Complex division, real part

Percentage Accurate: 62.4% → 83.7%

Time: 6.3s

Alternatives: 10

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.46 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d b (* a c)) (fma d d (* c c))))
        (t_1 (/ (fma (/ b c) d a) c)))
   (if (<= c -6.2e+120)
     t_1
     (if (<= c -5.8e-116)
       t_0
       (if (<= c 1.3e-129)
         (/ (fma (/ c d) a b) d)
         (if (<= c 1.46e+85) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, b, (a * c)) / fma(d, d, (c * c));
	double t_1 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -6.2e+120) {
		tmp = t_1;
	} else if (c <= -5.8e-116) {
		tmp = t_0;
	} else if (c <= 1.3e-129) {
		tmp = fma((c / d), a, b) / d;
	} else if (c <= 1.46e+85) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, b, Float64(a * c)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -6.2e+120)
		tmp = t_1;
	elseif (c <= -5.8e-116)
		tmp = t_0;
	elseif (c <= 1.3e-129)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	elseif (c <= 1.46e+85)
		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[(a * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.2e+120], t$95$1, If[LessEqual[c, -5.8e-116], t$95$0, If[LessEqual[c, 1.3e-129], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.46e+85], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.19999999999999947e120 or 1.46e85 < c

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -6.19999999999999947e120 < c < -5.7999999999999996e-116 or 1.3e-129 < c < 1.46e85

   1. Initial program 84.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 84.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 84.1

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

   4. Applied rewrites 84.1%

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

   if -5.7999999999999996e-116 < c < 1.3e-129

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 72.2

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 72.2

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

   4. Applied rewrites 72.2%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 91.1

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

   7. Applied rewrites 91.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{elif}\;c \leq -5.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{elif}\;c \leq 1.46 \cdot 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{d \cdot d}\\ \mathbf{elif}\;c \leq 1700:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e+120)
   (/ a c)
   (if (<= c -5.5e-39)
     (* (/ a (fma d d (* c c))) c)
     (if (<= c 1.25e-193)
       (/ (fma d b (* a c)) (* d d))
       (if (<= c 1700.0)
         (/ b d)
         (if (<= c 3.85e+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 <= -6.8e+120) {
		tmp = a / c;
	} else if (c <= -5.5e-39) {
		tmp = (a / fma(d, d, (c * c))) * c;
	} else if (c <= 1.25e-193) {
		tmp = fma(d, b, (a * c)) / (d * d);
	} else if (c <= 1700.0) {
		tmp = b / d;
	} else if (c <= 3.85e+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 <= -6.8e+120)
		tmp = Float64(a / c);
	elseif (c <= -5.5e-39)
		tmp = Float64(Float64(a / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1.25e-193)
		tmp = Float64(fma(d, b, Float64(a * c)) / Float64(d * d));
	elseif (c <= 1700.0)
		tmp = Float64(b / d);
	elseif (c <= 3.85e+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, -6.8e+120], N[(a / c), $MachinePrecision], If[LessEqual[c, -5.5e-39], N[(N[(a / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1.25e-193], N[(N[(d * b + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1700.0], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.85e+142], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -6.79999999999999998e120 or 3.85000000000000024e142 < c

   1. Initial program 35.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -6.79999999999999998e120 < c < -5.50000000000000018e-39

   1. Initial program 77.6%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 62.1

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

   5. Applied rewrites 62.1%

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

   if -5.50000000000000018e-39 < c < 1.2500000000000001e-193

   1. Initial program 78.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.4

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.4

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 78.4

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 73.5

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

   7. Applied rewrites 73.5%

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

   if 1.2500000000000001e-193 < c < 1700

   1. Initial program 76.2%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 1700 < c < 3.85000000000000024e142

   1. Initial program 78.4%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 70.3

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

   7. Applied rewrites 70.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{c \cdot c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-193}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{d \cdot d}\\ \mathbf{elif}\;c \leq 1700:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 1700:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e+120)
   (/ a c)
   (if (<= c -2.6e-50)
     (* (/ a (fma d d (* c c))) c)
     (if (<= c 1700.0)
       (/ b d)
       (if (<= c 3.85e+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 <= -6.8e+120) {
		tmp = a / c;
	} else if (c <= -2.6e-50) {
		tmp = (a / fma(d, d, (c * c))) * c;
	} else if (c <= 1700.0) {
		tmp = b / d;
	} else if (c <= 3.85e+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 <= -6.8e+120)
		tmp = Float64(a / c);
	elseif (c <= -2.6e-50)
		tmp = Float64(Float64(a / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1700.0)
		tmp = Float64(b / d);
	elseif (c <= 3.85e+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, -6.8e+120], N[(a / c), $MachinePrecision], If[LessEqual[c, -2.6e-50], N[(N[(a / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1700.0], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.85e+142], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.79999999999999998e120 or 3.85000000000000024e142 < c

   1. Initial program 35.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -6.79999999999999998e120 < c < -2.6000000000000001e-50

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -2.6000000000000001e-50 < c < 1700

   1. Initial program 77.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 68.7

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

   5. Applied rewrites 68.7%

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

   if 1700 < c < 3.85000000000000024e142

   1. Initial program 78.4%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 70.3

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

   7. Applied rewrites 70.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 1700:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 18000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.2e+120)
   (/ a c)
   (if (<= c 18000000000000.0)
     (/ (fma (/ a d) c b) d)
     (if (<= c 3.85e+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 <= -5.2e+120) {
		tmp = a / c;
	} else if (c <= 18000000000000.0) {
		tmp = fma((a / d), c, b) / d;
	} else if (c <= 3.85e+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 <= -5.2e+120)
		tmp = Float64(a / c);
	elseif (c <= 18000000000000.0)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (c <= 3.85e+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, -5.2e+120], N[(a / c), $MachinePrecision], If[LessEqual[c, 18000000000000.0], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.85e+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 < -5.1999999999999998e120 or 3.85000000000000024e142 < c

   1. Initial program 35.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -5.1999999999999998e120 < c < 1.8e13

   1. Initial program 77.8%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if 1.8e13 < c < 3.85000000000000024e142

   1. Initial program 77.8%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 72.1

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

   7. Applied rewrites 72.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 18000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;c \leq 3.85 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, b, a\right)}{c}\\ \mathbf{elif}\;c \leq 18000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.8)
   (/ (fma (/ d c) b a) c)
   (if (<= c 18000000000000.0)
     (/ (fma (/ c d) a b) d)
     (/ (fma (/ b c) d a) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8) {
		tmp = fma((d / c), b, a) / c;
	} else if (c <= 18000000000000.0) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = fma((b / c), d, a) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.8)
		tmp = Float64(fma(Float64(d / c), b, a) / c);
	elseif (c <= 18000000000000.0)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.8], N[(N[(N[(d / c), $MachinePrecision] * b + a), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 18000000000000.0], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.79999999999999982

   1. Initial program 51.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 51.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 51.0

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -5.79999999999999982 < c < 1.8e13

   1. Initial program 78.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if 1.8e13 < c

   1. Initial program 57.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 86.5

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

   5. Applied rewrites 86.5%

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

Alternative 6: 78.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -5.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 18000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -5.8)
     t_0
     (if (<= c 18000000000000.0) (/ (fma (/ c d) a b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -5.8) {
		tmp = t_0;
	} else if (c <= 18000000000000.0) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -5.8)
		tmp = t_0;
	elseif (c <= 18000000000000.0)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -5.8], t$95$0, If[LessEqual[c, 18000000000000.0], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.79999999999999982 or 1.8e13 < c

   1. Initial program 54.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -5.79999999999999982 < c < 1.8e13

   1. Initial program 78.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 78.1

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 78.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 78.1

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

Alternative 7: 77.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -5.8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 18000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -5.8)
     t_0
     (if (<= c 18000000000000.0) (/ (fma (/ a d) c b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -5.8) {
		tmp = t_0;
	} else if (c <= 18000000000000.0) {
		tmp = fma((a / d), c, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -5.8)
		tmp = t_0;
	elseif (c <= 18000000000000.0)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -5.8], t$95$0, If[LessEqual[c, 18000000000000.0], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.79999999999999982 or 1.8e13 < c

   1. Initial program 54.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -5.79999999999999982 < c < 1.8e13

   1. Initial program 78.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

Alternative 8: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+120}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 48000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e+120)
   (/ a c)
   (if (<= c -2.6e-50)
     (* (/ a (fma d d (* c c))) c)
     (if (<= c 48000.0) (/ b d) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.8e+120) {
		tmp = a / c;
	} else if (c <= -2.6e-50) {
		tmp = (a / fma(d, d, (c * c))) * c;
	} else if (c <= 48000.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.8e+120)
		tmp = Float64(a / c);
	elseif (c <= -2.6e-50)
		tmp = Float64(Float64(a / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 48000.0)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.8e+120], N[(a / c), $MachinePrecision], If[LessEqual[c, -2.6e-50], N[(N[(a / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 48000.0], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.79999999999999998e120 or 48000 < c

   1. Initial program 49.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -6.79999999999999998e120 < c < -2.6000000000000001e-50

   1. Initial program 79.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.1

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

   5. Applied rewrites 60.1%

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

   if -2.6000000000000001e-50 < c < 48000

   1. Initial program 77.1%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 68.7

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

   5. Applied rewrites 68.7%

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

Alternative 9: 63.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 48000:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.4e-34) (/ a c) (if (<= c 48000.0) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.4e-34) {
		tmp = a / c;
	} else if (c <= 48000.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.4d-34)) then
        tmp = a / c
    else if (c <= 48000.0d0) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.4e-34) {
		tmp = a / c;
	} else if (c <= 48000.0) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.4e-34:
		tmp = a / c
	elif c <= 48000.0:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.4e-34)
		tmp = Float64(a / c);
	elseif (c <= 48000.0)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.4e-34)
		tmp = a / c;
	elseif (c <= 48000.0)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.4e-34], N[(a / c), $MachinePrecision], If[LessEqual[c, 48000.0], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.39999999999999998e-34 or 48000 < c

   1. Initial program 56.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -1.39999999999999998e-34 < c < 48000

   1. Initial program 77.4%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

Alternative 10: 43.6% 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 66.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 44.3

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

5. Applied rewrites 44.3%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024296 
(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))))