Complex division, real part

Percentage Accurate: 61.0% → 84.9%

Time: 11.1s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 2e+223)
   (/ (/ (fma c a (* b d)) (hypot c d)) (hypot c d))
   (/ (+ b (/ a (/ d c))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 2e+223) {
		tmp = (fma(c, a, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (b + (a / (d / c))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+223)
		tmp = Float64(Float64(fma(c, a, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+223], N[(N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $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))) < 2.00000000000000009e223

   1. Initial program 79.9%

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

      1. *-un-lft-identity 79.9%

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

      2. add-sqr-sqrt 79.9%

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

      3. times-frac 79.8%

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

      4. hypot-define 79.8%

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

      5. fma-define 79.8%

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

      6. hypot-define 96.1%

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

   4. Applied egg-rr 96.1%

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

      1. associate-*l/ 96.3%

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

      2. *-un-lft-identity 96.3%

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

      3. fma-undefine 96.3%

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

      4. *-commutative 96.3%

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

      5. *-commutative 96.3%

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

      6. fma-define 96.3%

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

   6. Applied egg-rr 96.3%

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

   if 2.00000000000000009e223 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 17.1%

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

   3. Taylor expanded in d around inf 60.0%

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

      1. *-commutative 60.0%

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

      2. *-un-lft-identity 60.0%

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

      3. times-frac 67.4%

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

   5. Applied egg-rr 67.4%

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

      1. clear-num 67.3%

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

      2. frac-times 68.9%

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

      3. *-un-lft-identity 68.9%

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

      4. associate-/r/ 68.9%

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

      5. clear-num 68.9%

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

   7. Applied egg-rr 68.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + \left(a \cdot \frac{c}{d} - b \cdot {\left(\frac{c}{d}\right)}^{2}\right)}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-114}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1850000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{1}{\frac{c}{d}}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -9e+63)
   (/ (+ b (- (* a (/ c d)) (* b (pow (/ c d) 2.0)))) d)
   (if (<= d -1.15e-114)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 1850000000000.0)
       (/ (fma b (/ 1.0 (/ c d)) a) c)
       (/ (+ b (* c (/ a d))) (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9e+63) {
		tmp = (b + ((a * (c / d)) - (b * pow((c / d), 2.0)))) / d;
	} else if (d <= -1.15e-114) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 1850000000000.0) {
		tmp = fma(b, (1.0 / (c / d)), a) / c;
	} else {
		tmp = (b + (c * (a / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9e+63)
		tmp = Float64(Float64(b + Float64(Float64(a * Float64(c / d)) - Float64(b * (Float64(c / d) ^ 2.0)))) / d);
	elseif (d <= -1.15e-114)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1850000000000.0)
		tmp = Float64(fma(b, Float64(1.0 / Float64(c / d)), a) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9e+63], N[(N[(b + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] - N[(b * N[Power[N[(c / d), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.15e-114], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1850000000000.0], N[(N[(b * N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.00000000000000034e63

   1. Initial program 36.9%

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

      1. *-un-lft-identity 36.9%

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

      2. add-sqr-sqrt 36.9%

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

      3. times-frac 36.9%

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

      4. hypot-define 36.9%

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

      5. fma-define 36.9%

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

      6. hypot-define 54.0%

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

   4. Applied egg-rr 54.0%

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

      1. add-sqr-sqrt 53.9%

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

      2. pow2 53.9%

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

      3. inv-pow 53.9%

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

      4. sqrt-pow1 53.9%

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

      5. metadata-eval 53.9%

         \[\leadsto {\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{\color{blue}{-0.5}}\right)}^{2} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{hypot}\left(c, d\right)\right)}^{-0.5}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)} \]

   7. Taylor expanded in d around inf 65.0%

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

      1. +-commutative 65.0%

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

      2. associate-/l* 65.0%

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

      3. *-commutative 65.0%

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

      4. mul-1-neg 65.0%

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

      5. unsub-neg 65.0%

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

      6. *-commutative 65.0%

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

      7. associate-/l* 68.6%

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

      8. unpow2 68.6%

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

      9. unpow2 68.6%

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

      10. times-frac 83.0%

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

      11. unpow2 83.0%

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

   9. Simplified 83.0%

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

   if -9.00000000000000034e63 < d < -1.15e-114

   1. Initial program 81.3%

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

   if -1.15e-114 < d < 1.85e12

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-/l* 89.1%

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

      3. fma-define 89.1%

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

   5. Simplified 89.1%

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

      1. clear-num 89.2%

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

      2. inv-pow 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. unpow-1 89.2%

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

   9. Simplified 89.2%

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

   if 1.85e12 < d

   1. Initial program 54.1%

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

      1. *-un-lft-identity 54.1%

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

      2. add-sqr-sqrt 54.1%

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

      3. times-frac 54.1%

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

      4. hypot-define 54.1%

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

      5. fma-define 54.1%

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

      6. hypot-define 75.1%

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

   4. Applied egg-rr 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. fma-undefine 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

      6. fma-define 75.1%

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in c around 0 82.4%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   9. Applied egg-rr 87.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + \left(a \cdot \frac{c}{d} - b \cdot {\left(\frac{c}{d}\right)}^{2}\right)}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-114}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1850000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{1}{\frac{c}{d}}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 10000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{1}{\frac{c}{d}}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -8.2e+63)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d -3e-113)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 10000000000000.0)
       (/ (fma b (/ 1.0 (/ c d)) a) c)
       (/ (+ b (* c (/ a d))) (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8.2e+63) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -3e-113) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 10000000000000.0) {
		tmp = fma(b, (1.0 / (c / d)), a) / c;
	} else {
		tmp = (b + (c * (a / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -8.2e+63)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -3e-113)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 10000000000000.0)
		tmp = Float64(fma(b, Float64(1.0 / Float64(c / d)), a) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -8.2e+63], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3e-113], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 10000000000000.0], N[(N[(b * N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.19999999999999985e63

   1. Initial program 36.9%

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

   3. Taylor expanded in d around inf 79.3%

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

      1. associate-/l* 82.8%

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

   5. Simplified 82.8%

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

   if -8.19999999999999985e63 < d < -3.0000000000000001e-113

   1. Initial program 81.3%

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

   if -3.0000000000000001e-113 < d < 1e13

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-/l* 89.1%

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

      3. fma-define 89.1%

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

   5. Simplified 89.1%

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

      1. clear-num 89.2%

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

      2. inv-pow 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. unpow-1 89.2%

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

   9. Simplified 89.2%

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

   if 1e13 < d

   1. Initial program 54.1%

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

      1. *-un-lft-identity 54.1%

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

      2. add-sqr-sqrt 54.1%

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

      3. times-frac 54.1%

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

      4. hypot-define 54.1%

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

      5. fma-define 54.1%

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

      6. hypot-define 75.1%

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

   4. Applied egg-rr 75.1%

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

      1. associate-*l/ 75.1%

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

      2. *-un-lft-identity 75.1%

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

      3. fma-undefine 75.1%

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

      4. *-commutative 75.1%

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

      5. *-commutative 75.1%

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

      6. fma-define 75.1%

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

   6. Applied egg-rr 75.1%

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

   7. Taylor expanded in c around 0 82.4%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   9. Applied egg-rr 87.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-113}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 10000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{1}{\frac{c}{d}}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{1}{\frac{c}{d}}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -7.2e+63)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d -2.5e-118)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 2.5e+14)
       (/ (fma b (/ 1.0 (/ c d)) a) c)
       (/ (+ b (* c (/ a d))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -7.2e+63) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -2.5e-118) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 2.5e+14) {
		tmp = fma(b, (1.0 / (c / d)), a) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -7.2e+63)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -2.5e-118)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.5e+14)
		tmp = Float64(fma(b, Float64(1.0 / Float64(c / d)), a) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -7.2e+63], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.5e-118], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.5e+14], N[(N[(b * N[(1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.19999999999999998e63

   1. Initial program 36.9%

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

   3. Taylor expanded in d around inf 79.3%

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

      1. associate-/l* 82.8%

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

   5. Simplified 82.8%

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

   if -7.19999999999999998e63 < d < -2.50000000000000007e-118

   1. Initial program 81.3%

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

   if -2.50000000000000007e-118 < d < 2.5e14

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-/l* 89.1%

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

      3. fma-define 89.1%

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

   5. Simplified 89.1%

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

      1. clear-num 89.2%

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

      2. inv-pow 89.2%

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

   7. Applied egg-rr 89.2%

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

      1. unpow-1 89.2%

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

   9. Simplified 89.2%

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

   if 2.5e14 < d

   1. Initial program 54.1%

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

   3. Taylor expanded in d around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in c around 0 86.0%

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

Alternative 5: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.62 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7000000000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.62e+63)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d -5e-119)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= d 7000000000000.0)
       (/ (+ a (* b (/ d c))) c)
       (/ (+ b (* c (/ a d))) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.62e+63) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5e-119) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 7000000000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.62d+63)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= (-5d-119)) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else if (d <= 7000000000000.0d0) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (c * (a / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.62e+63) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= -5e-119) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (d <= 7000000000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.62e+63:
		tmp = (b + (a * (c / d))) / d
	elif d <= -5e-119:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	elif d <= 7000000000000.0:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (c * (a / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.62e+63)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= -5e-119)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 7000000000000.0)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.62e+63)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= -5e-119)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	elseif (d <= 7000000000000.0)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (c * (a / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.62e+63], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -5e-119], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7000000000000.0], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.62e63

   1. Initial program 36.9%

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

   3. Taylor expanded in d around inf 79.3%

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

      1. associate-/l* 82.8%

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

   5. Simplified 82.8%

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

   if -1.62e63 < d < -4.99999999999999993e-119

   1. Initial program 81.3%

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

   if -4.99999999999999993e-119 < d < 7e12

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf 89.0%

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

      1. +-commutative 89.0%

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

      2. associate-/l* 89.1%

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

      3. fma-define 89.1%

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

   5. Simplified 89.1%

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

      1. fma-undefine 89.1%

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

   7. Applied egg-rr 89.1%

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

   if 7e12 < d

   1. Initial program 54.1%

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

   3. Taylor expanded in d around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in c around 0 86.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.62 \cdot 10^{+63}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7000000000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-179}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6e-75)
   (/ b d)
   (if (<= d 8.8e-179)
     (/ a c)
     (if (<= d 2.1e-14) (/ (* b (/ d c)) c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6e-75) {
		tmp = b / d;
	} else if (d <= 8.8e-179) {
		tmp = a / c;
	} else if (d <= 2.1e-14) {
		tmp = (b * (d / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-6d-75)) then
        tmp = b / d
    else if (d <= 8.8d-179) then
        tmp = a / c
    else if (d <= 2.1d-14) then
        tmp = (b * (d / c)) / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6e-75) {
		tmp = b / d;
	} else if (d <= 8.8e-179) {
		tmp = a / c;
	} else if (d <= 2.1e-14) {
		tmp = (b * (d / c)) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6e-75:
		tmp = b / d
	elif d <= 8.8e-179:
		tmp = a / c
	elif d <= 2.1e-14:
		tmp = (b * (d / c)) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6e-75)
		tmp = Float64(b / d);
	elseif (d <= 8.8e-179)
		tmp = Float64(a / c);
	elseif (d <= 2.1e-14)
		tmp = Float64(Float64(b * Float64(d / c)) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6e-75)
		tmp = b / d;
	elseif (d <= 8.8e-179)
		tmp = a / c;
	elseif (d <= 2.1e-14)
		tmp = (b * (d / c)) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6e-75], N[(b / d), $MachinePrecision], If[LessEqual[d, 8.8e-179], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.1e-14], N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.9999999999999997e-75 or 2.0999999999999999e-14 < d

   1. Initial program 54.2%

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

   3. Taylor expanded in c around 0 65.5%

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

   if -5.9999999999999997e-75 < d < 8.80000000000000018e-179

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 78.6%

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

   if 8.80000000000000018e-179 < d < 2.0999999999999999e-14

   1. Initial program 84.1%

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

   3. Taylor expanded in c around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. associate-/l* 85.9%

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

      3. fma-define 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in b around inf 62.2%

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

      1. associate-*r/ 62.3%

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

   8. Simplified 62.3%

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

Alternative 7: 59.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-74}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-180}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.7e-74)
   (/ b d)
   (if (<= d 4e-180) (/ a c) (if (<= d 4.4e-14) (* (/ d c) (/ b c)) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.7e-74) {
		tmp = b / d;
	} else if (d <= 4e-180) {
		tmp = a / c;
	} else if (d <= 4.4e-14) {
		tmp = (d / c) * (b / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-2.7d-74)) then
        tmp = b / d
    else if (d <= 4d-180) then
        tmp = a / c
    else if (d <= 4.4d-14) then
        tmp = (d / c) * (b / c)
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.7e-74) {
		tmp = b / d;
	} else if (d <= 4e-180) {
		tmp = a / c;
	} else if (d <= 4.4e-14) {
		tmp = (d / c) * (b / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.7e-74:
		tmp = b / d
	elif d <= 4e-180:
		tmp = a / c
	elif d <= 4.4e-14:
		tmp = (d / c) * (b / c)
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.7e-74)
		tmp = Float64(b / d);
	elseif (d <= 4e-180)
		tmp = Float64(a / c);
	elseif (d <= 4.4e-14)
		tmp = Float64(Float64(d / c) * Float64(b / c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.7e-74)
		tmp = b / d;
	elseif (d <= 4e-180)
		tmp = a / c;
	elseif (d <= 4.4e-14)
		tmp = (d / c) * (b / c);
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.7e-74], N[(b / d), $MachinePrecision], If[LessEqual[d, 4e-180], N[(a / c), $MachinePrecision], If[LessEqual[d, 4.4e-14], N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.70000000000000018e-74 or 4.4000000000000002e-14 < d

   1. Initial program 54.2%

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

   3. Taylor expanded in c around 0 65.5%

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

   if -2.70000000000000018e-74 < d < 4.0000000000000001e-180

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 78.6%

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

   if 4.0000000000000001e-180 < d < 4.4000000000000002e-14

   1. Initial program 84.1%

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

   3. Taylor expanded in c around inf 85.9%

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

      1. +-commutative 85.9%

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

      2. associate-/l* 85.9%

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

      3. fma-define 85.9%

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

   5. Simplified 85.9%

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

   6. Taylor expanded in b around inf 62.2%

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

      1. associate-/l/ 57.1%

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

      2. *-commutative 57.1%

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

      3. times-frac 57.5%

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

   8. Applied egg-rr 57.5%

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

Alternative 8: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{-75}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.05e-75)
   (/ b d)
   (if (<= d 2.3e-132)
     (/ a c)
     (if (<= d 8.5e-14) (* d (/ (/ b c) c)) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.05e-75) {
		tmp = b / d;
	} else if (d <= 2.3e-132) {
		tmp = a / c;
	} else if (d <= 8.5e-14) {
		tmp = d * ((b / c) / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-2.05d-75)) then
        tmp = b / d
    else if (d <= 2.3d-132) then
        tmp = a / c
    else if (d <= 8.5d-14) then
        tmp = d * ((b / c) / c)
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.05e-75) {
		tmp = b / d;
	} else if (d <= 2.3e-132) {
		tmp = a / c;
	} else if (d <= 8.5e-14) {
		tmp = d * ((b / c) / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.05e-75:
		tmp = b / d
	elif d <= 2.3e-132:
		tmp = a / c
	elif d <= 8.5e-14:
		tmp = d * ((b / c) / c)
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.05e-75)
		tmp = Float64(b / d);
	elseif (d <= 2.3e-132)
		tmp = Float64(a / c);
	elseif (d <= 8.5e-14)
		tmp = Float64(d * Float64(Float64(b / c) / c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.05e-75)
		tmp = b / d;
	elseif (d <= 2.3e-132)
		tmp = a / c;
	elseif (d <= 8.5e-14)
		tmp = d * ((b / c) / c);
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.05e-75], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.3e-132], N[(a / c), $MachinePrecision], If[LessEqual[d, 8.5e-14], N[(d * N[(N[(b / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.05000000000000001e-75 or 8.50000000000000038e-14 < d

   1. Initial program 54.2%

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

   3. Taylor expanded in c around 0 65.5%

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

   if -2.05000000000000001e-75 < d < 2.30000000000000003e-132

   1. Initial program 73.6%

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

   3. Taylor expanded in c around inf 72.6%

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

   if 2.30000000000000003e-132 < d < 8.50000000000000038e-14

   1. Initial program 84.6%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. +-commutative 77.7%

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

      2. associate-/l* 77.8%

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

      3. fma-define 77.8%

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

   5. Simplified 77.8%

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

   6. Taylor expanded in b around inf 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-/l* 62.6%

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

   8. Applied egg-rr 62.6%

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

      1. associate-/l* 62.7%

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

   10. Applied egg-rr 62.7%

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

Alternative 9: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -68000000 \lor \neg \left(d \leq 38000000000\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -68000000.0) (not (<= d 38000000000.0)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -68000000.0) || !(d <= 38000000000.0)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-68000000.0d0)) .or. (.not. (d <= 38000000000.0d0))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + ((b * d) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -68000000.0) || !(d <= 38000000000.0)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -68000000.0) or not (d <= 38000000000.0):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -68000000.0) || !(d <= 38000000000.0))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -68000000.0) || ~((d <= 38000000000.0)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -68000000.0], N[Not[LessEqual[d, 38000000000.0]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.8e7 or 3.8e10 < d

   1. Initial program 49.4%

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

   3. Taylor expanded in d around inf 78.5%

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

      1. associate-/l* 80.6%

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

   5. Simplified 80.6%

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

   if -6.8e7 < d < 3.8e10

   1. Initial program 76.8%

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

   3. Taylor expanded in c around inf 83.9%

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

      1. *-commutative 83.9%

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

   5. Simplified 83.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -68000000 \lor \neg \left(d \leq 38000000000\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -13000000 \lor \neg \left(d \leq 6.4 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -13000000.0) (not (<= d 6.4e+15)))
   (/ b d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -13000000.0) || !(d <= 6.4e+15)) {
		tmp = b / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-13000000.0d0)) .or. (.not. (d <= 6.4d+15))) then
        tmp = b / d
    else
        tmp = (a + ((b * d) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -13000000.0) || !(d <= 6.4e+15)) {
		tmp = b / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -13000000.0) or not (d <= 6.4e+15):
		tmp = b / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -13000000.0) || !(d <= 6.4e+15))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -13000000.0) || ~((d <= 6.4e+15)))
		tmp = b / d;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -13000000.0], N[Not[LessEqual[d, 6.4e+15]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.3e7 or 6.4e15 < d

   1. Initial program 49.4%

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

   3. Taylor expanded in c around 0 68.0%

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

   if -1.3e7 < d < 6.4e15

   1. Initial program 76.8%

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

   3. Taylor expanded in c around inf 83.9%

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

      1. *-commutative 83.9%

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

   5. Simplified 83.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -13000000 \lor \neg \left(d \leq 6.4 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -48000000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1400000000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -48000000.0)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 1400000000000.0)
     (/ (+ a (* b (/ d c))) c)
     (/ (+ b (* c (/ a d))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -48000000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1400000000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-48000000.0d0)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 1400000000000.0d0) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (c * (a / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -48000000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1400000000000.0) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -48000000.0:
		tmp = (b + (a * (c / d))) / d
	elif d <= 1400000000000.0:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (c * (a / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -48000000.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 1400000000000.0)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -48000000.0)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 1400000000000.0)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (c * (a / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -48000000.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1400000000000.0], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8e7

   1. Initial program 45.8%

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

   3. Taylor expanded in d around inf 78.2%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   if -4.8e7 < d < 1.4e12

   1. Initial program 76.8%

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

   3. Taylor expanded in c around inf 83.9%

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

      1. +-commutative 83.9%

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

      2. associate-/l* 84.0%

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

      3. fma-define 84.0%

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

   5. Simplified 84.0%

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

      1. fma-undefine 84.0%

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

   7. Applied egg-rr 84.0%

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

   if 1.4e12 < d

   1. Initial program 54.1%

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

   3. Taylor expanded in d around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in c around 0 86.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -48000000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1400000000000:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -56000000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 16000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -56000000.0)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 16000000000.0)
     (/ (+ a (/ (* b d) c)) c)
     (/ (+ b (* c (/ a d))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -56000000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 16000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-56000000.0d0)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 16000000000.0d0) then
        tmp = (a + ((b * d) / c)) / c
    else
        tmp = (b + (c * (a / d))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -56000000.0) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 16000000000.0) {
		tmp = (a + ((b * d) / c)) / c;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -56000000.0:
		tmp = (b + (a * (c / d))) / d
	elif d <= 16000000000.0:
		tmp = (a + ((b * d) / c)) / c
	else:
		tmp = (b + (c * (a / d))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -56000000.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 16000000000.0)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -56000000.0)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 16000000000.0)
		tmp = (a + ((b * d) / c)) / c;
	else
		tmp = (b + (c * (a / d))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -56000000.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 16000000000.0], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.6e7

   1. Initial program 45.8%

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

   3. Taylor expanded in d around inf 78.2%

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

      1. associate-/l* 81.0%

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

   5. Simplified 81.0%

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

   if -5.6e7 < d < 1.6e10

   1. Initial program 76.8%

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

   3. Taylor expanded in c around inf 83.9%

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

      1. *-commutative 83.9%

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

   5. Simplified 83.9%

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

   if 1.6e10 < d

   1. Initial program 54.1%

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

   3. Taylor expanded in d around inf 78.9%

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

      1. *-commutative 78.9%

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

      2. *-un-lft-identity 78.9%

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

      3. times-frac 86.0%

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

   5. Applied egg-rr 86.0%

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

   6. Taylor expanded in c around 0 86.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -56000000:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 16000000000:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-74} \lor \neg \left(d \leq 0.25\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.65e-74) (not (<= d 0.25))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.65e-74) || !(d <= 0.25)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.65d-74)) .or. (.not. (d <= 0.25d0))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.65e-74) || !(d <= 0.25)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.65e-74) or not (d <= 0.25):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.65e-74) || !(d <= 0.25))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.65e-74) || ~((d <= 0.25)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.65e-74], N[Not[LessEqual[d, 0.25]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.64999999999999994e-74 or 0.25 < d

   1. Initial program 53.9%

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

   3. Taylor expanded in c around 0 65.5%

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

   if -2.64999999999999994e-74 < d < 0.25

   1. Initial program 75.3%

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

   3. Taylor expanded in c around inf 66.8%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.65 \cdot 10^{-74} \lor \neg \left(d \leq 0.25\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.6% 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 63.2%

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

3. Taylor expanded in c around inf 40.9%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
4. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024166 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))