Complex division, real part

Percentage Accurate: 61.9% → 80.9%

Time: 6.4s

Alternatives: 8

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: 61.9% 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: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{t\_0}, d \cdot \frac{b}{t\_0}\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma (/ a d) c b) d)))
   (if (<= d -5e+156)
     t_1
     (if (<= d -4e-73)
       (fma a (/ c t_0) (* d (/ b t_0)))
       (if (<= d 4.6e+48) (/ (fma (/ d c) b a) c) t_1)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -5e+156) {
		tmp = t_1;
	} else if (d <= -4e-73) {
		tmp = fma(a, (c / t_0), (d * (b / t_0)));
	} else if (d <= 4.6e+48) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -5e+156)
		tmp = t_1;
	elseif (d <= -4e-73)
		tmp = fma(a, Float64(c / t_0), Float64(d * Float64(b / t_0)));
	elseif (d <= 4.6e+48)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -5e+156], t$95$1, If[LessEqual[d, -4e-73], N[(a * N[(c / t$95$0), $MachinePrecision] + N[(d * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+48], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.99999999999999992e156 or 4.6e48 < d

   1. Initial program 35.9%

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

   3. Taylor expanded in c around 0

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

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if -4.99999999999999992e156 < d < -3.99999999999999999e-73

   1. Initial program 72.5%

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

   4. Applied rewrites 77.8%

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

   if -3.99999999999999999e-73 < d < 4.6e48

   1. Initial program 73.1%

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. 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 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -4 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, d \cdot \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{if}\;d \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ a d) c b) d)))
   (if (<= d -1.12e+83)
     t_0
     (if (<= d -1.9e-74)
       (/ (fma c a (* d b)) (+ (* c c) (* d d)))
       (if (<= d 4.6e+48) (/ (fma (/ d c) b a) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((a / d), c, b) / d;
	double tmp;
	if (d <= -1.12e+83) {
		tmp = t_0;
	} else if (d <= -1.9e-74) {
		tmp = fma(c, a, (d * b)) / ((c * c) + (d * d));
	} else if (d <= 4.6e+48) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(a / d), c, b) / d)
	tmp = 0.0
	if (d <= -1.12e+83)
		tmp = t_0;
	elseif (d <= -1.9e-74)
		tmp = Float64(fma(c, a, Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 4.6e+48)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.12e+83], t$95$0, If[LessEqual[d, -1.9e-74], N[(N[(c * a + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.6e+48], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.12e83 or 4.6e48 < d

   1. Initial program 40.6%

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

   3. Taylor expanded in c around 0

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

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -1.12e83 < d < -1.8999999999999998e-74

   1. Initial program 78.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sign-sub-inv N/A

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

      5. fp-cancel-sub-sign-inv N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 78.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 78.2

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

   4. Applied rewrites 78.2%

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

   if -1.8999999999999998e-74 < d < 4.6e48

   1. Initial program 73.1%

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

   4. Applied rewrites 69.2%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. 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 85.6

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

   7. Applied rewrites 85.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5e+156)
   (/ b d)
   (if (<= d -1.1e-26)
     (* (/ d (fma d d (* c c))) b)
     (if (<= d 4.4e+145) (/ (fma (/ d c) b a) c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5e+156) {
		tmp = b / d;
	} else if (d <= -1.1e-26) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 4.4e+145) {
		tmp = fma((d / c), b, a) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5e+156)
		tmp = Float64(b / d);
	elseif (d <= -1.1e-26)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 4.4e+145)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5e+156], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.1e-26], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 4.4e+145], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.99999999999999992e156 or 4.40000000000000017e145 < d

   1. Initial program 32.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -4.99999999999999992e156 < d < -1.1e-26

   1. Initial program 75.0%

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 69.8

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

   7. Applied rewrites 69.8%

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

   if -1.1e-26 < d < 4.40000000000000017e145

   1. Initial program 70.0%

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

   4. Applied rewrites 69.0%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. 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 79.1

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

   7. Applied rewrites 79.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-26}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+145}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-26} \lor \neg \left(d \leq 4.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.5e-26) (not (<= d 4.6e+48)))
   (/ (fma (/ a d) c b) d)
   (/ (fma (/ d c) b a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.5e-26) || !(d <= 4.6e+48)) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = fma((d / c), b, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.5e-26) || !(d <= 4.6e+48))
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.5e-26], N[Not[LessEqual[d, 4.6e+48]], $MachinePrecision]], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.4999999999999999e-26 or 4.6e48 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in c around 0

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

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if -4.4999999999999999e-26 < d < 4.6e48

   1. Initial program 72.3%

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. 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 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-26} \lor \neg \left(d \leq 4.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-26} \lor \neg \left(d \leq 4.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.5e-26) (not (<= d 4.6e+48)))
   (/ (fma a (/ c d) b) d)
   (/ (fma (/ d c) b a) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.5e-26) || !(d <= 4.6e+48)) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = fma((d / c), b, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.5e-26) || !(d <= 4.6e+48))
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.5e-26], N[Not[LessEqual[d, 4.6e+48]], $MachinePrecision]], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.4999999999999999e-26 or 4.6e48 < d

   1. Initial program 46.3%

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

   3. Taylor expanded in c around 0

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

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

   7. Applied rewrites 78.1%

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

   if -4.4999999999999999e-26 < d < 4.6e48

   1. Initial program 72.3%

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \frac{b \cdot d}{c}}{c}} \]
   6. 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 83.1

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

   7. Applied rewrites 83.1%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-26} \lor \neg \left(d \leq 4.6 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5e+156)
   (/ b d)
   (if (<= d -2.8e-38)
     (* (/ d (fma d d (* c c))) b)
     (if (<= d 2.3e-10) (/ a c) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5e+156) {
		tmp = b / d;
	} else if (d <= -2.8e-38) {
		tmp = (d / fma(d, d, (c * c))) * b;
	} else if (d <= 2.3e-10) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5e+156)
		tmp = Float64(b / d);
	elseif (d <= -2.8e-38)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * b);
	elseif (d <= 2.3e-10)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5e+156], N[(b / d), $MachinePrecision], If[LessEqual[d, -2.8e-38], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[d, 2.3e-10], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.99999999999999992e156 or 2.30000000000000007e-10 < d

   1. Initial program 38.8%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -4.99999999999999992e156 < d < -2.8e-38

   1. Initial program 75.7%

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

   4. Applied rewrites 79.8%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if -2.8e-38 < d < 2.30000000000000007e-10

   1. Initial program 73.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-33} \lor \neg \left(d \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.95e-33) (not (<= d 2.3e-10))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.95e-33) || !(d <= 2.3e-10)) {
		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 <= (-1.95d-33)) .or. (.not. (d <= 2.3d-10))) 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 <= -1.95e-33) || !(d <= 2.3e-10)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.95e-33) or not (d <= 2.3e-10):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.95e-33) || !(d <= 2.3e-10))
		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 <= -1.95e-33) || ~((d <= 2.3e-10)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.95e-33], N[Not[LessEqual[d, 2.3e-10]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.94999999999999987e-33 or 2.30000000000000007e-10 < d

   1. Initial program 47.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -1.94999999999999987e-33 < d < 2.30000000000000007e-10

   1. Initial program 73.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.6

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

   5. Applied rewrites 68.6%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-33} \lor \neg \left(d \leq 2.3 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.4% 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 59.0%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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: 99.4% 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 2024343 
(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))))