Complex division, real part

Percentage Accurate: 61.0% → 83.7%

Time: 12.3s

Alternatives: 12

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: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(b, \frac{d}{t\_0}, \frac{\mathsf{fma}\left(a, c, 0\right)}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -7.2 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-159}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-52}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma b (/ d t_0) (/ (fma a c 0.0) t_0)))
        (t_2 (/ (fma d (/ b c) a) c)))
   (if (<= c -7.2e+102)
     t_2
     (if (<= c -2.05e-159)
       t_1
       (if (<= c 1.35e-52)
         (/ (fma a (/ c d) b) d)
         (if (<= c 1.9e+110) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(b, (d / t_0), (fma(a, c, 0.0) / t_0));
	double t_2 = fma(d, (b / c), a) / c;
	double tmp;
	if (c <= -7.2e+102) {
		tmp = t_2;
	} else if (c <= -2.05e-159) {
		tmp = t_1;
	} else if (c <= 1.35e-52) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.9e+110) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(b, Float64(d / t_0), Float64(fma(a, c, 0.0) / t_0))
	t_2 = Float64(fma(d, Float64(b / c), a) / c)
	tmp = 0.0
	if (c <= -7.2e+102)
		tmp = t_2;
	elseif (c <= -2.05e-159)
		tmp = t_1;
	elseif (c <= 1.35e-52)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.9e+110)
		tmp = t_1;
	else
		tmp = t_2;
	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[(b * N[(d / t$95$0), $MachinePrecision] + N[(N[(a * c + 0.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.2e+102], t$95$2, If[LessEqual[c, -2.05e-159], t$95$1, If[LessEqual[c, 1.35e-52], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.9e+110], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.2000000000000003e102 or 1.89999999999999994e110 < c

   1. Initial program 40.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{a \cdot c + 0}}{{c}^{2} + {d}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{\mathsf{fma}\left(a, c, 0\right)}}{{c}^{2} + {d}^{2}}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

      17. *-lowering-*.f64 37.8

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

   5. Simplified 37.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)} \]

   6. Taylor expanded in c around inf

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 82.3

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

   8. Simplified 82.3%

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

   if -7.2000000000000003e102 < c < -2.05000000000000007e-159 or 1.35000000000000005e-52 < c < 1.89999999999999994e110

   1. Initial program 81.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{a \cdot c + 0}}{{c}^{2} + {d}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{\mathsf{fma}\left(a, c, 0\right)}}{{c}^{2} + {d}^{2}}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

      17. *-lowering-*.f64 86.5

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)} \]

   if -2.05000000000000007e-159 < c < 1.35000000000000005e-52

   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 d around inf

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 93.9

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

   5. Simplified 93.9%

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

Alternative 2: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(a, c, d \cdot b\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{a + \frac{\mathsf{fma}\left(b, d, \frac{a \cdot \left(d \cdot d\right)}{0 - c}\right)}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -3.8e+48)
     t_0
     (if (<= d -2.6e-141)
       (* (fma a c (* d b)) (/ 1.0 (fma c c (* d d))))
       (if (<= d 9.2e-39)
         (/ (+ a (/ (fma b d (/ (* a (* d d)) (- 0.0 c))) c)) c)
         (if (<= d 2.5e+144)
           (/ (fma d b (* c a)) (+ (* c c) (* d d)))
           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 <= -3.8e+48) {
		tmp = t_0;
	} else if (d <= -2.6e-141) {
		tmp = fma(a, c, (d * b)) * (1.0 / fma(c, c, (d * d)));
	} else if (d <= 9.2e-39) {
		tmp = (a + (fma(b, d, ((a * (d * d)) / (0.0 - c))) / c)) / c;
	} else if (d <= 2.5e+144) {
		tmp = fma(d, b, (c * a)) / ((c * c) + (d * d));
	} 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 <= -3.8e+48)
		tmp = t_0;
	elseif (d <= -2.6e-141)
		tmp = Float64(fma(a, c, Float64(d * b)) * Float64(1.0 / fma(c, c, Float64(d * d))));
	elseif (d <= 9.2e-39)
		tmp = Float64(Float64(a + Float64(fma(b, d, Float64(Float64(a * Float64(d * d)) / Float64(0.0 - c))) / c)) / c);
	elseif (d <= 2.5e+144)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)));
	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, -3.8e+48], t$95$0, If[LessEqual[d, -2.6e-141], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-39], N[(N[(a + N[(N[(b * d + N[(N[(a * N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(0.0 - c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.5e+144], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.8e48 or 2.5e144 < d

   1. Initial program 27.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

   if -3.8e48 < d < -2.60000000000000011e-141

   1. Initial program 86.4%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. *-lowering-*.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. clear-num N/A

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

      8. flip-+ N/A

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

      9. /-lowering-/.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 86.4

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000011e-141 < d < 9.20000000000000033e-39

   1. Initial program 68.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{a + \left(-1 \cdot \frac{a \cdot {d}^{2}}{{c}^{2}} + \frac{b \cdot d}{c}\right)}{c}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{a + \color{blue}{\left(\frac{b \cdot d}{c} + -1 \cdot \frac{a \cdot {d}^{2}}{{c}^{2}}\right)}}{c} \]

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

         \[\leadsto \frac{a + \color{blue}{\left(\frac{b \cdot d}{c} - \frac{a \cdot {d}^{2}}{{c}^{2}}\right)}}{c} \]

      4. unpow2 N/A

         \[\leadsto \frac{a + \left(\frac{b \cdot d}{c} - \frac{a \cdot {d}^{2}}{\color{blue}{c \cdot c}}\right)}{c} \]

      5. associate-/r* N/A

         \[\leadsto \frac{a + \left(\frac{b \cdot d}{c} - \color{blue}{\frac{\frac{a \cdot {d}^{2}}{c}}{c}}\right)}{c} \]

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

         \[\leadsto \frac{a + \frac{b \cdot d + \color{blue}{-1 \cdot \frac{a \cdot {d}^{2}}{c}}}{c}}{c} \]

      9. +-commutative N/A

         \[\leadsto \frac{a + \frac{\color{blue}{-1 \cdot \frac{a \cdot {d}^{2}}{c} + b \cdot d}}{c}}{c} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{a + \frac{-1 \cdot \frac{a \cdot {d}^{2}}{c} + b \cdot d}{c}}{c}} \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{a + \frac{\mathsf{fma}\left(b, d, 0 - \frac{a \cdot \left(d \cdot d\right)}{c}\right)}{c}}{c}} \]

   if 9.20000000000000033e-39 < d < 2.5e144

   1. Initial program 91.8%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 91.8

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

   4. Applied egg-rr 91.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(a, c, d \cdot b\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-39}:\\ \;\;\;\;\frac{a + \frac{\mathsf{fma}\left(b, d, \frac{a \cdot \left(d \cdot d\right)}{0 - c}\right)}{c}}{c}\\ \mathbf{elif}\;d \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c + d \cdot d}\\ t_1 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* c a)) (+ (* c c) (* d d))))
        (t_1 (/ (fma a (/ c d) b) d)))
   (if (<= d -3.4e+48)
     t_1
     (if (<= d -2e-193)
       t_0
       (if (<= d 1.02e-36)
         (/ (fma b (/ d c) a) c)
         (if (<= d 1.15e+144) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (c * a)) / ((c * c) + (d * d));
	double t_1 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -3.4e+48) {
		tmp = t_1;
	} else if (d <= -2e-193) {
		tmp = t_0;
	} else if (d <= 1.02e-36) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 1.15e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -3.4e+48)
		tmp = t_1;
	elseif (d <= -2e-193)
		tmp = t_0;
	elseif (d <= 1.02e-36)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 1.15e+144)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.4e+48], t$95$1, If[LessEqual[d, -2e-193], t$95$0, If[LessEqual[d, 1.02e-36], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.15e+144], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.4000000000000003e48 or 1.1500000000000001e144 < d

   1. Initial program 27.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

   if -3.4000000000000003e48 < d < -2.0000000000000001e-193 or 1.02e-36 < d < 1.1500000000000001e144

   1. Initial program 87.5%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 87.5

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

   4. Applied egg-rr 87.5%

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

   if -2.0000000000000001e-193 < d < 1.02e-36

   1. Initial program 66.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 88.7

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

   5. Simplified 88.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -2 \cdot 10^{-193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{-36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-193}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+144}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a c (* d b)) (fma c c (* d d))))
        (t_1 (/ (fma a (/ c d) b) d)))
   (if (<= d -3.8e+48)
     t_1
     (if (<= d -3.5e-193)
       t_0
       (if (<= d 5.6e-39)
         (/ (fma b (/ d c) a) c)
         (if (<= d 1.26e+144) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, c, (d * b)) / fma(c, c, (d * d));
	double t_1 = fma(a, (c / d), b) / d;
	double tmp;
	if (d <= -3.8e+48) {
		tmp = t_1;
	} else if (d <= -3.5e-193) {
		tmp = t_0;
	} else if (d <= 5.6e-39) {
		tmp = fma(b, (d / c), a) / c;
	} else if (d <= 1.26e+144) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(a, Float64(c / d), b) / d)
	tmp = 0.0
	if (d <= -3.8e+48)
		tmp = t_1;
	elseif (d <= -3.5e-193)
		tmp = t_0;
	elseif (d <= 5.6e-39)
		tmp = Float64(fma(b, Float64(d / c), a) / c);
	elseif (d <= 1.26e+144)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.8e+48], t$95$1, If[LessEqual[d, -3.5e-193], t$95$0, If[LessEqual[d, 5.6e-39], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.26e+144], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.8e48 or 1.26000000000000001e144 < d

   1. Initial program 27.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 82.9

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

   5. Simplified 82.9%

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

   if -3.8e48 < d < -3.50000000000000005e-193 or 5.6000000000000003e-39 < d < 1.26000000000000001e144

   1. Initial program 87.5%

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

      1. /-lowering-/.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   if -3.50000000000000005e-193 < d < 5.6000000000000003e-39

   1. Initial program 66.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 88.7

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

   5. Simplified 88.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;d \leq 1.26 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-81}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.28e+92)
   (/ b d)
   (if (<= d -1.1e-81)
     (* b (/ d (fma d d (* c c))))
     (if (<= d 8.2e-13)
       (/ a c)
       (if (<= d 1.15e+144) (/ (fma b d (* c a)) (* d d)) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.28e+92) {
		tmp = b / d;
	} else if (d <= -1.1e-81) {
		tmp = b * (d / fma(d, d, (c * c)));
	} else if (d <= 8.2e-13) {
		tmp = a / c;
	} else if (d <= 1.15e+144) {
		tmp = fma(b, d, (c * a)) / (d * d);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.28e+92)
		tmp = Float64(b / d);
	elseif (d <= -1.1e-81)
		tmp = Float64(b * Float64(d / fma(d, d, Float64(c * c))));
	elseif (d <= 8.2e-13)
		tmp = Float64(a / c);
	elseif (d <= 1.15e+144)
		tmp = Float64(fma(b, d, Float64(c * a)) / Float64(d * d));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.28e+92], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.1e-81], N[(b * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.2e-13], N[(a / c), $MachinePrecision], If[LessEqual[d, 1.15e+144], N[(N[(b * d + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.27999999999999996e92 or 1.1500000000000001e144 < d

   1. Initial program 27.0%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 78.9

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

   5. Simplified 78.9%

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

   if -1.27999999999999996e92 < d < -1.1e-81

   1. Initial program 67.6%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + 0}}{{c}^{2} + {d}^{2}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, d, 0\right)}}{{c}^{2} + {d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

      8. *-lowering-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 64.9

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

   7. Applied egg-rr 64.9%

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

   if -1.1e-81 < d < 8.2000000000000004e-13

   1. Initial program 70.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 70.9

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

   5. Simplified 70.9%

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

   if 8.2000000000000004e-13 < d < 1.1500000000000001e144

   1. Initial program 93.0%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 81.2

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

   5. Simplified 81.2%

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

   6. Taylor expanded in d around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 81.2

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

   8. Simplified 81.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-81}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 1.15 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, d, c \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-89}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.28e+92)
   (/ b d)
   (if (<= d -1.1e-89)
     (* b (/ d (fma d d (* c c))))
     (if (<= d 1.4e-12)
       (/ a c)
       (if (<= d 3.2e+144) (/ (fma a c (* d b)) (* d d)) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.28e+92) {
		tmp = b / d;
	} else if (d <= -1.1e-89) {
		tmp = b * (d / fma(d, d, (c * c)));
	} else if (d <= 1.4e-12) {
		tmp = a / c;
	} else if (d <= 3.2e+144) {
		tmp = fma(a, c, (d * b)) / (d * d);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.28e+92)
		tmp = Float64(b / d);
	elseif (d <= -1.1e-89)
		tmp = Float64(b * Float64(d / fma(d, d, Float64(c * c))));
	elseif (d <= 1.4e-12)
		tmp = Float64(a / c);
	elseif (d <= 3.2e+144)
		tmp = Float64(fma(a, c, Float64(d * b)) / Float64(d * d));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.28e+92], N[(b / d), $MachinePrecision], If[LessEqual[d, -1.1e-89], N[(b * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-12], N[(a / c), $MachinePrecision], If[LessEqual[d, 3.2e+144], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.27999999999999996e92 or 3.2000000000000001e144 < d

   1. Initial program 27.0%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 78.9

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

   5. Simplified 78.9%

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

   if -1.27999999999999996e92 < d < -1.10000000000000006e-89

   1. Initial program 67.6%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + 0}}{{c}^{2} + {d}^{2}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, d, 0\right)}}{{c}^{2} + {d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

      8. *-lowering-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 64.9

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

   7. Applied egg-rr 64.9%

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

   if -1.10000000000000006e-89 < d < 1.4000000000000001e-12

   1. Initial program 70.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 70.9

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

   5. Simplified 70.9%

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

   if 1.4000000000000001e-12 < d < 3.2000000000000001e144

   1. Initial program 93.0%

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

      1. /-lowering-/.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 93.0

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in c around 0

      \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{{d}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.1

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

   7. Simplified 81.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -1.1 \cdot 10^{-89}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-90}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-38}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 10^{+27}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.28e+92)
   (/ b d)
   (if (<= d -3e-90)
     (* b (/ d (fma d d (* c c))))
     (if (<= d 1.75e-38)
       (/ a c)
       (if (<= d 1e+27) (* a (/ c (fma c c (* d d)))) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.28e+92) {
		tmp = b / d;
	} else if (d <= -3e-90) {
		tmp = b * (d / fma(d, d, (c * c)));
	} else if (d <= 1.75e-38) {
		tmp = a / c;
	} else if (d <= 1e+27) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.28e+92)
		tmp = Float64(b / d);
	elseif (d <= -3e-90)
		tmp = Float64(b * Float64(d / fma(d, d, Float64(c * c))));
	elseif (d <= 1.75e-38)
		tmp = Float64(a / c);
	elseif (d <= 1e+27)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.28e+92], N[(b / d), $MachinePrecision], If[LessEqual[d, -3e-90], N[(b * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.75e-38], N[(a / c), $MachinePrecision], If[LessEqual[d, 1e+27], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.27999999999999996e92 or 1e27 < d

   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 c around 0

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

      1. /-lowering-/.f64 75.0

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

   5. Simplified 75.0%

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

   if -1.27999999999999996e92 < d < -3.0000000000000002e-90

   1. Initial program 67.6%

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

   3. Taylor expanded in a around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{b \cdot d + 0}}{{c}^{2} + {d}^{2}} \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, d, 0\right)}}{{c}^{2} + {d}^{2}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

      8. *-lowering-*.f64 51.4

         \[\leadsto \frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)} \]

   5. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(b, d, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 64.9

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

   7. Applied egg-rr 64.9%

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

   if -3.0000000000000002e-90 < d < 1.7500000000000001e-38

   1. Initial program 69.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 71.6

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

   5. Simplified 71.6%

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

   if 1.7500000000000001e-38 < d < 1e27

   1. Initial program 89.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{a \cdot c + 0}}{{c}^{2} + {d}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{\mathsf{fma}\left(a, c, 0\right)}}{{c}^{2} + {d}^{2}}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

      17. *-lowering-*.f64 89.3

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)} \]

   6. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.1

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

   8. Simplified 63.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.28 \cdot 10^{+92}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-90}:\\ \;\;\;\;b \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-38}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 10^{+27}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -24000:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -24000.0)
   (/ a c)
   (if (<= c 4000000000000.0)
     (/ (fma a (/ c d) b) d)
     (if (<= c 1.9e+142) (/ (fma a c (* d b)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -24000.0) {
		tmp = a / c;
	} else if (c <= 4000000000000.0) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 1.9e+142) {
		tmp = fma(a, c, (d * b)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -24000.0)
		tmp = Float64(a / c);
	elseif (c <= 4000000000000.0)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 1.9e+142)
		tmp = Float64(fma(a, c, Float64(d * b)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -24000.0], N[(a / c), $MachinePrecision], If[LessEqual[c, 4000000000000.0], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.9e+142], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -24000 or 1.89999999999999995e142 < c

   1. Initial program 43.0%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 69.4

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

   5. Simplified 69.4%

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

   if -24000 < c < 4e12

   1. Initial program 72.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 85.2

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

   5. Simplified 85.2%

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

   if 4e12 < c < 1.89999999999999995e142

   1. Initial program 74.4%

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

      1. /-lowering-/.f64 N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 74.4

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

   4. Applied egg-rr 74.4%

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

   5. Taylor expanded in c around inf

      \[\leadsto \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\color{blue}{{c}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. *-lowering-*.f64 70.6

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

   7. Simplified 70.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -24000:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-36}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+26}:\\ \;\;\;\;a \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.9e-8)
   (/ b d)
   (if (<= d 1.7e-36)
     (/ a c)
     (if (<= d 7e+26) (* a (/ c (fma c c (* d d)))) (/ b d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.9e-8) {
		tmp = b / d;
	} else if (d <= 1.7e-36) {
		tmp = a / c;
	} else if (d <= 7e+26) {
		tmp = a * (c / fma(c, c, (d * d)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.9e-8)
		tmp = Float64(b / d);
	elseif (d <= 1.7e-36)
		tmp = Float64(a / c);
	elseif (d <= 7e+26)
		tmp = Float64(a * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.9e-8], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.7e-36], N[(a / c), $MachinePrecision], If[LessEqual[d, 7e+26], N[(a * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.90000000000000014e-8 or 6.9999999999999998e26 < d

   1. Initial program 44.7%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 72.0

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

   5. Simplified 72.0%

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

   if -1.90000000000000014e-8 < d < 1.7000000000000001e-36

   1. Initial program 71.2%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 68.3

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

   5. Simplified 68.3%

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

   if 1.7000000000000001e-36 < d < 6.9999999999999998e26

   1. Initial program 89.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{a \cdot c + 0}}{{c}^{2} + {d}^{2}}\right) \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\color{blue}{\mathsf{fma}\left(a, c, 0\right)}}{{c}^{2} + {d}^{2}}\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{{d}^{2} + {c}^{2}}}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{d \cdot d} + {c}^{2}}\right) \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}}\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

      17. *-lowering-*.f64 89.3

          \[\leadsto \mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}\right) \]

   5. Simplified 89.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{\mathsf{fma}\left(a, c, 0\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)} \]

   6. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 63.1

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

   8. Simplified 63.1%

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

Alternative 10: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -3.2e-10) t_0 (if (<= d 4.8e-16) (/ (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 <= -3.2e-10) {
		tmp = t_0;
	} else if (d <= 4.8e-16) {
		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 <= -3.2e-10)
		tmp = t_0;
	elseif (d <= 4.8e-16)
		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, -3.2e-10], t$95$0, If[LessEqual[d, 4.8e-16], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.19999999999999981e-10 or 4.8000000000000001e-16 < d

   1. Initial program 49.6%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 80.4

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

   5. Simplified 80.4%

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

   if -3.19999999999999981e-10 < d < 4.8000000000000001e-16

   1. Initial program 72.1%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. /-lowering-/.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. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 82.4

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

   5. Simplified 82.4%

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

Alternative 11: 62.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -9.5e-9) (/ b d) (if (<= d 3.8e-12) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.5e-9) {
		tmp = b / d;
	} else if (d <= 3.8e-12) {
		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 <= (-9.5d-9)) then
        tmp = b / d
    else if (d <= 3.8d-12) 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 <= -9.5e-9) {
		tmp = b / d;
	} else if (d <= 3.8e-12) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -9.5e-9:
		tmp = b / d
	elif d <= 3.8e-12:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9.5e-9)
		tmp = Float64(b / d);
	elseif (d <= 3.8e-12)
		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 <= -9.5e-9)
		tmp = b / d;
	elseif (d <= 3.8e-12)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9.5e-9], N[(b / d), $MachinePrecision], If[LessEqual[d, 3.8e-12], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000007e-9 or 3.79999999999999996e-12 < d

   1. Initial program 49.1%

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

   3. Taylor expanded in c around 0

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

      1. /-lowering-/.f64 67.9

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

   5. Simplified 67.9%

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

   if -9.5000000000000007e-9 < d < 3.79999999999999996e-12

   1. Initial program 72.0%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 67.8

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

   5. Simplified 67.8%

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

Alternative 12: 41.9% 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 60.8%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 43.0

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

5. Simplified 43.0%

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

Developer Target 1: 99.3% 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 2024195 
(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))))