Complex division, real part

Percentage Accurate: 61.8% → 85.3%

Time: 9.6s

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: 61.8% 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: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(a, c, d \cdot b\right)\\ t_1 := \frac{-1}{\mathsf{fma}\left(\frac{-d}{t\_0}, d, -\frac{c}{a}\right)}\\ \mathbf{if}\;a \leq -4.3 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 52000000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(c \cdot \frac{-1}{t\_0}, c, -\frac{d}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma a c (* d b)))
        (t_1 (/ -1.0 (fma (/ (- d) t_0) d (- (/ c a))))))
   (if (<= a -4.3e-23)
     t_1
     (if (<= a 52000000000.0)
       (/ -1.0 (fma (* c (/ -1.0 t_0)) c (- (/ d b))))
       t_1))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(a, c, (d * b));
	double t_1 = -1.0 / fma((-d / t_0), d, -(c / a));
	double tmp;
	if (a <= -4.3e-23) {
		tmp = t_1;
	} else if (a <= 52000000000.0) {
		tmp = -1.0 / fma((c * (-1.0 / t_0)), c, -(d / b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(a, c, Float64(d * b))
	t_1 = Float64(-1.0 / fma(Float64(Float64(-d) / t_0), d, Float64(-Float64(c / a))))
	tmp = 0.0
	if (a <= -4.3e-23)
		tmp = t_1;
	elseif (a <= 52000000000.0)
		tmp = Float64(-1.0 / fma(Float64(c * Float64(-1.0 / t_0)), c, Float64(-Float64(d / b))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[(N[((-d) / t$95$0), $MachinePrecision] * d + (-N[(c / a), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -4.3e-23], t$95$1, If[LessEqual[a, 52000000000.0], N[(-1.0 / N[(N[(c * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] * c + (-N[(d / b), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if a < -4.30000000000000002e-23 or 5.2e10 < a

   1. Initial program 56.3%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 55.8%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-\mathsf{fma}\left(c, c, d \cdot d\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

   6. Applied rewrites 59.1%

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-c\right), c, \frac{d \cdot \left(-d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}\right)}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-lft-identity N/A

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

      14. lift-neg.f64 N/A

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

      15. distribute-lft-neg-out N/A

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

      16. lift-*.f64 N/A

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

   8. Applied rewrites 63.0%

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

   9. Taylor expanded in c around inf

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{\mathsf{neg}\left(d\right)}{\mathsf{fma}\left(a, c, b \cdot d\right)}, d, \mathsf{neg}\left(\color{blue}{\frac{c}{a}}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 87.5

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

   11. Applied rewrites 87.5%

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

   if -4.30000000000000002e-23 < a < 5.2e10

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-\mathsf{fma}\left(c, c, d \cdot d\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

   6. Applied rewrites 77.2%

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

   7. Taylor expanded in d around inf

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(\mathsf{neg}\left(c\right)\right), c, \color{blue}{-1 \cdot \frac{d}{b}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 91.7

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

   9. Applied rewrites 91.7%

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-c\right), c, \color{blue}{\frac{-d}{b}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{-d}{\mathsf{fma}\left(a, c, d \cdot b\right)}, d, -\frac{c}{a}\right)}\\ \mathbf{elif}\;a \leq 52000000000:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(c \cdot \frac{-1}{\mathsf{fma}\left(a, c, d \cdot b\right)}, c, -\frac{d}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\frac{-d}{\mathsf{fma}\left(a, c, d \cdot b\right)}, d, -\frac{c}{a}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{b + \frac{\mathsf{fma}\left(b, -\frac{c \cdot c}{d}, a \cdot c\right)}{d}}{d}\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a c (* d b)) (fma c c (* d d))))
        (t_1 (/ (fma b (/ d c) a) c)))
   (if (<= c -1.1e+151)
     t_1
     (if (<= c -5.5e-62)
       t_0
       (if (<= c 9.4e-94)
         (/ (+ b (/ (fma b (- (/ (* c c) d)) (* a c)) d)) d)
         (if (<= c 5.9e+119) 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(b, (d / c), a) / c;
	double tmp;
	if (c <= -1.1e+151) {
		tmp = t_1;
	} else if (c <= -5.5e-62) {
		tmp = t_0;
	} else if (c <= 9.4e-94) {
		tmp = (b + (fma(b, -((c * c) / d), (a * c)) / d)) / d;
	} else if (c <= 5.9e+119) {
		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(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -1.1e+151)
		tmp = t_1;
	elseif (c <= -5.5e-62)
		tmp = t_0;
	elseif (c <= 9.4e-94)
		tmp = Float64(Float64(b + Float64(fma(b, Float64(-Float64(Float64(c * c) / d)), Float64(a * c)) / d)) / d);
	elseif (c <= 5.9e+119)
		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[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.1e+151], t$95$1, If[LessEqual[c, -5.5e-62], t$95$0, If[LessEqual[c, 9.4e-94], N[(N[(b + N[(N[(b * (-N[(N[(c * c), $MachinePrecision] / d), $MachinePrecision]) + N[(a * c), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.9e+119], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.10000000000000003e151 or 5.9000000000000001e119 < c

   1. Initial program 30.6%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -1.10000000000000003e151 < c < -5.50000000000000022e-62 or 9.40000000000000007e-94 < c < 5.9000000000000001e119

   1. Initial program 79.4%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   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. lift-*.f64 N/A

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

      3. lower-fma.f64 79.4

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 79.4

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

   4. Applied rewrites 79.4%

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

   if -5.50000000000000022e-62 < c < 9.40000000000000007e-94

   1. Initial program 69.5%

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

   3. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. unsub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 92.2%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{b + \frac{\mathsf{fma}\left(b, -\frac{c \cdot c}{d}, a \cdot c\right)}{d}}{d}\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a c (* d b)) (fma c c (* d d))))
        (t_1 (/ (fma b (/ d c) a) c)))
   (if (<= c -1.1e+151)
     t_1
     (if (<= c -2.35e-66)
       t_0
       (if (<= c 9.4e-94)
         (/ (fma a (/ c d) b) d)
         (if (<= c 5.9e+119) 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(b, (d / c), a) / c;
	double tmp;
	if (c <= -1.1e+151) {
		tmp = t_1;
	} else if (c <= -2.35e-66) {
		tmp = t_0;
	} else if (c <= 9.4e-94) {
		tmp = fma(a, (c / d), b) / d;
	} else if (c <= 5.9e+119) {
		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(b, Float64(d / c), a) / c)
	tmp = 0.0
	if (c <= -1.1e+151)
		tmp = t_1;
	elseif (c <= -2.35e-66)
		tmp = t_0;
	elseif (c <= 9.4e-94)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	elseif (c <= 5.9e+119)
		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[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.1e+151], t$95$1, If[LessEqual[c, -2.35e-66], t$95$0, If[LessEqual[c, 9.4e-94], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.9e+119], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.10000000000000003e151 or 5.9000000000000001e119 < c

   1. Initial program 30.6%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if -1.10000000000000003e151 < c < -2.35e-66 or 9.40000000000000007e-94 < c < 5.9000000000000001e119

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 79.6

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -2.35e-66 < c < 9.40000000000000007e-94

   1. Initial program 69.1%

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

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{d}{c}, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma a (/ c d) b) d)))
   (if (<= d -1.15e+99) t_0 (if (<= d 2.25e+15) (/ (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 <= -1.15e+99) {
		tmp = t_0;
	} else if (d <= 2.25e+15) {
		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 <= -1.15e+99)
		tmp = t_0;
	elseif (d <= 2.25e+15)
		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, -1.15e+99], t$95$0, If[LessEqual[d, 2.25e+15], N[(N[(b * N[(d / c), $MachinePrecision] + a), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.1500000000000001e99 or 2.25e15 < d

   1. Initial program 51.0%

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -1.1500000000000001e99 < d < 2.25e15

   1. Initial program 70.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 79.5

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

   5. Applied rewrites 79.5%

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

Alternative 5: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\mathsf{fma}\left(\frac{d}{c}, d, c\right)}\\ \mathbf{if}\;c \leq -4.2 \cdot 10^{-60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{c}{d}, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (fma (/ d c) d c))))
   (if (<= c -4.2e-60) t_0 (if (<= c 2.5e-25) (/ (fma a (/ c d) b) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / fma((d / c), d, c);
	double tmp;
	if (c <= -4.2e-60) {
		tmp = t_0;
	} else if (c <= 2.5e-25) {
		tmp = fma(a, (c / d), b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / fma(Float64(d / c), d, c))
	tmp = 0.0
	if (c <= -4.2e-60)
		tmp = t_0;
	elseif (c <= 2.5e-25)
		tmp = Float64(fma(a, Float64(c / d), b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / N[(N[(d / c), $MachinePrecision] * d + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.2e-60], t$95$0, If[LessEqual[c, 2.5e-25], N[(N[(a * N[(c / d), $MachinePrecision] + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.19999999999999982e-60 or 2.49999999999999981e-25 < c

   1. Initial program 56.6%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 56.5%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-\mathsf{fma}\left(c, c, d \cdot d\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

   6. Applied rewrites 65.8%

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

   7. Taylor expanded in a around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 71.7

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

   9. Applied rewrites 71.7%

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

   11. Applied rewrites 73.6%

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

   if -4.19999999999999982e-60 < c < 2.49999999999999981e-25

   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 d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 6: 67.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{\mathsf{fma}\left(\frac{d}{c}, d, c\right)}\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-109}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (fma (/ d c) d c))))
   (if (<= c -2.1e-65) t_0 (if (<= c 1.25e-109) (/ b d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / fma((d / c), d, c);
	double tmp;
	if (c <= -2.1e-65) {
		tmp = t_0;
	} else if (c <= 1.25e-109) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / fma(Float64(d / c), d, c))
	tmp = 0.0
	if (c <= -2.1e-65)
		tmp = t_0;
	elseif (c <= 1.25e-109)
		tmp = Float64(b / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / N[(N[(d / c), $MachinePrecision] * d + c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.1e-65], t$95$0, If[LessEqual[c, 1.25e-109], N[(b / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.10000000000000003e-65 or 1.25000000000000005e-109 < c

   1. Initial program 59.0%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. lift-+.f64 N/A

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

      6. flip-+ N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{\mathsf{fma}\left(a, c, b \cdot d\right)} \cdot \left(-\mathsf{fma}\left(c, c, d \cdot d\right)\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-neg-in N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

   6. Applied rewrites 67.2%

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

   7. Taylor expanded in a around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.5

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

   9. Applied rewrites 70.5%

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

   11. Applied rewrites 72.2%

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

   if -2.10000000000000003e-65 < c < 1.25000000000000005e-109

   1. Initial program 68.6%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

Alternative 7: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, a \cdot c\right)}{c \cdot c}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.6e+100)
   (/ a c)
   (if (<= c -7.2e-65)
     (/ (fma d b (* a c)) (* c c))
     (if (<= c 4.8e-42) (/ b d) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.6e+100) {
		tmp = a / c;
	} else if (c <= -7.2e-65) {
		tmp = fma(d, b, (a * c)) / (c * c);
	} else if (c <= 4.8e-42) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.6e+100)
		tmp = Float64(a / c);
	elseif (c <= -7.2e-65)
		tmp = Float64(fma(d, b, Float64(a * c)) / Float64(c * c));
	elseif (c <= 4.8e-42)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.6e+100], N[(a / c), $MachinePrecision], If[LessEqual[c, -7.2e-65], N[(N[(d * b + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e-42], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.6000000000000002e100 or 4.80000000000000005e-42 < c

   1. Initial program 51.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -2.6000000000000002e100 < c < -7.1999999999999996e-65

   1. Initial program 75.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{c}^{2}}} \]
   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 54.5

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

   5. Applied rewrites 54.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 54.5

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

   7. Applied rewrites 54.5%

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

   if -7.1999999999999996e-65 < c < 4.80000000000000005e-42

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 8: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+150}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-65}:\\ \;\;\;\;\frac{a \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8e+150)
   (/ a c)
   (if (<= c -2.1e-65)
     (/ (* a c) (fma d d (* c c)))
     (if (<= c 4.8e-42) (/ b d) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8e+150) {
		tmp = a / c;
	} else if (c <= -2.1e-65) {
		tmp = (a * c) / fma(d, d, (c * c));
	} else if (c <= 4.8e-42) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8e+150)
		tmp = Float64(a / c);
	elseif (c <= -2.1e-65)
		tmp = Float64(Float64(a * c) / fma(d, d, Float64(c * c)));
	elseif (c <= 4.8e-42)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8e+150], N[(a / c), $MachinePrecision], If[LessEqual[c, -2.1e-65], N[(N[(a * c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e-42], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.99999999999999985e150 or 4.80000000000000005e-42 < c

   1. Initial program 46.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -7.99999999999999985e150 < c < -2.10000000000000003e-65

   1. Initial program 78.8%

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

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   if -2.10000000000000003e-65 < c < 4.80000000000000005e-42

   1. Initial program 70.9%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 71.1

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

   5. Applied rewrites 71.1%

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

Alternative 9: 63.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.7e+14) (/ a c) (if (<= c 4.8e-42) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.7e+14) {
		tmp = a / c;
	} else if (c <= 4.8e-42) {
		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 <= (-3.7d+14)) then
        tmp = a / c
    else if (c <= 4.8d-42) 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 <= -3.7e+14) {
		tmp = a / c;
	} else if (c <= 4.8e-42) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.7e+14:
		tmp = a / c
	elif c <= 4.8e-42:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.7e+14)
		tmp = Float64(a / c);
	elseif (c <= 4.8e-42)
		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 <= -3.7e+14)
		tmp = a / c;
	elseif (c <= 4.8e-42)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.7e+14], N[(a / c), $MachinePrecision], If[LessEqual[c, 4.8e-42], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.7e14 or 4.80000000000000005e-42 < c

   1. Initial program 53.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.1

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

   5. Applied rewrites 67.1%

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

   if -3.7e14 < c < 4.80000000000000005e-42

   1. Initial program 73.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

Alternative 10: 41.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{a}{c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 43.7

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

5. Applied rewrites 43.7%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
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 2024233 
(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))))