Complex division, real part

Percentage Accurate: 61.7% → 83.6%

Time: 5.8s

Alternatives: 9

Speedup: 1.6×

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: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -9.8 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-101}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* b d) (* c a)) (+ (* d d) (* c c))))
        (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -9.8e+128)
     t_1
     (if (<= d -1.5e-101)
       t_0
       (if (<= d 4e-131)
         (/ (fma b (/ d c) a) c)
         (if (<= d 1.15e+89) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c));
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -9.8e+128) {
		tmp = t_1;
	} else if (d <= -1.5e-101) {
		tmp = t_0;
	} else if (d <= 4e-131) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 1.15e+89) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -9.8e+128)
		tmp = t_1;
	elseif (d <= -1.5e-101)
		tmp = t_0;
	elseif (d <= 4e-131)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 1.15e+89)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.8e+128], t$95$1, If[LessEqual[d, -1.5e-101], t$95$0, If[LessEqual[d, 4e-131], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.15e+89], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.80000000000000035e128 or 1.1499999999999999e89 < d

   1. Initial program 33.2%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 85.0

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

   5. Applied rewrites 85.0%

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

   if -9.80000000000000035e128 < d < -1.5000000000000002e-101 or 3.9999999999999999e-131 < d < 1.1499999999999999e89

   1. Initial program 79.8%

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

   if -1.5000000000000002e-101 < d < 3.9999999999999999e-131

   1. Initial program 70.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 94.7%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.8 \cdot 10^{+128}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-131}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-34)
   (/ (fma b (/ d c) a) c)
   (if (<= c 1.25e-51) (/ (fma (/ c d) a b) d) (/ (fma (/ b c) d a) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-34) {
		tmp = fma(b, (d / c), a) / c;
	} else if (c <= 1.25e-51) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-34)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (c <= 1.25e-51)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-34], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.25e-51], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3e-34

   1. Initial program 51.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.5%

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

   if -3e-34 < c < 1.25000000000000001e-51

   1. Initial program 75.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.1

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

   8. Applied rewrites 82.1%

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

   if 1.25000000000000001e-51 < c

   1. Initial program 55.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

Alternative 3: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-34)
   (/ (fma b (/ d c) a) c)
   (if (<= c 1.25e-51) (/ (fma (/ a d) c b) d) (/ (fma (/ b c) d a) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-34) {
		tmp = fma(b, (d / c), a) / c;
	} else if (c <= 1.25e-51) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-34)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (c <= 1.25e-51)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-34], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.25e-51], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3e-34

   1. Initial program 51.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 76.5%

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

   if -3e-34 < c < 1.25000000000000001e-51

   1. Initial program 75.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if 1.25000000000000001e-51 < c

   1. Initial program 55.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

Alternative 4: 76.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -3 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma b (/ d c) a) c)))
   (if (<= c -3e-34) t_0 (if (<= c 1.25e-51) (/ (fma (/ a d) c b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(b, (d / c), a) / c;
	double tmp;
	if (c <= -3e-34) {
		tmp = t_0;
	} else if (c <= 1.25e-51) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -3e-34)
		tmp = t_0;
	elseif (c <= 1.25e-51)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3e-34], t$95$0, If[LessEqual[c, 1.25e-51], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3e-34 or 1.25000000000000001e-51 < c

   1. Initial program 53.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 77.9%

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

   if -3e-34 < c < 1.25000000000000001e-51

   1. Initial program 75.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.9

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

   5. Applied rewrites 80.9%

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

Alternative 5: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.25 \cdot 10^{+39}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+124}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.25e+39)
   (/ b d)
   (if (<= d 2.05e+124) (/ (fma b (/ d c) a) c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.25e+39) {
		tmp = b / d;
	} else if (d <= 2.05e+124) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.25e+39)
		tmp = Float64(b / d);
	elseif (d <= 2.05e+124)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.25e+39], N[(b / d), $MachinePrecision], If[LessEqual[d, 2.05e+124], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.25000000000000025e39 or 2.05000000000000001e124 < d

   1. Initial program 45.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -5.25000000000000025e39 < d < 2.05000000000000001e124

   1. Initial program 73.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 75.2%

      \[\leadsto \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 64.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2e+41)
   (/ b d)
   (if (<= d -1.4e-61)
     (/ (fma c a (* b d)) (* d d))
     (if (<= d 1.75e+91) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2e+41) {
		tmp = b / d;
	} else if (d <= -1.4e-61) {
		tmp = fma(c, a, (b * d)) / (d * d);
	} else if (d <= 1.75e+91) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2e+41)
		tmp = Float64(b / d);
	elseif (d <= -1.4e-61)
		tmp = Float64(fma(c, a, Float64(b * d)) / Float64(d * d));
	elseif (d <= 1.75e+91)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2e+41], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.4e-61], N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e+91], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.00000000000000001e41 or 1.75e91 < d

   1. Initial program 45.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -2.00000000000000001e41 < d < -1.4000000000000001e-61

   1. Initial program 78.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 41.1

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in c around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 61.9

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

   11. Applied rewrites 61.9%

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

   if -1.4000000000000001e-61 < d < 1.75e91

   1. Initial program 73.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 63.1

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

   5. Applied rewrites 63.1%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{b \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.85e+163)
   (/ b d)
   (if (<= d -1.6e-57)
     (/ (* b d) (fma d d (* c c)))
     (if (<= d 1.75e+91) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.85e+163) {
		tmp = b / d;
	} else if (d <= -1.6e-57) {
		tmp = (b * d) / fma(d, d, (c * c));
	} else if (d <= 1.75e+91) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.85e+163)
		tmp = Float64(b / d);
	elseif (d <= -1.6e-57)
		tmp = Float64(Float64(b * d) / fma(d, d, Float64(c * c)));
	elseif (d <= 1.75e+91)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.85e+163], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.6e-57], N[(N[(b * d), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e+91], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.8499999999999999e163 or 1.75e91 < d

   1. Initial program 33.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if -2.8499999999999999e163 < d < -1.6e-57

   1. Initial program 74.6%

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

   3. Taylor expanded in c around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 61.1

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

   5. Applied rewrites 61.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 61.1

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

   7. Applied rewrites 61.1%

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

   if -1.6e-57 < d < 1.75e91

   1. Initial program 73.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 62.7

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

   5. Applied rewrites 62.7%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.85 \cdot 10^{+163}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-57}:\\ \;\;\;\;\frac{b \cdot d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{+91}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.65 \cdot 10^{-38}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-51}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.65e-38) (/ a c) (if (<= c 1.22e-51) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.65e-38) {
		tmp = a / c;
	} else if (c <= 1.22e-51) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.65d-38)) then
        tmp = a / c
    else if (c <= 1.22d-51) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.65e-38) {
		tmp = a / c;
	} else if (c <= 1.22e-51) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.65e-38:
		tmp = a / c
	elif c <= 1.22e-51:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.65e-38)
		tmp = Float64(a / c);
	elseif (c <= 1.22e-51)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.65e-38)
		tmp = a / c;
	elseif (c <= 1.22e-51)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.65e-38], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.22e-51], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.65e-38 or 1.21999999999999998e-51 < c

   1. Initial program 53.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if -3.65e-38 < c < 1.21999999999999998e-51

   1. Initial program 74.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 69.6

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

   5. Applied rewrites 69.6%

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

Alternative 9: 42.7% accurate, 3.2× 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 63.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.7

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

5. Applied rewrites 42.7%

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

Developer Target 1: 99.2% accurate, 0.6× 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 2024249 
(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))))