Complex division, real part

Percentage Accurate: 61.4% → 80.1%

Time: 7.9s

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: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -1 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot d + a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (/ c d) (/ a d) (/ b d))))
   (if (<= d -1e-7)
     t_0
     (if (<= d 2.2e-112)
       (/ (fma (/ d c) b a) c)
       (if (<= d 2.3e+154) (/ (+ (* b d) (* a c)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / d), (a / d), (b / d));
	double tmp;
	if (d <= -1e-7) {
		tmp = t_0;
	} else if (d <= 2.2e-112) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 2.3e+154) {
		tmp = ((b * d) + (a * c)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / d), Float64(a / d), Float64(b / d))
	tmp = 0.0
	if (d <= -1e-7)
		tmp = t_0;
	elseif (d <= 2.2e-112)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 2.3e+154)
		tmp = Float64(Float64(Float64(b * d) + Float64(a * c)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1e-7], t$95$0, If[LessEqual[d, 2.2e-112], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.3e+154], N[(N[(N[(b * d), $MachinePrecision] + N[(a * c), $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 < -9.9999999999999995e-8 or 2.3e154 < d

   1. Initial program 34.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 81.2%

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

   if -9.9999999999999995e-8 < d < 2.20000000000000021e-112

   1. Initial program 67.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if 2.20000000000000021e-112 < d < 2.3e154

   1. Initial program 80.9%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot d + a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{a}{d}, \frac{b}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot d + a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1e-7)
   (/ (fma (/ a d) c b) d)
   (if (<= d 2.2e-112)
     (/ (fma (/ d c) b a) c)
     (if (<= d 2.3e+154)
       (/ (+ (* b d) (* a c)) (+ (* d d) (* c c)))
       (/ (fma (/ c d) a b) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1e-7) {
		tmp = fma((a / d), c, b) / d;
	} else if (d <= 2.2e-112) {
		tmp = fma((d / c), b, a) / c;
	} else if (d <= 2.3e+154) {
		tmp = ((b * d) + (a * c)) / ((d * d) + (c * c));
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1e-7)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (d <= 2.2e-112)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (d <= 2.3e+154)
		tmp = Float64(Float64(Float64(b * d) + Float64(a * c)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1e-7], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.2e-112], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.3e+154], N[(N[(N[(b * d), $MachinePrecision] + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.9999999999999995e-8

   1. Initial program 41.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

   if -9.9999999999999995e-8 < d < 2.20000000000000021e-112

   1. Initial program 67.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   7. 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. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 90.7

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

   8. Applied rewrites 90.7%

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

   if 2.20000000000000021e-112 < d < 2.3e154

   1. Initial program 80.9%

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

   if 2.3e154 < d

   1. Initial program 23.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 24.7

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

   5. Applied rewrites 24.7%

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

   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. associate-/l* N/A

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

      4. *-commutative 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 81.5

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

   8. Applied rewrites 81.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{b \cdot d + a \cdot c}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e+102)
   (/ a c)
   (if (<= c -3.1e-226)
     (/ b d)
     (if (<= c 1.18e-20) (/ (fma c a (* b d)) (* d d)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+102) {
		tmp = a / c;
	} else if (c <= -3.1e-226) {
		tmp = b / d;
	} else if (c <= 1.18e-20) {
		tmp = fma(c, a, (b * d)) / (d * d);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e+102)
		tmp = Float64(a / c);
	elseif (c <= -3.1e-226)
		tmp = Float64(b / d);
	elseif (c <= 1.18e-20)
		tmp = Float64(fma(c, a, Float64(b * d)) / Float64(d * d));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e+102], N[(a / c), $MachinePrecision], If[LessEqual[c, -3.1e-226], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.18e-20], N[(N[(c * a + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5e102 or 1.1800000000000001e-20 < c

   1. Initial program 46.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -7.5e102 < c < -3.09999999999999989e-226

   1. Initial program 61.3%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 52.2

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

   5. Applied rewrites 52.2%

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

   if -3.09999999999999989e-226 < c < 1.1800000000000001e-20

   1. Initial program 79.0%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.3

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

   8. Applied rewrites 32.3%

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

   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 68.9

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

   11. Applied rewrites 68.9%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-226}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, b \cdot d\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-140}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+49}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot a\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e+102)
   (/ a c)
   (if (<= c 5e-140)
     (/ b d)
     (if (<= c 2e+49) (* (/ c (fma c c (* d d))) a) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+102) {
		tmp = a / c;
	} else if (c <= 5e-140) {
		tmp = b / d;
	} else if (c <= 2e+49) {
		tmp = (c / fma(c, c, (d * d))) * a;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e+102)
		tmp = Float64(a / c);
	elseif (c <= 5e-140)
		tmp = Float64(b / d);
	elseif (c <= 2e+49)
		tmp = Float64(Float64(c / fma(c, c, Float64(d * d))) * a);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e+102], N[(a / c), $MachinePrecision], If[LessEqual[c, 5e-140], N[(b / d), $MachinePrecision], If[LessEqual[c, 2e+49], N[(N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5e102 or 1.99999999999999989e49 < c

   1. Initial program 43.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -7.5e102 < c < 5.00000000000000015e-140

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 5.00000000000000015e-140 < c < 1.99999999999999989e49

   1. Initial program 79.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 56.6

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

   8. Applied rewrites 56.6%

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

Alternative 5: 77.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -4.4 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -4.4e+102) t_0 (if (<= c 8e+46) (/ (fma (/ c d) a b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -4.4e+102) {
		tmp = t_0;
	} else if (c <= 8e+46) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -4.4e+102)
		tmp = t_0;
	elseif (c <= 8e+46)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.4e+102], t$95$0, If[LessEqual[c, 8e+46], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.40000000000000015e102 or 7.9999999999999999e46 < c

   1. Initial program 43.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   if -4.40000000000000015e102 < c < 7.9999999999999999e46

   1. Initial program 70.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 23.2

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

   5. Applied rewrites 23.2%

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

   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. associate-/l* N/A

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

      4. *-commutative 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 76.5

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

   8. Applied rewrites 76.5%

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

Alternative 6: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -4.4 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -4.4e+102) t_0 (if (<= c 8e+46) (/ (fma (/ a d) c b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -4.4e+102) {
		tmp = t_0;
	} else if (c <= 8e+46) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -4.4e+102)
		tmp = t_0;
	elseif (c <= 8e+46)
		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[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.4e+102], t$95$0, If[LessEqual[c, 8e+46], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.40000000000000015e102 or 7.9999999999999999e46 < c

   1. Initial program 43.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   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 85.5

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

   5. Applied rewrites 85.5%

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

   if -4.40000000000000015e102 < c < 7.9999999999999999e46

   1. Initial program 70.9%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

Alternative 7: 70.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e+102)
   (/ a c)
   (if (<= c 6.5e+132) (/ (fma (/ a d) c b) d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+102) {
		tmp = a / c;
	} else if (c <= 6.5e+132) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e+102)
		tmp = Float64(a / c);
	elseif (c <= 6.5e+132)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e+102], N[(a / c), $MachinePrecision], If[LessEqual[c, 6.5e+132], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.5e102 or 6.4999999999999994e132 < c

   1. Initial program 41.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

   if -7.5e102 < c < 6.4999999999999994e132

   1. Initial program 69.7%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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 71.7

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

   5. Applied rewrites 71.7%

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

Alternative 8: 62.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e+102) (/ a c) (if (<= c 7.8e-22) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e+102) {
		tmp = a / c;
	} else if (c <= 7.8e-22) {
		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 <= (-7.5d+102)) then
        tmp = a / c
    else if (c <= 7.8d-22) 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 <= -7.5e+102) {
		tmp = a / c;
	} else if (c <= 7.8e-22) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.5e+102:
		tmp = a / c
	elif c <= 7.8e-22:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e+102)
		tmp = Float64(a / c);
	elseif (c <= 7.8e-22)
		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 <= -7.5e+102)
		tmp = a / c;
	elseif (c <= 7.8e-22)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e+102], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.8e-22], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.5e102 or 7.79999999999999996e-22 < c

   1. Initial program 46.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

   if -7.5e102 < c < 7.79999999999999996e-22

   1. Initial program 70.7%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 56.4

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

   5. Applied rewrites 56.4%

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

Alternative 9: 43.8% 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 59.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 45.0

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

5. Applied rewrites 45.0%

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

Developer Target 1: 99.4% 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 2024271 
(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))))