Complex division, real part

Percentage Accurate: 61.6% → 79.3%

Time: 7.2s

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: 61.6% 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: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.0001107)
   (/ (fma a (/ c d) b) d)
   (if (<= d 9.5e-64)
     (/ (fma d (/ b c) a) c)
     (if (<= d 1.85e+75)
       (/ (+ (* a c) (* d b)) (+ (* c c) (* d d)))
       (/ (fma (/ a d) c b) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = fma(a, (c / d), b) / d;
	} else if (d <= 9.5e-64) {
		tmp = fma(d, (b / c), a) / c;
	} else if (d <= 1.85e+75) {
		tmp = ((a * c) + (d * b)) / ((c * c) + (d * d));
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (d <= 9.5e-64)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	elseif (d <= 1.85e+75)
		tmp = Float64(Float64(Float64(a * c) + Float64(d * b)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.0001107], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 9.5e-64], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.85e+75], N[(N[(N[(a * c), $MachinePrecision] + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.10699999999999994e-4

   1. Initial program 57.4%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -1.10699999999999994e-4 < d < 9.50000000000000043e-64

   1. Initial program 70.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 90.9

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

   8. Applied rewrites 90.9%

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

   if 9.50000000000000043e-64 < d < 1.85000000000000005e75

   1. Initial program 91.3%

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

   if 1.85000000000000005e75 < d

   1. Initial program 52.3%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 90.7%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+75}:\\ \;\;\;\;\frac{a \cdot c + d \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.0001107)
   (/ b d)
   (if (<= d 8.8e-244)
     (/ a c)
     (if (<= d 2.15e-44)
       (/ (fma b d (* a c)) (* c c))
       (if (<= d 4.4e+155) (/ (fma c a (* d b)) (* d d)) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = b / d;
	} else if (d <= 8.8e-244) {
		tmp = a / c;
	} else if (d <= 2.15e-44) {
		tmp = fma(b, d, (a * c)) / (c * c);
	} else if (d <= 4.4e+155) {
		tmp = fma(c, a, (d * b)) / (d * d);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = Float64(b / d);
	elseif (d <= 8.8e-244)
		tmp = Float64(a / c);
	elseif (d <= 2.15e-44)
		tmp = Float64(fma(b, d, Float64(a * c)) / Float64(c * c));
	elseif (d <= 4.4e+155)
		tmp = Float64(fma(c, a, Float64(d * b)) / Float64(d * d));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.0001107], N[(b / d), $MachinePrecision], If[LessEqual[d, 8.8e-244], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.15e-44], N[(N[(b * d + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e+155], N[(N[(c * a + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.10699999999999994e-4 or 4.4000000000000005e155 < d

   1. Initial program 53.5%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.10699999999999994e-4 < d < 8.79999999999999939e-244

   1. Initial program 64.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 8.79999999999999939e-244 < d < 2.15000000000000007e-44

   1. Initial program 80.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 89.0

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

   8. Applied rewrites 89.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 73.9%

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

   if 2.15000000000000007e-44 < d < 4.4000000000000005e155

   1. Initial program 79.7%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 65.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, a, d \cdot b\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -0.0001107)
     t_0
     (if (<= d 8.8e-244)
       (/ a c)
       (if (<= d 2.15e-44) (/ (fma b d (* a c)) (* c c)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -0.0001107) {
		tmp = t_0;
	} else if (d <= 8.8e-244) {
		tmp = a / c;
	} else if (d <= 2.15e-44) {
		tmp = fma(b, d, (a * c)) / (c * c);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = t_0;
	elseif (d <= 8.8e-244)
		tmp = Float64(a / c);
	elseif (d <= 2.15e-44)
		tmp = Float64(fma(b, d, Float64(a * c)) / Float64(c * c));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -0.0001107], t$95$0, If[LessEqual[d, 8.8e-244], N[(a / c), $MachinePrecision], If[LessEqual[d, 2.15e-44], N[(N[(b * d + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10699999999999994e-4 or 2.15000000000000007e-44 < d

   1. Initial program 59.9%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.10699999999999994e-4 < d < 8.79999999999999939e-244

   1. Initial program 64.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 8.79999999999999939e-244 < d < 2.15000000000000007e-44

   1. Initial program 80.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 89.0

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

   8. Applied rewrites 89.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 73.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.0001107)
   (/ b d)
   (if (<= d 8.8e-244)
     (/ a c)
     (if (<= d 1.2e-42) (/ (fma b d (* a c)) (* c c)) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = b / d;
	} else if (d <= 8.8e-244) {
		tmp = a / c;
	} else if (d <= 1.2e-42) {
		tmp = fma(b, d, (a * c)) / (c * c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = Float64(b / d);
	elseif (d <= 8.8e-244)
		tmp = Float64(a / c);
	elseif (d <= 1.2e-42)
		tmp = Float64(fma(b, d, Float64(a * c)) / Float64(c * c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.0001107], N[(b / d), $MachinePrecision], If[LessEqual[d, 8.8e-244], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.2e-42], N[(N[(b * d + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10699999999999994e-4 or 1.20000000000000001e-42 < d

   1. Initial program 59.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -1.10699999999999994e-4 < d < 8.79999999999999939e-244

   1. Initial program 64.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if 8.79999999999999939e-244 < d < 1.20000000000000001e-42

   1. Initial program 80.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 64.2

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

   5. Applied rewrites 64.2%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 89.0

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

   8. Applied rewrites 89.0%

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

   9. Taylor expanded in c around 0

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

   11. Applied rewrites 73.9%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 8.8 \cdot 10^{-244}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.0001107)
   (/ (fma a (/ c d) b) d)
   (if (<= d 1.2e-42) (/ (fma d (/ b c) a) c) (/ (fma (/ a d) c b) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = fma(a, (c / d), b) / d;
	} else if (d <= 1.2e-42) {
		tmp = fma(d, (b / c), a) / c;
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (d <= 1.2e-42)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.0001107], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.2e-42], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.10699999999999994e-4

   1. Initial program 57.4%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if -1.10699999999999994e-4 < d < 1.20000000000000001e-42

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

   if 1.20000000000000001e-42 < d

   1. Initial program 62.4%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.4

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

   5. Applied rewrites 82.4%

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

   7. Applied rewrites 83.7%

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

Alternative 6: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -0.0001107) t_0 (if (<= d 1.2e-42) (/ (fma d (/ b c) a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -0.0001107) {
		tmp = t_0;
	} else if (d <= 1.2e-42) {
		tmp = fma(d, (b / c), a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = t_0;
	elseif (d <= 1.2e-42)
		tmp = Float64(fma(d, Float64(b / c), a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -0.0001107], t$95$0, If[LessEqual[d, 1.2e-42], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.10699999999999994e-4 or 1.20000000000000001e-42 < d

   1. Initial program 59.9%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.10699999999999994e-4 < d < 1.20000000000000001e-42

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.4

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

   5. Applied rewrites 69.4%

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

   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. *-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. lower-fma.f64 N/A

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

      6. lower-/.f64 90.0

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

   8. Applied rewrites 90.0%

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

Alternative 7: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -0.0001107) t_0 (if (<= d 1.2e-42) (/ (fma b (/ d c) a) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -0.0001107) {
		tmp = t_0;
	} else if (d <= 1.2e-42) {
		tmp = fma(b, (d / c), a) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = t_0;
	elseif (d <= 1.2e-42)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -0.0001107], t$95$0, If[LessEqual[d, 1.2e-42], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.10699999999999994e-4 or 1.20000000000000001e-42 < d

   1. Initial program 59.9%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 82.9

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

   5. Applied rewrites 82.9%

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

   if -1.10699999999999994e-4 < d < 1.20000000000000001e-42

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.2

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

   5. Applied rewrites 89.2%

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

Alternative 8: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0001107:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -0.0001107) (/ b d) (if (<= d 1.15e-54) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = b / d;
	} else if (d <= 1.15e-54) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-0.0001107d0)) then
        tmp = b / d
    else if (d <= 1.15d-54) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -0.0001107) {
		tmp = b / d;
	} else if (d <= 1.15e-54) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -0.0001107:
		tmp = b / d
	elif d <= 1.15e-54:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -0.0001107)
		tmp = Float64(b / d);
	elseif (d <= 1.15e-54)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -0.0001107)
		tmp = b / d;
	elseif (d <= 1.15e-54)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -0.0001107], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.15e-54], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.10699999999999994e-4 or 1.1499999999999999e-54 < d

   1. Initial program 60.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   if -1.10699999999999994e-4 < d < 1.1499999999999999e-54

   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 70.0

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

   5. Applied rewrites 70.0%

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

Alternative 9: 42.6% 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 65.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.5

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

5. Applied rewrites 42.5%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024235 
(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))))