Complex division, real part

Percentage Accurate: 61.4% → 80.3%

Time: 11.0s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 80.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -4.6 \cdot 10^{+107}:\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{{d}^{2}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -4.6e+107)
     (* (+ b (* c (/ a d))) (/ -1.0 (hypot c d)))
     (if (<= d -1.28e+84)
       (+ (/ a c) (/ (* d (/ b c)) c))
       (if (<= d -1.7e-35)
         t_0
         (if (<= d 9.5e-87)
           (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
           (if (<= d 6.5e+78) t_0 (+ (/ b d) (/ a (/ (pow d 2.0) c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4.6e+107) {
		tmp = (b + (c * (a / d))) * (-1.0 / hypot(c, d));
	} else if (d <= -1.28e+84) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= -1.7e-35) {
		tmp = t_0;
	} else if (d <= 9.5e-87) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 6.5e+78) {
		tmp = t_0;
	} else {
		tmp = (b / d) + (a / (pow(d, 2.0) / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -4.6e+107) {
		tmp = (b + (c * (a / d))) * (-1.0 / Math.hypot(c, d));
	} else if (d <= -1.28e+84) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= -1.7e-35) {
		tmp = t_0;
	} else if (d <= 9.5e-87) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 6.5e+78) {
		tmp = t_0;
	} else {
		tmp = (b / d) + (a / (Math.pow(d, 2.0) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -4.6e+107:
		tmp = (b + (c * (a / d))) * (-1.0 / math.hypot(c, d))
	elif d <= -1.28e+84:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif d <= -1.7e-35:
		tmp = t_0
	elif d <= 9.5e-87:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	elif d <= 6.5e+78:
		tmp = t_0
	else:
		tmp = (b / d) + (a / (math.pow(d, 2.0) / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -4.6e+107)
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -1.28e+84)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (d <= -1.7e-35)
		tmp = t_0;
	elseif (d <= 9.5e-87)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 6.5e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(a / Float64((d ^ 2.0) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -4.6e+107)
		tmp = (b + (c * (a / d))) * (-1.0 / hypot(c, d));
	elseif (d <= -1.28e+84)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (d <= -1.7e-35)
		tmp = t_0;
	elseif (d <= 9.5e-87)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	elseif (d <= 6.5e+78)
		tmp = t_0;
	else
		tmp = (b / d) + (a / ((d ^ 2.0) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.6e+107], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.28e+84], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.7e-35], t$95$0, If[LessEqual[d, 9.5e-87], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.5e+78], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(a / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.6000000000000001e107

   1. Initial program 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. add-sqr-sqrt 37.3%

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

      3. times-frac 37.3%

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

      4. hypot-def 37.3%

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

      5. fma-def 37.3%

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

      6. hypot-def 68.2%

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

   4. Applied egg-rr 68.2%

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

   5. Taylor expanded in d around -inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. unsub-neg 89.9%

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

      3. neg-mul-1 89.9%

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

      4. associate-/l* 92.4%

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

      5. associate-/r/ 92.5%

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

   7. Simplified 92.5%

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

   if -4.6000000000000001e107 < d < -1.28e84

   1. Initial program 23.7%

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

   3. Taylor expanded in c around inf 67.3%

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

      1. *-commutative 67.3%

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

      2. pow2 67.3%

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

      3. times-frac 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*l/ 100.0%

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

   7. Applied egg-rr 100.0%

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

   if -1.28e84 < d < -1.7000000000000001e-35 or 9.5e-87 < d < 6.50000000000000036e78

   1. Initial program 87.5%

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

   if -1.7000000000000001e-35 < d < 9.5e-87

   1. Initial program 68.5%

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

   3. Taylor expanded in c around inf 83.0%

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

      1. *-un-lft-identity 83.0%

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

      2. pow2 83.0%

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

      3. times-frac 93.2%

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

   5. Applied egg-rr 93.2%

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

   if 6.50000000000000036e78 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 71.4%

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

      1. associate-/l* 76.1%

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

   5. Simplified 76.1%

      \[\leadsto \color{blue}{\frac{b}{d} + \frac{a}{\frac{{d}^{2}}{c}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.6 \cdot 10^{+107}:\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.28 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{{d}^{2}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.8% accurate, 0.0× speedup

Math

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

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)))
   (+ (/ a c) (* (/ d c) (/ b 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 = (a / c) + ((d / c) * (b / 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(Float64(a / c) + Float64(Float64(d / c) * Float64(b / 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[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $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 74.9%

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

      1. *-un-lft-identity 74.9%

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

      2. add-sqr-sqrt 74.9%

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

      3. times-frac 74.9%

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

      4. hypot-def 74.9%

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

      5. fma-def 74.9%

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

      6. hypot-def 93.9%

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

   4. Applied egg-rr 93.9%

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

   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. Add Preprocessing

   3. Taylor expanded in c around inf 43.7%

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

      1. *-commutative 43.7%

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

      2. pow2 43.7%

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

      3. times-frac 56.2%

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

   5. Applied egg-rr 56.2%

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

4. Final simplification 87.7%

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

Alternative 3: 78.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 60000000:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ d (hypot c d)) (/ b (hypot c d)))))
   (if (<= d -1.2e+84)
     t_0
     (if (<= d -2.1e-35)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= d 60000000.0) (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (d / hypot(c, d)) * (b / hypot(c, d));
	double tmp;
	if (d <= -1.2e+84) {
		tmp = t_0;
	} else if (d <= -2.1e-35) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 60000000.0) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	double tmp;
	if (d <= -1.2e+84) {
		tmp = t_0;
	} else if (d <= -2.1e-35) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 60000000.0) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	tmp = 0
	if d <= -1.2e+84:
		tmp = t_0
	elif d <= -2.1e-35:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 60000000.0:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)))
	tmp = 0.0
	if (d <= -1.2e+84)
		tmp = t_0;
	elseif (d <= -2.1e-35)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 60000000.0)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (d / hypot(c, d)) * (b / hypot(c, d));
	tmp = 0.0;
	if (d <= -1.2e+84)
		tmp = t_0;
	elseif (d <= -2.1e-35)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 60000000.0)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.2e+84], t$95$0, If[LessEqual[d, -2.1e-35], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 60000000.0], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2e84 or 6e7 < d

   1. Initial program 46.9%

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

   3. Taylor expanded in a around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. fma-def 40.5%

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

      3. add-sqr-sqrt 40.5%

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

      4. fma-def 40.5%

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

      5. hypot-udef 40.5%

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

      6. fma-def 40.5%

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

      7. hypot-udef 40.5%

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

      8. times-frac 82.5%

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

   5. Applied egg-rr 82.5%

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

   if -1.2e84 < d < -2.1e-35

   1. Initial program 88.4%

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

   if -2.1e-35 < d < 6e7

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. pow2 81.8%

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

      3. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+84}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 60000000:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{{d}^{2}}{c}}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{+43}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (* d (/ b c)) c))))
   (if (<= c -3.4e-51)
     t_0
     (if (<= c 8e-121)
       (+ (/ b d) (/ a (/ (pow d 2.0) c)))
       (if (<= c 2.75e+43) (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + ((d * (b / c)) / c);
	double tmp;
	if (c <= -3.4e-51) {
		tmp = t_0;
	} else if (c <= 8e-121) {
		tmp = (b / d) + (a / (pow(d, 2.0) / c));
	} else if (c <= 2.75e+43) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (a / c) + ((d * (b / c)) / c)
    if (c <= (-3.4d-51)) then
        tmp = t_0
    else if (c <= 8d-121) then
        tmp = (b / d) + (a / ((d ** 2.0d0) / c))
    else if (c <= 2.75d+43) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + ((d * (b / c)) / c);
	double tmp;
	if (c <= -3.4e-51) {
		tmp = t_0;
	} else if (c <= 8e-121) {
		tmp = (b / d) + (a / (Math.pow(d, 2.0) / c));
	} else if (c <= 2.75e+43) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + ((d * (b / c)) / c)
	tmp = 0
	if c <= -3.4e-51:
		tmp = t_0
	elif c <= 8e-121:
		tmp = (b / d) + (a / (math.pow(d, 2.0) / c))
	elif c <= 2.75e+43:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c))
	tmp = 0.0
	if (c <= -3.4e-51)
		tmp = t_0;
	elseif (c <= 8e-121)
		tmp = Float64(Float64(b / d) + Float64(a / Float64((d ^ 2.0) / c)));
	elseif (c <= 2.75e+43)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + ((d * (b / c)) / c);
	tmp = 0.0;
	if (c <= -3.4e-51)
		tmp = t_0;
	elseif (c <= 8e-121)
		tmp = (b / d) + (a / ((d ^ 2.0) / c));
	elseif (c <= 2.75e+43)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.4e-51], t$95$0, If[LessEqual[c, 8e-121], N[(N[(b / d), $MachinePrecision] + N[(a / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.75e+43], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.40000000000000003e-51 or 2.74999999999999995e43 < c

   1. Initial program 51.8%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. *-commutative 77.7%

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

      2. pow2 77.7%

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

      3. times-frac 83.6%

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

   5. Applied egg-rr 83.6%

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

      1. associate-*l/ 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -3.40000000000000003e-51 < c < 7.9999999999999998e-121

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0 82.1%

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

      1. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   if 7.9999999999999998e-121 < c < 2.74999999999999995e43

   1. Initial program 90.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{d} + \frac{a}{\frac{{d}^{2}}{c}}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{+43}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 29000000:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.4e+108)
   (/ b d)
   (if (<= d -9.2e+58)
     (+ (/ a c) (/ (* d (/ b c)) c))
     (if (<= d -7.2e-35)
       (/ (* b d) (+ (* c c) (* d d)))
       (if (<= d 29000000.0)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.4e+108) {
		tmp = b / d;
	} else if (d <= -9.2e+58) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= -7.2e-35) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 29000000.0) {
		tmp = (a / c) + ((1.0 / c) * ((b * 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 <= (-2.4d+108)) then
        tmp = b / d
    else if (d <= (-9.2d+58)) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else if (d <= (-7.2d-35)) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 29000000.0d0) then
        tmp = (a / c) + ((1.0d0 / c) * ((b * 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 <= -2.4e+108) {
		tmp = b / d;
	} else if (d <= -9.2e+58) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else if (d <= -7.2e-35) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 29000000.0) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.4e+108:
		tmp = b / d
	elif d <= -9.2e+58:
		tmp = (a / c) + ((d * (b / c)) / c)
	elif d <= -7.2e-35:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 29000000.0:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.4e+108)
		tmp = Float64(b / d);
	elseif (d <= -9.2e+58)
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	elseif (d <= -7.2e-35)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 29000000.0)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * 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 <= -2.4e+108)
		tmp = b / d;
	elseif (d <= -9.2e+58)
		tmp = (a / c) + ((d * (b / c)) / c);
	elseif (d <= -7.2e-35)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 29000000.0)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.4e+108], N[(b / d), $MachinePrecision], If[LessEqual[d, -9.2e+58], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -7.2e-35], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 29000000.0], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.40000000000000019e108 or 2.9e7 < d

   1. Initial program 48.4%

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

   3. Taylor expanded in c around 0 73.5%

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

   if -2.40000000000000019e108 < d < -9.2000000000000001e58

   1. Initial program 49.1%

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

   3. Taylor expanded in c around inf 56.0%

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

      1. *-commutative 56.0%

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

      2. pow2 56.0%

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

      3. times-frac 73.5%

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

   5. Applied egg-rr 73.5%

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

      1. associate-*l/ 73.8%

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

   7. Applied egg-rr 73.8%

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

   if -9.2000000000000001e58 < d < -7.20000000000000038e-35

   1. Initial program 90.4%

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

   3. Taylor expanded in a around 0 61.5%

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

   if -7.20000000000000038e-35 < d < 2.9e7

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 81.8%

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

      1. *-un-lft-identity 81.8%

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

      2. pow2 81.8%

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

      3. times-frac 90.9%

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

   5. Applied egg-rr 90.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 29000000:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2600:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (* d (/ b c)) c))))
   (if (<= d -1.65e+108)
     (/ b d)
     (if (<= d -9e+58)
       t_0
       (if (<= d -2.5e-34)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 2600.0) t_0 (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + ((d * (b / c)) / c);
	double tmp;
	if (d <= -1.65e+108) {
		tmp = b / d;
	} else if (d <= -9e+58) {
		tmp = t_0;
	} else if (d <= -2.5e-34) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2600.0) {
		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 = (a / c) + ((d * (b / c)) / c)
    if (d <= (-1.65d+108)) then
        tmp = b / d
    else if (d <= (-9d+58)) then
        tmp = t_0
    else if (d <= (-2.5d-34)) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 2600.0d0) 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 = (a / c) + ((d * (b / c)) / c);
	double tmp;
	if (d <= -1.65e+108) {
		tmp = b / d;
	} else if (d <= -9e+58) {
		tmp = t_0;
	} else if (d <= -2.5e-34) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2600.0) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + ((d * (b / c)) / c)
	tmp = 0
	if d <= -1.65e+108:
		tmp = b / d
	elif d <= -9e+58:
		tmp = t_0
	elif d <= -2.5e-34:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 2600.0:
		tmp = t_0
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c))
	tmp = 0.0
	if (d <= -1.65e+108)
		tmp = Float64(b / d);
	elseif (d <= -9e+58)
		tmp = t_0;
	elseif (d <= -2.5e-34)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2600.0)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + ((d * (b / c)) / c);
	tmp = 0.0;
	if (d <= -1.65e+108)
		tmp = b / d;
	elseif (d <= -9e+58)
		tmp = t_0;
	elseif (d <= -2.5e-34)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 2600.0)
		tmp = t_0;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.65e+108], N[(b / d), $MachinePrecision], If[LessEqual[d, -9e+58], t$95$0, If[LessEqual[d, -2.5e-34], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2600.0], t$95$0, N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.6500000000000001e108 or 2600 < d

   1. Initial program 48.4%

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

   3. Taylor expanded in c around 0 73.5%

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

   if -1.6500000000000001e108 < d < -8.9999999999999996e58 or -2.5000000000000001e-34 < d < 2600

   1. Initial program 68.2%

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

   3. Taylor expanded in c around inf 79.8%

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

      1. *-commutative 79.8%

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

      2. pow2 79.8%

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

      3. times-frac 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. associate-*l/ 89.0%

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

   7. Applied egg-rr 89.0%

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

   if -8.9999999999999996e58 < d < -2.5000000000000001e-34

   1. Initial program 90.4%

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

   3. Taylor expanded in a around 0 61.5%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2600:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+118} \lor \neg \left(c \leq 6.4 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.65e+118) (not (<= c 6.4e+42)))
   (+ (/ a c) (/ (* d (/ b c)) c))
   (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.65e+118) || !(c <= 6.4e+42)) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.65d+118)) .or. (.not. (c <= 6.4d+42))) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.65e+118) || !(c <= 6.4e+42)) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.65e+118) or not (c <= 6.4e+42):
		tmp = (a / c) + ((d * (b / c)) / c)
	else:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.65e+118) || !(c <= 6.4e+42))
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	else
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.65e+118) || ~((c <= 6.4e+42)))
		tmp = (a / c) + ((d * (b / c)) / c);
	else
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.65e+118], N[Not[LessEqual[c, 6.4e+42]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.65e118 or 6.40000000000000004e42 < c

   1. Initial program 44.3%

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

   3. Taylor expanded in c around inf 80.6%

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

      1. *-commutative 80.6%

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

      2. pow2 80.6%

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

      3. times-frac 88.3%

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

   5. Applied egg-rr 88.3%

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

      1. associate-*l/ 89.2%

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

   7. Applied egg-rr 89.2%

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

   if -1.65e118 < c < 6.40000000000000004e42

   1. Initial program 75.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+118} \lor \neg \left(c \leq 6.4 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.3 \cdot 10^{+107} \lor \neg \left(d \leq 850000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.3e+107) (not (<= d 850000000000.0)))
   (/ b d)
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.3e+107) || !(d <= 850000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.3d+107)) .or. (.not. (d <= 850000000000.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.3e+107) || !(d <= 850000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.3e+107) or not (d <= 850000000000.0):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.3e+107) || !(d <= 850000000000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.3e+107) || ~((d <= 850000000000.0)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.3e+107], N[Not[LessEqual[d, 850000000000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.3e107 or 8.5e11 < d

   1. Initial program 48.4%

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

   3. Taylor expanded in c around 0 73.5%

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

   if -5.3e107 < d < 8.5e11

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

      2. pow2 74.6%

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

      3. times-frac 81.9%

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

   5. Applied egg-rr 81.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.3 \cdot 10^{+107} \lor \neg \left(d \leq 850000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+109} \lor \neg \left(d \leq 12500000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.35e+109) (not (<= d 12500000.0)))
   (/ b d)
   (+ (/ a c) (/ (* d (/ b c)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.35e+109) || !(d <= 12500000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.35d+109)) .or. (.not. (d <= 12500000.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.35e+109) || !(d <= 12500000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.35e+109) or not (d <= 12500000.0):
		tmp = b / d
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.35e+109) || !(d <= 12500000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.35e+109) || ~((d <= 12500000.0)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.35e+109], N[Not[LessEqual[d, 12500000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.35000000000000001e109 or 1.25e7 < d

   1. Initial program 48.4%

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

   3. Taylor expanded in c around 0 73.5%

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

   if -1.35000000000000001e109 < d < 1.25e7

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

      2. pow2 74.6%

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

      3. times-frac 81.9%

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

   5. Applied egg-rr 81.9%

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

      1. associate-*l/ 82.6%

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

   7. Applied egg-rr 82.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+109} \lor \neg \left(d \leq 12500000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-35} \lor \neg \left(d \leq 12000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.2e-35) (not (<= d 12000000.0))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e-35) || !(d <= 12000000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-9.2d-35)) .or. (.not. (d <= 12000000.0d0))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e-35) || !(d <= 12000000.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9.2e-35) or not (d <= 12000000.0):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9.2e-35) || !(d <= 12000000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -9.2e-35) || ~((d <= 12000000.0)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9.2e-35], N[Not[LessEqual[d, 12000000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.1999999999999996e-35 or 1.2e7 < d

   1. Initial program 55.3%

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

   3. Taylor expanded in c around 0 64.0%

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

   if -9.1999999999999996e-35 < d < 1.2e7

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 79.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{-35} \lor \neg \left(d \leq 12000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.1% 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 62.6%

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

3. Taylor expanded in c around inf 49.7%

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

4. Final simplification 49.7%

   \[\leadsto \frac{a}{c} \]
5. Add Preprocessing

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 2024096 
(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))))