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (+ (/ a c) (/ 1.0 (/ c (/ b (/ c d)))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 1.3e-136) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= 1e+57) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.2d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 1.3d-136) then
        tmp = (b / d) + ((c * (a / d)) / d)
    else if (c <= 1d+57) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = (a / c) + (1.0d0 / (c / (b / (c / d))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 1.3e-136) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else if (c <= 1e+57) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.2e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 1.3e-136:
		tmp = (b / d) + ((c * (a / d)) / d)
	elif c <= 1e+57:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (a / c) + (1.0 / (c / (b / (c / d))))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.2e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 1.3e-136)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	elseif (c <= 1e+57)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(c / Float64(b / Float64(c / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.2e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 1.3e-136)
		tmp = (b / d) + ((c * (a / d)) / d);
	elseif (c <= 1e+57)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.2e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-136], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1e+57], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c / N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.19999999999999991e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.19999999999999991e91 < c < 1.29999999999999998e-136

   1. Initial program 66.8%

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

   2. Taylor expanded in c around 0 70.9%

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

      1. unpow2 70.9%

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

      2. times-frac 77.8%

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

   4. Simplified 77.8%

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

      1. associate-*l/ 78.7%

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

   6. Applied egg-rr 78.7%

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

   if 1.29999999999999998e-136 < c < 1.00000000000000005e57

   1. Initial program 80.4%

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

   if 1.00000000000000005e57 < c

   1. Initial program 37.8%

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

   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. associate-/l* 69.9%

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

   4. Simplified 69.9%

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

      1. clear-num 69.9%

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

      2. inv-pow 69.9%

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

   6. Applied egg-rr 69.9%

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

      1. unpow-1 69.9%

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

      2. unpow2 69.9%

         \[\leadsto \frac{a}{c} + \frac{1}{\frac{\frac{\color{blue}{{c}^{2}}}{b}}{d}} \]

      3. associate-/r* 67.5%

         \[\leadsto \frac{a}{c} + \frac{1}{\color{blue}{\frac{{c}^{2}}{b \cdot d}}} \]

      4. unpow2 67.5%

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

      5. times-frac 82.6%

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

   8. Simplified 82.6%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{\frac{c}{b} \cdot \frac{c}{d}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 79.2%

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

      2. associate-/l* 82.6%

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

   10. Applied egg-rr 82.6%

       \[\leadsto \frac{a}{c} + \frac{1}{\color{blue}{\frac{c}{\frac{b}{\frac{c}{d}}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-136}:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;c \leq 10^{+57}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \end{array} \]

Alternative 6: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 450000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{d} \cdot \frac{c}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 450000000.0)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (+ (/ a c) (/ 1.0 (* (/ c d) (/ c b)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 450000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + (1.0 / ((c / d) * (c / b)));
	}
	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 <= (-1.15d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 450000000.0d0) then
        tmp = (b / d) + ((c * (a / d)) / d)
    else
        tmp = (a / c) + (1.0d0 / ((c / d) * (c / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 450000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + (1.0 / ((c / d) * (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 450000000.0:
		tmp = (b / d) + ((c * (a / d)) / d)
	else:
		tmp = (a / c) + (1.0 / ((c / d) * (c / b)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 450000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	else
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(Float64(c / d) * Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 450000000.0)
		tmp = (b / d) + ((c * (a / d)) / d);
	else
		tmp = (a / c) + (1.0 / ((c / d) * (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 450000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(N[(c / d), $MachinePrecision] * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 4.5e8

   1. Initial program 68.9%

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

   2. Taylor expanded in c around 0 69.2%

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

      1. unpow2 69.2%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 4.5e8 < c

   1. Initial program 43.4%

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

   2. Taylor expanded in c around inf 68.0%

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

      1. unpow2 68.0%

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

      2. associate-/l* 70.1%

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

   4. Simplified 70.1%

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

      1. clear-num 70.2%

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

      2. inv-pow 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. unpow-1 70.2%

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

      2. unpow2 70.2%

         \[\leadsto \frac{a}{c} + \frac{1}{\frac{\frac{\color{blue}{{c}^{2}}}{b}}{d}} \]

      3. associate-/r* 68.0%

         \[\leadsto \frac{a}{c} + \frac{1}{\color{blue}{\frac{{c}^{2}}{b \cdot d}}} \]

      4. unpow2 68.0%

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

      5. times-frac 81.4%

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

   8. Simplified 81.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 450000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{d} \cdot \frac{c}{b}}\\ \end{array} \]

Alternative 7: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 31000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 31000000.0)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (+ (/ a c) (/ 1.0 (/ c (/ b (/ c d))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 31000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.15d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 31000000.0d0) then
        tmp = (b / d) + ((c * (a / d)) / d)
    else
        tmp = (a / c) + (1.0d0 / (c / (b / (c / d))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 31000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 31000000.0:
		tmp = (b / d) + ((c * (a / d)) / d)
	else:
		tmp = (a / c) + (1.0 / (c / (b / (c / d))))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 31000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	else
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(c / Float64(b / Float64(c / d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 31000000.0)
		tmp = (b / d) + ((c * (a / d)) / d);
	else
		tmp = (a / c) + (1.0 / (c / (b / (c / d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 31000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c / N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 3.1e7

   1. Initial program 68.9%

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

   2. Taylor expanded in c around 0 69.2%

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

      1. unpow2 69.2%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 3.1e7 < c

   1. Initial program 43.4%

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

   2. Taylor expanded in c around inf 68.0%

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

      1. unpow2 68.0%

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

      2. associate-/l* 70.1%

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

   4. Simplified 70.1%

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

      1. clear-num 70.2%

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

      2. inv-pow 70.2%

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

   6. Applied egg-rr 70.2%

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

      1. unpow-1 70.2%

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

      2. unpow2 70.2%

         \[\leadsto \frac{a}{c} + \frac{1}{\frac{\frac{\color{blue}{{c}^{2}}}{b}}{d}} \]

      3. associate-/r* 68.0%

         \[\leadsto \frac{a}{c} + \frac{1}{\color{blue}{\frac{{c}^{2}}{b \cdot d}}} \]

      4. unpow2 68.0%

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

      5. times-frac 81.4%

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

   8. Simplified 81.4%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{\frac{c}{b} \cdot \frac{c}{d}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 78.4%

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

      2. associate-/l* 81.5%

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

   10. Applied egg-rr 81.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 31000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{\frac{b}{\frac{c}{d}}}}\\ \end{array} \]

Alternative 8: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+66}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.5e+142)
   (/ b d)
   (if (<= d 3.05e+66) (+ (/ a c) (* (/ b c) (/ d c))) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.5e+142) {
		tmp = b / d;
	} else if (d <= 3.05e+66) {
		tmp = (a / c) + ((b / c) * (d / 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 <= (-6.5d+142)) then
        tmp = b / d
    else if (d <= 3.05d+66) then
        tmp = (a / c) + ((b / c) * (d / 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 <= -6.5e+142) {
		tmp = b / d;
	} else if (d <= 3.05e+66) {
		tmp = (a / c) + ((b / c) * (d / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6.5e+142:
		tmp = b / d
	elif d <= 3.05e+66:
		tmp = (a / c) + ((b / c) * (d / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6.5e+142)
		tmp = Float64(b / d);
	elseif (d <= 3.05e+66)
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6.5e+142)
		tmp = b / d;
	elseif (d <= 3.05e+66)
		tmp = (a / c) + ((b / c) * (d / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6.5e+142], N[(b / d), $MachinePrecision], If[LessEqual[d, 3.05e+66], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.4999999999999997e142 or 3.0500000000000001e66 < d

   1. Initial program 34.6%

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

   2. Taylor expanded in c around 0 69.3%

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

   if -6.4999999999999997e142 < d < 3.0500000000000001e66

   1. Initial program 71.3%

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

   2. Taylor expanded in c around inf 57.8%

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

      1. unpow2 57.8%

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

      2. times-frac 65.3%

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

   4. Simplified 65.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+142}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+66}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 9: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 15000000:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 15000000.0)
     (+ (/ b d) (* (/ a d) (/ c d)))
     (+ (/ a c) (* (/ b c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 15000000.0) {
		tmp = (b / d) + ((a / d) * (c / d));
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.15d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 15000000.0d0) then
        tmp = (b / d) + ((a / d) * (c / d))
    else
        tmp = (a / c) + ((b / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 15000000.0) {
		tmp = (b / d) + ((a / d) * (c / d));
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 15000000.0:
		tmp = (b / d) + ((a / d) * (c / d))
	else:
		tmp = (a / c) + ((b / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 15000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(a / d) * Float64(c / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 15000000.0)
		tmp = (b / d) + ((a / d) * (c / d));
	else
		tmp = (a / c) + ((b / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 15000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(a / d), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 1.5e7

   1. Initial program 68.9%

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

   2. Taylor expanded in c around 0 69.2%

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

      1. unpow2 69.2%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

   if 1.5e7 < c

   1. Initial program 43.4%

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

   2. Taylor expanded in c around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 15000000:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d} \cdot \frac{c}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 10: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 5.2e-51)
     (+ (/ b d) (/ a (* d (/ d c))))
     (+ (/ a c) (* (/ b c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 5.2e-51) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.15d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 5.2d-51) then
        tmp = (b / d) + (a / (d * (d / c)))
    else
        tmp = (a / c) + ((b / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 5.2e-51) {
		tmp = (b / d) + (a / (d * (d / c)));
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 5.2e-51:
		tmp = (b / d) + (a / (d * (d / c)))
	else:
		tmp = (a / c) + ((b / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 5.2e-51)
		tmp = Float64(Float64(b / d) + Float64(a / Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 5.2e-51)
		tmp = (b / d) + (a / (d * (d / c)));
	else
		tmp = (a / c) + ((b / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.2e-51], N[(N[(b / d), $MachinePrecision] + N[(a / N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 5.2e-51

   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 71.4%

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

      1. unpow2 71.4%

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

      2. times-frac 77.7%

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

   4. Simplified 77.7%

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

      1. clear-num 76.6%

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

      2. frac-times 78.5%

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

      3. *-un-lft-identity 78.5%

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

   6. Applied egg-rr 78.5%

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

   if 5.2e-51 < c

   1. Initial program 47.5%

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

   2. Taylor expanded in c around inf 64.9%

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

      1. unpow2 64.9%

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

      2. times-frac 75.8%

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

   4. Simplified 75.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 11: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 175000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (+ (/ a c) (/ d (/ c (/ b c))))
   (if (<= c 175000000.0)
     (+ (/ b d) (/ (* c (/ a d)) d))
     (+ (/ a c) (* (/ b c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 175000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.15d+91)) then
        tmp = (a / c) + (d / (c / (b / c)))
    else if (c <= 175000000.0d0) then
        tmp = (b / d) + ((c * (a / d)) / d)
    else
        tmp = (a / c) + ((b / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = (a / c) + (d / (c / (b / c)));
	} else if (c <= 175000000.0) {
		tmp = (b / d) + ((c * (a / d)) / d);
	} else {
		tmp = (a / c) + ((b / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = (a / c) + (d / (c / (b / c)))
	elif c <= 175000000.0:
		tmp = (b / d) + ((c * (a / d)) / d)
	else:
		tmp = (a / c) + ((b / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c / Float64(b / c))));
	elseif (c <= 175000000.0)
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) / d));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = (a / c) + (d / (c / (b / c)));
	elseif (c <= 175000000.0)
		tmp = (b / d) + ((c * (a / d)) / d);
	else
		tmp = (a / c) + ((b / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 175000000.0], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91

   1. Initial program 35.6%

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

   2. Taylor expanded in c around inf 64.1%

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

      1. unpow2 64.1%

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

      2. associate-/l* 67.4%

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

   4. Simplified 67.4%

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

   5. Taylor expanded in c around 0 67.4%

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

      1. *-rgt-identity 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\frac{\color{blue}{{c}^{2} \cdot 1}}{b}} \]

      2. associate-*r/ 67.4%

         \[\leadsto \frac{a}{c} + \frac{d}{\color{blue}{{c}^{2} \cdot \frac{1}{b}}} \]

      3. unpow2 67.4%

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

      4. associate-*l* 77.1%

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

      5. associate-*r/ 77.0%

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

      6. *-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. clear-num 77.1%

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

      2. un-div-inv 77.1%

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

   9. Applied egg-rr 77.1%

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

   if -1.14999999999999996e91 < c < 1.75e8

   1. Initial program 68.9%

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

   2. Taylor expanded in c around 0 69.2%

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

      1. unpow2 69.2%

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

      2. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*l/ 76.2%

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

   6. Applied egg-rr 76.2%

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

   if 1.75e8 < c

   1. Initial program 43.4%

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

   2. Taylor expanded in c around inf 68.0%

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

      1. unpow2 68.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{\frac{c}{\frac{b}{c}}}\\ \mathbf{elif}\;c \leq 175000000:\\ \;\;\;\;\frac{b}{d} + \frac{c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 12: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e+91)
   (/ a c)
   (if (<= c 1.15e-144)
     (/ b d)
     (if (<= c 7.2e-110)
       (/ a (/ (* d d) c))
       (if (<= c 1.3e-50) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = a / c;
	} else if (c <= 1.15e-144) {
		tmp = b / d;
	} else if (c <= 7.2e-110) {
		tmp = a / ((d * d) / c);
	} else if (c <= 1.3e-50) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.15d+91)) then
        tmp = a / c
    else if (c <= 1.15d-144) then
        tmp = b / d
    else if (c <= 7.2d-110) then
        tmp = a / ((d * d) / c)
    else if (c <= 1.3d-50) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e+91) {
		tmp = a / c;
	} else if (c <= 1.15e-144) {
		tmp = b / d;
	} else if (c <= 7.2e-110) {
		tmp = a / ((d * d) / c);
	} else if (c <= 1.3e-50) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.15e+91:
		tmp = a / c
	elif c <= 1.15e-144:
		tmp = b / d
	elif c <= 7.2e-110:
		tmp = a / ((d * d) / c)
	elif c <= 1.3e-50:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e+91)
		tmp = Float64(a / c);
	elseif (c <= 1.15e-144)
		tmp = Float64(b / d);
	elseif (c <= 7.2e-110)
		tmp = Float64(a / Float64(Float64(d * d) / c));
	elseif (c <= 1.3e-50)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.15e+91)
		tmp = a / c;
	elseif (c <= 1.15e-144)
		tmp = b / d;
	elseif (c <= 7.2e-110)
		tmp = a / ((d * d) / c);
	elseif (c <= 1.3e-50)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e+91], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.15e-144], N[(b / d), $MachinePrecision], If[LessEqual[c, 7.2e-110], N[(a / N[(N[(d * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-50], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.14999999999999996e91 or 1.3000000000000001e-50 < c

   1. Initial program 43.2%

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

   2. Taylor expanded in c around inf 60.1%

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

   if -1.14999999999999996e91 < c < 1.15e-144 or 7.1999999999999999e-110 < c < 1.3000000000000001e-50

   1. Initial program 68.4%

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

   2. Taylor expanded in c around 0 60.1%

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

   if 1.15e-144 < c < 7.1999999999999999e-110

   1. Initial program 99.7%

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

   2. Taylor expanded in a around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/l* 99.7%

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

      3. unpow2 99.7%

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

      4. unpow2 99.7%

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

      5. fma-udef 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in d around inf 99.7%

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

      1. unpow2 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{a}{\frac{d \cdot d}{c}}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 13: 63.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.8e+91) (/ a c) (if (<= c 1.3e-50) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8e+91) {
		tmp = a / c;
	} else if (c <= 1.3e-50) {
		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 <= (-5.8d+91)) then
        tmp = a / c
    else if (c <= 1.3d-50) 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 <= -5.8e+91) {
		tmp = a / c;
	} else if (c <= 1.3e-50) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.8e+91:
		tmp = a / c
	elif c <= 1.3e-50:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.8e+91)
		tmp = Float64(a / c);
	elseif (c <= 1.3e-50)
		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 <= -5.8e+91)
		tmp = a / c;
	elseif (c <= 1.3e-50)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.8e+91], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.3e-50], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.80000000000000028e91 or 1.3000000000000001e-50 < c

   1. Initial program 43.2%

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

   2. Taylor expanded in c around inf 60.1%

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

   if -5.80000000000000028e91 < c < 1.3000000000000001e-50

   1. Initial program 69.5%

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

   2. Taylor expanded in c around 0 58.2%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-50}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 14: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.3%

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

2. Taylor expanded in c around inf 36.0%

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

3. Final simplification 36.0%

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

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 2023238 
(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))))