Complex division, real part

Percentage Accurate: 62.0% → 83.0%

Time: 11.8s

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: 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: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ t_1 := \frac{b}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -2.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-a\right) - t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{t\_0}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))) (t_1 (/ b (/ c d))))
   (if (<= c -2.06e+102)
     (/ (- (- a) t_1) (hypot c d))
     (if (<= c -2.3e-164)
       (/ t_0 (+ (* c c) (* d d)))
       (if (<= c 1.05e-105)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 6.6e+78)
           (/ t_0 (pow (hypot c d) 2.0))
           (/ (+ a t_1) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = b / (c / d);
	double tmp;
	if (c <= -2.06e+102) {
		tmp = (-a - t_1) / hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1.05e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 6.6e+78) {
		tmp = t_0 / pow(hypot(c, d), 2.0);
	} else {
		tmp = (a + t_1) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = b / (c / d);
	double tmp;
	if (c <= -2.06e+102) {
		tmp = (-a - t_1) / Math.hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1.05e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 6.6e+78) {
		tmp = t_0 / Math.pow(Math.hypot(c, d), 2.0);
	} else {
		tmp = (a + t_1) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	t_1 = b / (c / d)
	tmp = 0
	if c <= -2.06e+102:
		tmp = (-a - t_1) / math.hypot(c, d)
	elif c <= -2.3e-164:
		tmp = t_0 / ((c * c) + (d * d))
	elif c <= 1.05e-105:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 6.6e+78:
		tmp = t_0 / math.pow(math.hypot(c, d), 2.0)
	else:
		tmp = (a + t_1) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	t_1 = Float64(b / Float64(c / d))
	tmp = 0.0
	if (c <= -2.06e+102)
		tmp = Float64(Float64(Float64(-a) - t_1) / hypot(c, d));
	elseif (c <= -2.3e-164)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.05e-105)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 6.6e+78)
		tmp = Float64(t_0 / (hypot(c, d) ^ 2.0));
	else
		tmp = Float64(Float64(a + t_1) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	t_1 = b / (c / d);
	tmp = 0.0;
	if (c <= -2.06e+102)
		tmp = (-a - t_1) / hypot(c, d);
	elseif (c <= -2.3e-164)
		tmp = t_0 / ((c * c) + (d * d));
	elseif (c <= 1.05e-105)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 6.6e+78)
		tmp = t_0 / (hypot(c, d) ^ 2.0);
	else
		tmp = (a + t_1) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.06e+102], N[(N[((-a) - t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.3e-164], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.05e-105], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.6e+78], N[(t$95$0 / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a + t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.05999999999999999e102

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. add-sqr-sqrt 33.3%

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

      3. times-frac 33.2%

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

      4. hypot-def 33.2%

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

      5. fma-def 33.2%

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

      6. hypot-def 53.4%

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

   4. Applied egg-rr 53.4%

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

      1. fma-def 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*l/ 53.3%

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

      4. div-inv 53.5%

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

      5. +-commutative 53.5%

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

      6. fma-def 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in c around -inf 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unsub-neg 67.2%

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

      3. neg-mul-1 67.2%

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

      4. associate-/l* 85.1%

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

   9. Simplified 85.1%

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

   if -2.05999999999999999e102 < c < -2.29999999999999985e-164

   1. Initial program 83.0%

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

   if -2.29999999999999985e-164 < c < 1.05e-105

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

   if 1.05e-105 < c < 6.6e78

   1. Initial program 81.5%

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

      1. expm1-log1p-u 78.9%

         \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(c \cdot c + d \cdot d\right)\right)}} \]

      2. expm1-udef 50.8%

         \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{e^{\mathsf{log1p}\left(c \cdot c + d \cdot d\right)} - 1}} \]

      3. add-sqr-sqrt 50.8%

         \[\leadsto \frac{a \cdot c + b \cdot d}{e^{\mathsf{log1p}\left(\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\right)} - 1} \]

      4. pow2 50.8%

         \[\leadsto \frac{a \cdot c + b \cdot d}{e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt{c \cdot c + d \cdot d}\right)}^{2}}\right)} - 1} \]

      5. hypot-def 50.8%

         \[\leadsto \frac{a \cdot c + b \cdot d}{e^{\mathsf{log1p}\left({\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}\right)} - 1} \]

   4. Applied egg-rr 50.8%

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}\right)} - 1}} \]
   5. Step-by-step derivation

      1. expm1-def 78.9%

         \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}\right)\right)}} \]

      2. expm1-log1p 81.5%

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

   6. Simplified 81.5%

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

   if 6.6e78 < c

   1. Initial program 41.2%

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

      1. *-un-lft-identity 41.2%

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

      2. add-sqr-sqrt 41.2%

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

      3. times-frac 41.2%

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

      4. hypot-def 41.2%

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

      5. fma-def 41.2%

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

      6. hypot-def 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. fma-def 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*l/ 61.0%

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

      4. div-inv 61.1%

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

      5. +-commutative 61.1%

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

      6. fma-def 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in d around 0 82.9%

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

      1. associate-/l* 88.8%

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

   9. Simplified 88.8%

      \[\leadsto \frac{\color{blue}{a + \frac{b}{\frac{c}{d}}}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-a\right) - \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.3% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(b * d + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $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 77.9%

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

      1. *-un-lft-identity 77.9%

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

      2. add-sqr-sqrt 77.9%

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

      3. times-frac 77.8%

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

      4. hypot-def 77.8%

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

      5. fma-def 77.8%

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

      6. hypot-def 93.8%

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

   4. Applied egg-rr 93.8%

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

      1. fma-def 93.8%

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

      2. *-commutative 93.8%

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

      3. associate-*l/ 93.8%

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

      4. div-inv 93.9%

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

      5. +-commutative 93.9%

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

      6. fma-def 93.9%

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

   6. Applied egg-rr 93.9%

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

   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 a around inf 1.5%

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

      1. *-commutative 1.5%

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

   5. Simplified 1.5%

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

      1. add-sqr-sqrt 1.5%

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

      2. hypot-udef 1.5%

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

      3. hypot-udef 1.5%

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

      4. times-frac 54.5%

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

   7. Applied egg-rr 54.5%

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

4. Final simplification 85.8%

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

Alternative 3: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{if}\;c \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ a c) (* d (/ b (pow c 2.0))))))
   (if (<= c -3.7e+59)
     t_1
     (if (<= c -2.3e-164)
       t_0
       (if (<= c 3.3e-106)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 1.45e+78) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (a / c) + (d * (b / pow(c, 2.0)));
	double tmp;
	if (c <= -3.7e+59) {
		tmp = t_1;
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3.3e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.45e+78) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (a / c) + (d * (b / (c ** 2.0d0)))
    if (c <= (-3.7d+59)) then
        tmp = t_1
    else if (c <= (-2.3d-164)) then
        tmp = t_0
    else if (c <= 3.3d-106) then
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    else if (c <= 1.45d+78) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

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 t_1 = (a / c) + (d * (b / Math.pow(c, 2.0)));
	double tmp;
	if (c <= -3.7e+59) {
		tmp = t_1;
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3.3e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.45e+78) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (a / c) + (d * (b / math.pow(c, 2.0)))
	tmp = 0
	if c <= -3.7e+59:
		tmp = t_1
	elif c <= -2.3e-164:
		tmp = t_0
	elif c <= 3.3e-106:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 1.45e+78:
		tmp = t_0
	else:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(a / c) + Float64(d * Float64(b / (c ^ 2.0))))
	tmp = 0.0
	if (c <= -3.7e+59)
		tmp = t_1;
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3.3e-106)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 1.45e+78)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (a / c) + (d * (b / (c ^ 2.0)));
	tmp = 0.0;
	if (c <= -3.7e+59)
		tmp = t_1;
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3.3e-106)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 1.45e+78)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.7e+59], t$95$1, If[LessEqual[c, -2.3e-164], t$95$0, If[LessEqual[c, 3.3e-106], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.45e+78], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.69999999999999997e59 or 1.45000000000000008e78 < c

   1. Initial program 41.7%

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

   3. Taylor expanded in c around inf 71.6%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 77.6%

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

   5. Simplified 77.6%

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

   if -3.69999999999999997e59 < c < -2.29999999999999985e-164 or 3.30000000000000016e-106 < c < 1.45000000000000008e78

   1. Initial program 83.6%

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

   if -2.29999999999999985e-164 < c < 3.30000000000000016e-106

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.05e+60)
     (+ (/ a c) (* d (/ b (pow c 2.0))))
     (if (<= c -2.3e-164)
       t_0
       (if (<= c 1.45e-105)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 6.2e+78) t_0 (/ (+ a (/ b (/ c d))) (hypot c d))))))))

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 (c <= -1.05e+60) {
		tmp = (a / c) + (d * (b / pow(c, 2.0)));
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 1.45e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 6.2e+78) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	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 (c <= -1.05e+60) {
		tmp = (a / c) + (d * (b / Math.pow(c, 2.0)));
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 1.45e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 6.2e+78) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.05e+60:
		tmp = (a / c) + (d * (b / math.pow(c, 2.0)))
	elif c <= -2.3e-164:
		tmp = t_0
	elif c <= 1.45e-105:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 6.2e+78:
		tmp = t_0
	else:
		tmp = (a + (b / (c / d))) / math.hypot(c, d)
	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 (c <= -1.05e+60)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(b / (c ^ 2.0))));
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 1.45e-105)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 6.2e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	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 (c <= -1.05e+60)
		tmp = (a / c) + (d * (b / (c ^ 2.0)));
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 1.45e-105)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 6.2e+78)
		tmp = t_0;
	else
		tmp = (a + (b / (c / d))) / hypot(c, d);
	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[c, -1.05e+60], N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.3e-164], t$95$0, If[LessEqual[c, 1.45e-105], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.2e+78], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.0500000000000001e60

   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 inf 65.2%

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

      1. associate-/l* 69.4%

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

      2. associate-/r/ 72.7%

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

   5. Simplified 72.7%

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

   if -1.0500000000000001e60 < c < -2.29999999999999985e-164 or 1.45000000000000002e-105 < c < 6.2e78

   1. Initial program 83.6%

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

   if -2.29999999999999985e-164 < c < 1.45000000000000002e-105

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

   if 6.2e78 < c

   1. Initial program 41.2%

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

      1. *-un-lft-identity 41.2%

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

      2. add-sqr-sqrt 41.2%

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

      3. times-frac 41.2%

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

      4. hypot-def 41.2%

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

      5. fma-def 41.2%

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

      6. hypot-def 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. fma-def 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*l/ 61.0%

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

      4. div-inv 61.1%

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

      5. +-commutative 61.1%

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

      6. fma-def 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in d around 0 82.9%

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

      1. associate-/l* 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+60}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-a\right) - t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))) (t_1 (/ b (/ c d))))
   (if (<= c -2.2e+102)
     (/ (- (- a) t_1) (hypot c d))
     (if (<= c -2.3e-164)
       t_0
       (if (<= c 3.9e-106)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 4.2e+78) t_0 (/ (+ a t_1) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = b / (c / d);
	double tmp;
	if (c <= -2.2e+102) {
		tmp = (-a - t_1) / hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3.9e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 4.2e+78) {
		tmp = t_0;
	} else {
		tmp = (a + t_1) / hypot(c, d);
	}
	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 t_1 = b / (c / d);
	double tmp;
	if (c <= -2.2e+102) {
		tmp = (-a - t_1) / Math.hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3.9e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 4.2e+78) {
		tmp = t_0;
	} else {
		tmp = (a + t_1) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = b / (c / d)
	tmp = 0
	if c <= -2.2e+102:
		tmp = (-a - t_1) / math.hypot(c, d)
	elif c <= -2.3e-164:
		tmp = t_0
	elif c <= 3.9e-106:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 4.2e+78:
		tmp = t_0
	else:
		tmp = (a + t_1) / math.hypot(c, d)
	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)))
	t_1 = Float64(b / Float64(c / d))
	tmp = 0.0
	if (c <= -2.2e+102)
		tmp = Float64(Float64(Float64(-a) - t_1) / hypot(c, d));
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3.9e-106)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 4.2e+78)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + t_1) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = b / (c / d);
	tmp = 0.0;
	if (c <= -2.2e+102)
		tmp = (-a - t_1) / hypot(c, d);
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3.9e-106)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 4.2e+78)
		tmp = t_0;
	else
		tmp = (a + t_1) / hypot(c, d);
	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]}, Block[{t$95$1 = N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.2e+102], N[(N[((-a) - t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.3e-164], t$95$0, If[LessEqual[c, 3.9e-106], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.2e+78], t$95$0, N[(N[(a + t$95$1), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.20000000000000007e102

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. add-sqr-sqrt 33.3%

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

      3. times-frac 33.2%

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

      4. hypot-def 33.2%

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

      5. fma-def 33.2%

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

      6. hypot-def 53.4%

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

   4. Applied egg-rr 53.4%

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

      1. fma-def 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*l/ 53.3%

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

      4. div-inv 53.5%

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

      5. +-commutative 53.5%

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

      6. fma-def 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in c around -inf 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unsub-neg 67.2%

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

      3. neg-mul-1 67.2%

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

      4. associate-/l* 85.1%

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

   9. Simplified 85.1%

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

   if -2.20000000000000007e102 < c < -2.29999999999999985e-164 or 3.9000000000000001e-106 < c < 4.2000000000000002e78

   1. Initial program 82.5%

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

   if -2.29999999999999985e-164 < c < 3.9000000000000001e-106

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

   if 4.2000000000000002e78 < c

   1. Initial program 41.2%

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

      1. *-un-lft-identity 41.2%

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

      2. add-sqr-sqrt 41.2%

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

      3. times-frac 41.2%

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

      4. hypot-def 41.2%

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

      5. fma-def 41.2%

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

      6. hypot-def 61.1%

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

   4. Applied egg-rr 61.1%

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

      1. fma-def 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*l/ 61.0%

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

      4. div-inv 61.1%

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

      5. +-commutative 61.1%

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

      6. fma-def 61.2%

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

   6. Applied egg-rr 61.2%

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

   7. Taylor expanded in d around 0 82.9%

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

      1. associate-/l* 88.8%

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

   9. Simplified 88.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{\left(-a\right) - \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.15e+102)
     (/ (- a) (hypot c d))
     (if (<= c -2.3e-164)
       t_0
       (if (<= c 1.75e-105)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 1.05e+153) t_0 (/ a 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 (c <= -2.15e+102) {
		tmp = -a / hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 1.75e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.05e+153) {
		tmp = t_0;
	} else {
		tmp = a / 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 (c <= -2.15e+102) {
		tmp = -a / Math.hypot(c, d);
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 1.75e-105) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.05e+153) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.15e+102:
		tmp = -a / math.hypot(c, d)
	elif c <= -2.3e-164:
		tmp = t_0
	elif c <= 1.75e-105:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 1.05e+153:
		tmp = t_0
	else:
		tmp = a / 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 (c <= -2.15e+102)
		tmp = Float64(Float64(-a) / hypot(c, d));
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 1.75e-105)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 1.05e+153)
		tmp = t_0;
	else
		tmp = Float64(a / 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 (c <= -2.15e+102)
		tmp = -a / hypot(c, d);
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 1.75e-105)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 1.05e+153)
		tmp = t_0;
	else
		tmp = a / 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[c, -2.15e+102], N[((-a) / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.3e-164], t$95$0, If[LessEqual[c, 1.75e-105], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.05e+153], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.15e102

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. add-sqr-sqrt 33.3%

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

      3. times-frac 33.2%

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

      4. hypot-def 33.2%

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

      5. fma-def 33.2%

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

      6. hypot-def 53.4%

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

   4. Applied egg-rr 53.4%

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

      1. fma-def 53.4%

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

      2. *-commutative 53.4%

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

      3. associate-*l/ 53.3%

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

      4. div-inv 53.5%

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

      5. +-commutative 53.5%

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

      6. fma-def 53.5%

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

   6. Applied egg-rr 53.5%

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

   7. Taylor expanded in c around -inf 69.7%

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

      1. neg-mul-1 69.7%

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

   9. Simplified 69.7%

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

   if -2.15e102 < c < -2.29999999999999985e-164 or 1.75e-105 < c < 1.05000000000000008e153

   1. Initial program 81.2%

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

   if -2.29999999999999985e-164 < c < 1.75e-105

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

   if 1.05000000000000008e153 < c

   1. Initial program 29.2%

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

   3. Taylor expanded in c around inf 81.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{+102}:\\ \;\;\;\;\frac{-a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -2.1e+102)
     (/ a c)
     (if (<= c -2.3e-164)
       t_0
       (if (<= c 3e-106)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 9.2e+151) t_0 (/ a 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 (c <= -2.1e+102) {
		tmp = a / c;
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 9.2e+151) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-2.1d+102)) then
        tmp = a / c
    else if (c <= (-2.3d-164)) then
        tmp = t_0
    else if (c <= 3d-106) then
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    else if (c <= 9.2d+151) then
        tmp = t_0
    else
        tmp = a / c
    end if
    code = tmp
end function

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 (c <= -2.1e+102) {
		tmp = a / c;
	} else if (c <= -2.3e-164) {
		tmp = t_0;
	} else if (c <= 3e-106) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 9.2e+151) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.1e+102:
		tmp = a / c
	elif c <= -2.3e-164:
		tmp = t_0
	elif c <= 3e-106:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 9.2e+151:
		tmp = t_0
	else:
		tmp = a / 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 (c <= -2.1e+102)
		tmp = Float64(a / c);
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3e-106)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 9.2e+151)
		tmp = t_0;
	else
		tmp = Float64(a / 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 (c <= -2.1e+102)
		tmp = a / c;
	elseif (c <= -2.3e-164)
		tmp = t_0;
	elseif (c <= 3e-106)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 9.2e+151)
		tmp = t_0;
	else
		tmp = a / 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[c, -2.1e+102], N[(a / c), $MachinePrecision], If[LessEqual[c, -2.3e-164], t$95$0, If[LessEqual[c, 3e-106], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.2e+151], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.10000000000000001e102 or 9.2000000000000003e151 < c

   1. Initial program 31.4%

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

   3. Taylor expanded in c around inf 75.1%

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

   if -2.10000000000000001e102 < c < -2.29999999999999985e-164 or 3.00000000000000019e-106 < c < 9.2000000000000003e151

   1. Initial program 81.2%

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

   if -2.29999999999999985e-164 < c < 3.00000000000000019e-106

   1. Initial program 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 65.9%

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

      4. hypot-def 66.0%

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

      5. fma-def 65.9%

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

      6. hypot-def 82.6%

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

   4. Applied egg-rr 82.6%

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

   5. Taylor expanded in c around 0 50.3%

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

      1. associate-/l* 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in c around 0 92.1%

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

      1. clear-num 92.0%

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

      2. associate-/r/ 92.1%

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

      3. clear-num 92.1%

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

   10. Applied egg-rr 92.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+151}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{if}\;c \leq -1.35 \cdot 10^{-27}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{a \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* a (/ c d))))))
   (if (<= c -1.35e-27)
     (/ a c)
     (if (<= c 6e-101)
       t_0
       (if (<= c 8.6e+42)
         (/ (* a c) (+ (* c c) (* d d)))
         (if (<= c 2.1e+77) t_0 (/ a c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double tmp;
	if (c <= -1.35e-27) {
		tmp = a / c;
	} else if (c <= 6e-101) {
		tmp = t_0;
	} else if (c <= 8.6e+42) {
		tmp = (a * c) / ((c * c) + (d * d));
	} else if (c <= 2.1e+77) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / d) * (b + (a * (c / d)))
    if (c <= (-1.35d-27)) then
        tmp = a / c
    else if (c <= 6d-101) then
        tmp = t_0
    else if (c <= 8.6d+42) then
        tmp = (a * c) / ((c * c) + (d * d))
    else if (c <= 2.1d+77) then
        tmp = t_0
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double tmp;
	if (c <= -1.35e-27) {
		tmp = a / c;
	} else if (c <= 6e-101) {
		tmp = t_0;
	} else if (c <= 8.6e+42) {
		tmp = (a * c) / ((c * c) + (d * d));
	} else if (c <= 2.1e+77) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a * (c / d)))
	tmp = 0
	if c <= -1.35e-27:
		tmp = a / c
	elif c <= 6e-101:
		tmp = t_0
	elif c <= 8.6e+42:
		tmp = (a * c) / ((c * c) + (d * d))
	elif c <= 2.1e+77:
		tmp = t_0
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))))
	tmp = 0.0
	if (c <= -1.35e-27)
		tmp = Float64(a / c);
	elseif (c <= 6e-101)
		tmp = t_0;
	elseif (c <= 8.6e+42)
		tmp = Float64(Float64(a * c) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.1e+77)
		tmp = t_0;
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a * (c / d)));
	tmp = 0.0;
	if (c <= -1.35e-27)
		tmp = a / c;
	elseif (c <= 6e-101)
		tmp = t_0;
	elseif (c <= 8.6e+42)
		tmp = (a * c) / ((c * c) + (d * d));
	elseif (c <= 2.1e+77)
		tmp = t_0;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.35e-27], N[(a / c), $MachinePrecision], If[LessEqual[c, 6e-101], t$95$0, If[LessEqual[c, 8.6e+42], N[(N[(a * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.1e+77], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.34999999999999994e-27 or 2.0999999999999999e77 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf 68.8%

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

   if -1.34999999999999994e-27 < c < 6.0000000000000006e-101 or 8.5999999999999996e42 < c < 2.0999999999999999e77

   1. Initial program 72.5%

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

      1. *-un-lft-identity 72.5%

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

      2. add-sqr-sqrt 72.5%

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

      3. times-frac 72.4%

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

      4. hypot-def 72.5%

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

      5. fma-def 72.5%

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

      6. hypot-def 84.7%

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

   4. Applied egg-rr 84.7%

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

   5. Taylor expanded in c around 0 47.3%

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

      1. associate-/l* 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in c around 0 83.7%

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

      1. clear-num 83.7%

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

      2. associate-/r/ 83.7%

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

      3. clear-num 83.7%

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

   10. Applied egg-rr 83.7%

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

   if 6.0000000000000006e-101 < c < 8.5999999999999996e42

   1. Initial program 85.3%

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

   3. Taylor expanded in a around inf 68.7%

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

      1. *-commutative 68.7%

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

   5. Simplified 68.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-27}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{a \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-32} \lor \neg \left(c \leq 2.65 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.5e-32) (not (<= c 2.65e+78)))
   (/ a c)
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.5e-32) || !(c <= 2.65e+78)) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a * (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 <= (-5.5d-32)) .or. (.not. (c <= 2.65d+78))) then
        tmp = a / c
    else
        tmp = (1.0d0 / d) * (b + (a * (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 <= -5.5e-32) || !(c <= 2.65e+78)) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.5e-32) or not (c <= 2.65e+78):
		tmp = a / c
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.5e-32) || !(c <= 2.65e+78))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.5e-32) || ~((c <= 2.65e+78)))
		tmp = a / c;
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.5e-32], N[Not[LessEqual[c, 2.65e+78]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.50000000000000024e-32 or 2.64999999999999981e78 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf 68.8%

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

   if -5.50000000000000024e-32 < c < 2.64999999999999981e78

   1. Initial program 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. add-sqr-sqrt 74.4%

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

      3. times-frac 74.4%

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

      4. hypot-def 74.4%

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

      5. fma-def 74.4%

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

      6. hypot-def 84.8%

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

   4. Applied egg-rr 84.8%

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

   5. Taylor expanded in c around 0 42.7%

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

      1. associate-/l* 42.7%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in c around 0 78.3%

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

      1. clear-num 78.3%

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

      2. associate-/r/ 78.3%

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

      3. clear-num 78.3%

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

   10. Applied egg-rr 78.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-32} \lor \neg \left(c \leq 2.65 \cdot 10^{+78}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-27} \lor \neg \left(c \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.7e-27) (not (<= c 2.05e+77))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.7e-27) || !(c <= 2.05e+77)) {
		tmp = a / 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 ((c <= (-2.7d-27)) .or. (.not. (c <= 2.05d+77))) then
        tmp = a / 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 ((c <= -2.7e-27) || !(c <= 2.05e+77)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.7e-27) or not (c <= 2.05e+77):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.7e-27) || !(c <= 2.05e+77))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.7e-27) || ~((c <= 2.05e+77)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.7e-27], N[Not[LessEqual[c, 2.05e+77]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.69999999999999989e-27 or 2.05e77 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf 68.8%

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

   if -2.69999999999999989e-27 < c < 2.05e77

   1. Initial program 74.4%

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

   3. Taylor expanded in c around 0 61.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-27} \lor \neg \left(c \leq 2.05 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

3. Taylor expanded in c around inf 43.1%

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

4. Final simplification 43.1%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(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))))