Complex division, real part

Percentage Accurate: 61.8% → 80.8%

Time: 10.9s

Alternatives: 8

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ t_1 := \sqrt[3]{\mathsf{fma}\left(a, c, d \cdot b\right)}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+57}:\\ \;\;\;\;\frac{{t\_1}^{2}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d (+ (* c (/ a d)) b)))) (t_1 (cbrt (fma a c (* d b)))))
   (if (<= d -7.5e+25)
     t_0
     (if (<= d 1.45e-74)
       (/ (+ a (/ (* d b) c)) c)
       (if (<= d 1.02e+57)
         (* (/ (pow t_1 2.0) (hypot c d)) (/ t_1 (hypot c d)))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / (d / ((c * (a / d)) + b));
	double t_1 = cbrt(fma(a, c, (d * b)));
	double tmp;
	if (d <= -7.5e+25) {
		tmp = t_0;
	} else if (d <= 1.45e-74) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 1.02e+57) {
		tmp = (pow(t_1, 2.0) / hypot(c, d)) * (t_1 / hypot(c, d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / Float64(d / Float64(Float64(c * Float64(a / d)) + b)))
	t_1 = cbrt(fma(a, c, Float64(d * b)))
	tmp = 0.0
	if (d <= -7.5e+25)
		tmp = t_0;
	elseif (d <= 1.45e-74)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 1.02e+57)
		tmp = Float64(Float64((t_1 ^ 2.0) / hypot(c, d)) * Float64(t_1 / hypot(c, d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[(d / N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[d, -7.5e+25], t$95$0, If[LessEqual[d, 1.45e-74], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.02e+57], N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.49999999999999993e25 or 1.02e57 < d

   1. Initial program 53.2%

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

      1. fma-define 53.2%

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

      2. fma-define 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in d around inf 80.3%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. clear-num 83.9%

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

      2. inv-pow 83.9%

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

      3. +-commutative 83.9%

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

      4. fma-define 83.9%

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

   9. Applied egg-rr 83.9%

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

       1. unpow-1 83.9%

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

   11. Simplified 83.9%

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

       1. fma-undefine 83.9%

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

       2. associate-*r/ 80.1%

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

       3. *-commutative 80.1%

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

       4. associate-*r/ 85.3%

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

   13. Applied egg-rr 85.3%

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

   if -7.49999999999999993e25 < d < 1.45e-74

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in c around inf 91.6%

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 1.45e-74 < d < 1.02e57

   1. Initial program 80.6%

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

      1. fma-define 80.6%

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

      2. fma-define 80.7%

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

   3. Simplified 80.7%

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

      1. add-cube-cbrt 79.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}}{\mathsf{fma}\left(c, c, d \cdot d\right)} \]

      2. add-sqr-sqrt 79.6%

         \[\leadsto \frac{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\color{blue}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}}} \]

      3. times-frac 79.5%

         \[\leadsto \color{blue}{\frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}}} \]

      4. pow2 79.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}^{2}}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

      5. fma-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}^{2}}{\sqrt{\color{blue}{c \cdot c + d \cdot d}}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

      6. hypot-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}^{2}}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\mathsf{fma}\left(c, c, d \cdot d\right)}} \]

      7. fma-define 79.5%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}^{2}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\sqrt{\color{blue}{c \cdot c + d \cdot d}}} \]

      8. hypot-define 98.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}^{2}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, b \cdot d\right)}}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \]

   6. Applied egg-rr 98.1%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+57}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{\mathsf{fma}\left(a, c, d \cdot b\right)}\right)}^{2}}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\sqrt[3]{\mathsf{fma}\left(a, c, d \cdot b\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{if}\;d \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d (+ (* c (/ a d)) b)))))
   (if (<= d -2.7e+29)
     t_0
     (if (<= d 1.5e-76)
       (/ (+ a (/ (* d b) c)) c)
       (if (<= d 4.7e+55) (/ (fma a c (* d b)) (fma c c (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / (d / ((c * (a / d)) + b));
	double tmp;
	if (d <= -2.7e+29) {
		tmp = t_0;
	} else if (d <= 1.5e-76) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 4.7e+55) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / Float64(d / Float64(Float64(c * Float64(a / d)) + b)))
	tmp = 0.0
	if (d <= -2.7e+29)
		tmp = t_0;
	elseif (d <= 1.5e-76)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 4.7e+55)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[(d / N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.7e+29], t$95$0, If[LessEqual[d, 1.5e-76], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.7e+55], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.7e29 or 4.7000000000000001e55 < d

   1. Initial program 53.2%

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

      1. fma-define 53.2%

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

      2. fma-define 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in d around inf 80.3%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. clear-num 83.9%

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

      2. inv-pow 83.9%

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

      3. +-commutative 83.9%

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

      4. fma-define 83.9%

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

   9. Applied egg-rr 83.9%

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

       1. unpow-1 83.9%

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

   11. Simplified 83.9%

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

       1. fma-undefine 83.9%

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

       2. associate-*r/ 80.1%

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

       3. *-commutative 80.1%

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

       4. associate-*r/ 85.3%

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

   13. Applied egg-rr 85.3%

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

   if -2.7e29 < d < 1.50000000000000012e-76

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in c around inf 91.6%

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 1.50000000000000012e-76 < d < 4.7000000000000001e55

   1. Initial program 80.6%

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

      1. fma-define 80.6%

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

      2. fma-define 80.7%

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

   3. Simplified 80.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (/ d (+ (* c (/ a d)) b)))))
   (if (<= d -4.5e+28)
     t_0
     (if (<= d 4.7e-71)
       (/ (+ a (/ (* d b) c)) c)
       (if (<= d 2.15e+58) (/ (+ (* d b) (* c a)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / (d / ((c * (a / d)) + b));
	double tmp;
	if (d <= -4.5e+28) {
		tmp = t_0;
	} else if (d <= 4.7e-71) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.15e+58) {
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (d / ((c * (a / d)) + b))
    if (d <= (-4.5d+28)) then
        tmp = t_0
    else if (d <= 4.7d-71) then
        tmp = (a + ((d * b) / c)) / c
    else if (d <= 2.15d+58) then
        tmp = ((d * b) + (c * a)) / ((d * d) + (c * c))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / (d / ((c * (a / d)) + b));
	double tmp;
	if (d <= -4.5e+28) {
		tmp = t_0;
	} else if (d <= 4.7e-71) {
		tmp = (a + ((d * b) / c)) / c;
	} else if (d <= 2.15e+58) {
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 1.0 / (d / ((c * (a / d)) + b))
	tmp = 0
	if d <= -4.5e+28:
		tmp = t_0
	elif d <= 4.7e-71:
		tmp = (a + ((d * b) / c)) / c
	elif d <= 2.15e+58:
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / Float64(d / Float64(Float64(c * Float64(a / d)) + b)))
	tmp = 0.0
	if (d <= -4.5e+28)
		tmp = t_0;
	elseif (d <= 4.7e-71)
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	elseif (d <= 2.15e+58)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / (d / ((c * (a / d)) + b));
	tmp = 0.0;
	if (d <= -4.5e+28)
		tmp = t_0;
	elseif (d <= 4.7e-71)
		tmp = (a + ((d * b) / c)) / c;
	elseif (d <= 2.15e+58)
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[(d / N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.5e+28], t$95$0, If[LessEqual[d, 4.7e-71], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.15e+58], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.4999999999999997e28 or 2.14999999999999996e58 < d

   1. Initial program 53.2%

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

      1. fma-define 53.2%

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

      2. fma-define 53.2%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in d around inf 80.3%

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

      1. associate-/l* 84.2%

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

   7. Simplified 84.2%

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

      1. clear-num 83.9%

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

      2. inv-pow 83.9%

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

      3. +-commutative 83.9%

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

      4. fma-define 83.9%

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

   9. Applied egg-rr 83.9%

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

       1. unpow-1 83.9%

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

   11. Simplified 83.9%

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

       1. fma-undefine 83.9%

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

       2. associate-*r/ 80.1%

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

       3. *-commutative 80.1%

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

       4. associate-*r/ 85.3%

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

   13. Applied egg-rr 85.3%

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

   if -4.4999999999999997e28 < d < 4.69999999999999996e-71

   1. Initial program 69.4%

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

      1. fma-define 69.4%

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

      2. fma-define 69.4%

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

   3. Simplified 69.4%

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

   5. Taylor expanded in c around inf 91.6%

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

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   if 4.69999999999999996e-71 < d < 2.14999999999999996e58

   1. Initial program 80.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-71}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+58}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+28} \lor \neg \left(d \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.05e+28) (not (<= d 1.3e-53)))
   (/ 1.0 (/ d (+ (* c (/ a d)) b)))
   (/ (+ a (/ (* d b) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e+28) || !(d <= 1.3e-53)) {
		tmp = 1.0 / (d / ((c * (a / d)) + b));
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.05d+28)) .or. (.not. (d <= 1.3d-53))) then
        tmp = 1.0d0 / (d / ((c * (a / d)) + b))
    else
        tmp = (a + ((d * b) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.05e+28) || !(d <= 1.3e-53)) {
		tmp = 1.0 / (d / ((c * (a / d)) + b));
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.05e+28) or not (d <= 1.3e-53):
		tmp = 1.0 / (d / ((c * (a / d)) + b))
	else:
		tmp = (a + ((d * b) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.05e+28) || !(d <= 1.3e-53))
		tmp = Float64(1.0 / Float64(d / Float64(Float64(c * Float64(a / d)) + b)));
	else
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.05e+28) || ~((d <= 1.3e-53)))
		tmp = 1.0 / (d / ((c * (a / d)) + b));
	else
		tmp = (a + ((d * b) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.05e+28], N[Not[LessEqual[d, 1.3e-53]], $MachinePrecision]], N[(1.0 / N[(d / N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.04999999999999995e28 or 1.29999999999999998e-53 < d

   1. Initial program 58.6%

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

      1. fma-define 58.6%

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

      2. fma-define 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in d around inf 78.2%

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

      1. associate-/l* 81.4%

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

   7. Simplified 81.4%

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

      1. clear-num 81.2%

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

      2. inv-pow 81.2%

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

      3. +-commutative 81.2%

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

      4. fma-define 81.2%

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

   9. Applied egg-rr 81.2%

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

       1. unpow-1 81.2%

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

   11. Simplified 81.2%

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

       1. fma-undefine 81.2%

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

       2. associate-*r/ 78.0%

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

       3. *-commutative 78.0%

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

       4. associate-*r/ 82.3%

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

   13. Applied egg-rr 82.3%

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

   if -1.04999999999999995e28 < d < 1.29999999999999998e-53

   1. Initial program 69.0%

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

      1. fma-define 69.1%

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

      2. fma-define 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in c around inf 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+28} \lor \neg \left(d \leq 1.3 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{1}{\frac{d}{c \cdot \frac{a}{d} + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+28} \lor \neg \left(d \leq 10^{-53}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.4e+28) (not (<= d 1e-53)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (/ (* d b) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.4e+28) || !(d <= 1e-53)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-7.4d+28)) .or. (.not. (d <= 1d-53))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + ((d * b) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.4e+28) || !(d <= 1e-53)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.4e+28) or not (d <= 1e-53):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + ((d * b) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.4e+28) || !(d <= 1e-53))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.4e+28) || ~((d <= 1e-53)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + ((d * b) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.4e+28], N[Not[LessEqual[d, 1e-53]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.3999999999999998e28 or 1.00000000000000003e-53 < d

   1. Initial program 58.6%

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

      1. fma-define 58.6%

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

      2. fma-define 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in d around inf 78.2%

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

      1. associate-/l* 81.4%

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

   7. Simplified 81.4%

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

   if -7.3999999999999998e28 < d < 1.00000000000000003e-53

   1. Initial program 69.0%

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

      1. fma-define 69.1%

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

      2. fma-define 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in c around inf 90.3%

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

      1. *-commutative 90.3%

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

   7. Simplified 90.3%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.4 \cdot 10^{+28} \lor \neg \left(d \leq 10^{-53}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+28} \lor \neg \left(d \leq 3.1 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.6e+28) (not (<= d 3.1e+35)))
   (/ b d)
   (/ (+ a (/ (* d b) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+28) || !(d <= 3.1e+35)) {
		tmp = b / d;
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.6d+28)) .or. (.not. (d <= 3.1d+35))) then
        tmp = b / d
    else
        tmp = (a + ((d * b) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+28) || !(d <= 3.1e+35)) {
		tmp = b / d;
	} else {
		tmp = (a + ((d * b) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.6e+28) or not (d <= 3.1e+35):
		tmp = b / d
	else:
		tmp = (a + ((d * b) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.6e+28) || !(d <= 3.1e+35))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(d * b) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.6e+28) || ~((d <= 3.1e+35)))
		tmp = b / d;
	else
		tmp = (a + ((d * b) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.6e+28], N[Not[LessEqual[d, 3.1e+35]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6e28 or 3.09999999999999987e35 < d

   1. Initial program 54.4%

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

      1. fma-define 54.4%

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

      2. fma-define 54.4%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in c around 0 69.7%

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

   if -1.6e28 < d < 3.09999999999999987e35

   1. Initial program 71.1%

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

      1. fma-define 71.1%

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

      2. fma-define 71.1%

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

   3. Simplified 71.1%

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

   5. Taylor expanded in c around inf 82.3%

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

      1. *-commutative 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+28} \lor \neg \left(d \leq 3.1 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{d \cdot b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+16} \lor \neg \left(d \leq 1.25 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4e+16) (not (<= d 1.25e-53))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+16) || !(d <= 1.25e-53)) {
		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 <= (-4d+16)) .or. (.not. (d <= 1.25d-53))) 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 <= -4e+16) || !(d <= 1.25e-53)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4e+16) or not (d <= 1.25e-53):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4e+16) || !(d <= 1.25e-53))
		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 <= -4e+16) || ~((d <= 1.25e-53)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4e+16], N[Not[LessEqual[d, 1.25e-53]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4e16 or 1.25e-53 < d

   1. Initial program 59.1%

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

      1. fma-define 59.1%

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

      2. fma-define 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in c around 0 63.9%

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

   if -4e16 < d < 1.25e-53

   1. Initial program 68.5%

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

      1. fma-define 68.5%

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

      2. fma-define 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in c around inf 68.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+16} \lor \neg \left(d \leq 1.25 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 63.3%

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

   1. fma-define 63.3%

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

   2. fma-define 63.3%

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

3. Simplified 63.3%

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

5. Taylor expanded in c around inf 39.9%

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

Developer Target 1: 99.2% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024145 
(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))))