Complex division, imag part

Percentage Accurate: 61.1% → 82.6%

Time: 6.7s

Alternatives: 15

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;\frac{b - \left({c}^{-1} \cdot a\right) \cdot d}{c}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))))
   (if (<= c -8e+101)
     (/ (- b (* (* (pow c -1.0) a) d)) c)
     (if (<= c -9.5e-38)
       t_0
       (if (<= c 1.22e-119)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 5.6e+75)
           t_0
           (fma
            (fma
             (- (* (/ a (pow c 4.0)) d) (/ b (pow c 3.0)))
             d
             (/ (/ (- a) c) c))
            d
            (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -8e+101) {
		tmp = (b - ((pow(c, -1.0) * a) * d)) / c;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 5.6e+75) {
		tmp = t_0;
	} else {
		tmp = fma(fma((((a / pow(c, 4.0)) * d) - (b / pow(c, 3.0))), d, ((-a / c) / c)), d, (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -8e+101)
		tmp = Float64(Float64(b - Float64(Float64((c ^ -1.0) * a) * d)) / c);
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 5.6e+75)
		tmp = t_0;
	else
		tmp = fma(fma(Float64(Float64(Float64(a / (c ^ 4.0)) * d) - Float64(b / (c ^ 3.0))), d, Float64(Float64(Float64(-a) / c) / c)), d, Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8e+101], N[(N[(b - N[(N[(N[Power[c, -1.0], $MachinePrecision] * a), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -9.5e-38], t$95$0, If[LessEqual[c, 1.22e-119], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.6e+75], t$95$0, N[(N[(N[(N[(N[(a / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] - N[(b / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d + N[(N[((-a) / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * d + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.9999999999999998e101

   1. Initial program 39.1%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   7. Applied rewrites 86.1%

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

   if -7.9999999999999998e101 < c < -9.5000000000000009e-38 or 1.22e-119 < c < 5.60000000000000023e75

   1. Initial program 86.7%

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

   if -9.5000000000000009e-38 < c < 1.22e-119

   1. Initial program 71.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

   if 5.60000000000000023e75 < c

   1. Initial program 44.5%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) + \frac{b}{c}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) \cdot d} + \frac{b}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right), d, \frac{b}{c}\right)} \]

   5. Applied rewrites 86.9%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;\frac{b - \left({c}^{-1} \cdot a\right) \cdot d}{c}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{+75}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{b - \left({c}^{-1} \cdot a\right) \cdot d}{c}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* d d) (* c c))))
        (t_1 (/ (- b (* (* (pow c -1.0) a) d)) c)))
   (if (<= c -8e+101)
     t_1
     (if (<= c -9.5e-38)
       t_0
       (if (<= c 1.22e-119)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 2.6e+63) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double t_1 = (b - ((pow(c, -1.0) * a) * d)) / c;
	double tmp;
	if (c <= -8e+101) {
		tmp = t_1;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.6e+63) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
    t_1 = (b - (((c ** (-1.0d0)) * a) * d)) / c
    if (c <= (-8d+101)) then
        tmp = t_1
    else if (c <= (-9.5d-38)) then
        tmp = t_0
    else if (c <= 1.22d-119) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 2.6d+63) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double t_1 = (b - ((Math.pow(c, -1.0) * a) * d)) / c;
	double tmp;
	if (c <= -8e+101) {
		tmp = t_1;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.6e+63) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
	t_1 = (b - ((math.pow(c, -1.0) * a) * d)) / c
	tmp = 0
	if c <= -8e+101:
		tmp = t_1
	elif c <= -9.5e-38:
		tmp = t_0
	elif c <= 1.22e-119:
		tmp = (((b * c) / d) - a) / d
	elif c <= 2.6e+63:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(Float64(b - Float64(Float64((c ^ -1.0) * a) * d)) / c)
	tmp = 0.0
	if (c <= -8e+101)
		tmp = t_1;
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.6e+63)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	t_1 = (b - (((c ^ -1.0) * a) * d)) / c;
	tmp = 0.0;
	if (c <= -8e+101)
		tmp = t_1;
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 2.6e+63)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(N[(N[Power[c, -1.0], $MachinePrecision] * a), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -8e+101], t$95$1, If[LessEqual[c, -9.5e-38], t$95$0, If[LessEqual[c, 1.22e-119], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.6e+63], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.9999999999999998e101 or 2.6000000000000001e63 < c

   1. Initial program 42.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 84.9%

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

   if -7.9999999999999998e101 < c < -9.5000000000000009e-38 or 1.22e-119 < c < 2.6000000000000001e63

   1. Initial program 89.2%

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

   if -9.5000000000000009e-38 < c < 1.22e-119

   1. Initial program 71.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;\frac{b - \left({c}^{-1} \cdot a\right) \cdot d}{c}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \left({c}^{-1} \cdot a\right) \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{-1}{c} \cdot \mathsf{fma}\left(b, -1, \frac{a}{c} \cdot d\right)\\ \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* d d) (* c c))))
        (t_1 (* (/ -1.0 c) (fma b -1.0 (* (/ a c) d)))))
   (if (<= c -8e+101)
     t_1
     (if (<= c -9.5e-38)
       t_0
       (if (<= c 1.22e-119)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 4.4e+86) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double t_1 = (-1.0 / c) * fma(b, -1.0, ((a / c) * d));
	double tmp;
	if (c <= -8e+101) {
		tmp = t_1;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4.4e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(Float64(-1.0 / c) * fma(b, -1.0, Float64(Float64(a / c) * d)))
	tmp = 0.0
	if (c <= -8e+101)
		tmp = t_1;
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4.4e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-1.0 / c), $MachinePrecision] * N[(b * -1.0 + N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8e+101], t$95$1, If[LessEqual[c, -9.5e-38], t$95$0, If[LessEqual[c, 1.22e-119], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.4e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.9999999999999998e101 or 4.40000000000000006e86 < c

   1. Initial program 41.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   7. Applied rewrites 86.0%

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

   9. Applied rewrites 85.8%

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

   if -7.9999999999999998e101 < c < -9.5000000000000009e-38 or 1.22e-119 < c < 4.40000000000000006e86

   1. Initial program 86.9%

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

   if -9.5000000000000009e-38 < c < 1.22e-119

   1. Initial program 71.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{c} \cdot \mathsf{fma}\left(b, -1, \frac{a}{c} \cdot d\right)\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \mathsf{fma}\left(b, -1, \frac{a}{c} \cdot d\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* d d) (* c c))))
        (t_1 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -3.4e+111)
     t_1
     (if (<= c -9.5e-38)
       t_0
       (if (<= c 1.22e-119)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 2.6e+63) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double t_1 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.4e+111) {
		tmp = t_1;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.6e+63) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
    t_1 = (b - ((d * a) / c)) / c
    if (c <= (-3.4d+111)) then
        tmp = t_1
    else if (c <= (-9.5d-38)) then
        tmp = t_0
    else if (c <= 1.22d-119) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 2.6d+63) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double t_1 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.4e+111) {
		tmp = t_1;
	} else if (c <= -9.5e-38) {
		tmp = t_0;
	} else if (c <= 1.22e-119) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.6e+63) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
	t_1 = (b - ((d * a) / c)) / c
	tmp = 0
	if c <= -3.4e+111:
		tmp = t_1
	elif c <= -9.5e-38:
		tmp = t_0
	elif c <= 1.22e-119:
		tmp = (((b * c) / d) - a) / d
	elif c <= 2.6e+63:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -3.4e+111)
		tmp = t_1;
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.6e+63)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	t_1 = (b - ((d * a) / c)) / c;
	tmp = 0.0;
	if (c <= -3.4e+111)
		tmp = t_1;
	elseif (c <= -9.5e-38)
		tmp = t_0;
	elseif (c <= 1.22e-119)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 2.6e+63)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.4e+111], t$95$1, If[LessEqual[c, -9.5e-38], t$95$0, If[LessEqual[c, 1.22e-119], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.6e+63], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.4000000000000001e111 or 2.6000000000000001e63 < c

   1. Initial program 42.5%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -3.4000000000000001e111 < c < -9.5000000000000009e-38 or 1.22e-119 < c < 2.6000000000000001e63

   1. Initial program 86.5%

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

   if -9.5000000000000009e-38 < c < 1.22e-119

   1. Initial program 71.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 92.7

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

   5. Applied rewrites 92.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+63}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-d, a, b \cdot c\right)\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma (- d) a (* b c))))
   (if (<= c -1.25e+123)
     (/ 1.0 (/ c b))
     (if (<= c -2.1e-14)
       (* (/ b (fma d d (* c c))) c)
       (if (<= c 1e-67)
         (/ t_0 (* d d))
         (if (<= c 3.8e+88) (/ t_0 (* c c)) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-d, a, (b * c));
	double tmp;
	if (c <= -1.25e+123) {
		tmp = 1.0 / (c / b);
	} else if (c <= -2.1e-14) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 1e-67) {
		tmp = t_0 / (d * d);
	} else if (c <= 3.8e+88) {
		tmp = t_0 / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(-d), a, Float64(b * c))
	tmp = 0.0
	if (c <= -1.25e+123)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= -2.1e-14)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1e-67)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 3.8e+88)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+123], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.1e-14], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1e-67], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e+88], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.24999999999999994e123

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 77.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -1.24999999999999994e123 < c < -2.0999999999999999e-14

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.0999999999999999e-14 < c < 9.99999999999999943e-68

   1. Initial program 74.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 67.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

   if 9.99999999999999943e-68 < c < 3.7999999999999997e88

   1. Initial program 88.7%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 3.7999999999999997e88 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+123)
   (/ 1.0 (/ c b))
   (if (<= c -2.1e-14)
     (* (/ b (fma d d (* c c))) c)
     (if (<= c 1e-67)
       (/ (- (* b c) (* d a)) (* d d))
       (if (<= c 3.8e+88) (/ (fma (- d) a (* b c)) (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+123) {
		tmp = 1.0 / (c / b);
	} else if (c <= -2.1e-14) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 1e-67) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (c <= 3.8e+88) {
		tmp = fma(-d, a, (b * c)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+123)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= -2.1e-14)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1e-67)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (c <= 3.8e+88)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+123], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.1e-14], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1e-67], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e+88], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.24999999999999994e123

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 77.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -1.24999999999999994e123 < c < -2.0999999999999999e-14

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.0999999999999999e-14 < c < 9.99999999999999943e-68

   1. Initial program 74.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 9.99999999999999943e-68 < c < 3.7999999999999997e88

   1. Initial program 88.7%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lower-fma.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 3.7999999999999997e88 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - d \cdot a\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* d a))))
   (if (<= c -1.25e+123)
     (/ 1.0 (/ c b))
     (if (<= c -2.1e-14)
       (* (/ b (fma d d (* c c))) c)
       (if (<= c 1e-67)
         (/ t_0 (* d d))
         (if (<= c 3.8e+88) (/ t_0 (* c c)) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if (c <= -1.25e+123) {
		tmp = 1.0 / (c / b);
	} else if (c <= -2.1e-14) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 1e-67) {
		tmp = t_0 / (d * d);
	} else if (c <= 3.8e+88) {
		tmp = t_0 / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (c <= -1.25e+123)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= -2.1e-14)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1e-67)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 3.8e+88)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+123], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.1e-14], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1e-67], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e+88], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.24999999999999994e123

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 77.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -1.24999999999999994e123 < c < -2.0999999999999999e-14

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.0999999999999999e-14 < c < 9.99999999999999943e-68

   1. Initial program 74.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if 9.99999999999999943e-68 < c < 3.7999999999999997e88

   1. Initial program 88.7%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if 3.7999999999999997e88 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.25e+123)
   (/ 1.0 (/ c b))
   (if (<= c -2.65e-14)
     (* (/ b (fma d d (* c c))) c)
     (if (<= c 8e-74)
       (/ (- a) d)
       (if (<= c 3.8e+88) (/ (- (* b c) (* d a)) (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.25e+123) {
		tmp = 1.0 / (c / b);
	} else if (c <= -2.65e-14) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 8e-74) {
		tmp = -a / d;
	} else if (c <= 3.8e+88) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.25e+123)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= -2.65e-14)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 8e-74)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 3.8e+88)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.25e+123], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.65e-14], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 8e-74], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 3.8e+88], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.24999999999999994e123

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 77.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -1.24999999999999994e123 < c < -2.6500000000000001e-14

   1. Initial program 74.9%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -2.6500000000000001e-14 < c < 7.99999999999999966e-74

   1. Initial program 73.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.3

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if 7.99999999999999966e-74 < c < 3.7999999999999997e88

   1. Initial program 89.2%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 3.7999999999999997e88 < c

   1. Initial program 43.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.3

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

   5. Applied rewrites 78.3%

      \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+88}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-92}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.1 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ b (fma d d (* c c))) c)))
   (if (<= c -1.25e+123)
     (/ 1.0 (/ c b))
     (if (<= c -2.65e-14)
       t_0
       (if (<= c 1.12e-92) (/ (- a) d) (if (<= c 4.1e+119) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / fma(d, d, (c * c))) * c;
	double tmp;
	if (c <= -1.25e+123) {
		tmp = 1.0 / (c / b);
	} else if (c <= -2.65e-14) {
		tmp = t_0;
	} else if (c <= 1.12e-92) {
		tmp = -a / d;
	} else if (c <= 4.1e+119) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / fma(d, d, Float64(c * c))) * c)
	tmp = 0.0
	if (c <= -1.25e+123)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= -2.65e-14)
		tmp = t_0;
	elseif (c <= 1.12e-92)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 4.1e+119)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]}, If[LessEqual[c, -1.25e+123], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.65e-14], t$95$0, If[LessEqual[c, 1.12e-92], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 4.1e+119], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.24999999999999994e123

   1. Initial program 39.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.7

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

   5. Applied rewrites 76.7%

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

   7. Applied rewrites 77.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -1.24999999999999994e123 < c < -2.6500000000000001e-14 or 1.11999999999999999e-92 < c < 4.0999999999999997e119

   1. Initial program 77.2%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -2.6500000000000001e-14 < c < 1.11999999999999999e-92

   1. Initial program 73.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.0

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if 4.0999999999999997e119 < c

   1. Initial program 42.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 82.3

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

   5. Applied rewrites 82.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq -2.65 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-92}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.1 \cdot 10^{+119}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -7.6e+121)
     t_0
     (if (<= c -1.85e-8)
       (* (/ b (fma d d (* c c))) c)
       (if (<= c 2e-67) (/ (- (/ (* b c) d) a) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -7.6e+121) {
		tmp = t_0;
	} else if (c <= -1.85e-8) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 2e-67) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -7.6e+121)
		tmp = t_0;
	elseif (c <= -1.85e-8)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 2e-67)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.6e+121], t$95$0, If[LessEqual[c, -1.85e-8], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 2e-67], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.6e121 or 1.99999999999999989e-67 < c

   1. Initial program 49.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -7.6e121 < c < -1.85e-8

   1. Initial program 76.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.85e-8 < c < 1.99999999999999989e-67

   1. Initial program 74.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b}{d} \cdot c - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -7.6e+121)
     t_0
     (if (<= c -1.85e-8)
       (* (/ b (fma d d (* c c))) c)
       (if (<= c 2e-67) (/ (- (* (/ b d) c) a) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -7.6e+121) {
		tmp = t_0;
	} else if (c <= -1.85e-8) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 2e-67) {
		tmp = (((b / d) * c) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -7.6e+121)
		tmp = t_0;
	elseif (c <= -1.85e-8)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 2e-67)
		tmp = Float64(Float64(Float64(Float64(b / d) * c) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.6e+121], t$95$0, If[LessEqual[c, -1.85e-8], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 2e-67], N[(N[(N[(N[(b / d), $MachinePrecision] * c), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.6e121 or 1.99999999999999989e-67 < c

   1. Initial program 49.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -7.6e121 < c < -1.85e-8

   1. Initial program 76.4%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if -1.85e-8 < c < 1.99999999999999989e-67

   1. Initial program 74.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 89.1

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

   5. Applied rewrites 89.1%

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

   7. Applied rewrites 86.4%

      \[\leadsto \frac{c \cdot \frac{b}{d} - a}{d} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.85 \cdot 10^{-8}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 2 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b}{d} \cdot c - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -7.6e+121)
     t_0
     (if (<= c -2.1e-14)
       (* (/ b (fma d d (* c c))) c)
       (if (<= c 1e-67) (/ (fma (- d) a (* b c)) (* d d)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -7.6e+121) {
		tmp = t_0;
	} else if (c <= -2.1e-14) {
		tmp = (b / fma(d, d, (c * c))) * c;
	} else if (c <= 1e-67) {
		tmp = fma(-d, a, (b * c)) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -7.6e+121)
		tmp = t_0;
	elseif (c <= -2.1e-14)
		tmp = Float64(Float64(b / fma(d, d, Float64(c * c))) * c);
	elseif (c <= 1e-67)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / Float64(d * d));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.6e+121], t$95$0, If[LessEqual[c, -2.1e-14], N[(N[(b / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 1e-67], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.6e121 or 9.99999999999999943e-68 < c

   1. Initial program 49.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -7.6e121 < c < -2.0999999999999999e-14

   1. Initial program 77.0%

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

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if -2.0999999999999999e-14 < c < 9.99999999999999943e-68

   1. Initial program 74.2%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.7

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

   5. Applied rewrites 67.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 67.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 67.7

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

   7. Applied rewrites 67.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot c\\ \mathbf{elif}\;c \leq 10^{-67}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.3e-14) (/ 1.0 (/ c b)) (if (<= c 8e-74) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.3e-14) {
		tmp = 1.0 / (c / b);
	} else if (c <= 8e-74) {
		tmp = -a / d;
	} else {
		tmp = b / 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 <= (-5.3d-14)) then
        tmp = 1.0d0 / (c / b)
    else if (c <= 8d-74) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.3e-14) {
		tmp = 1.0 / (c / b);
	} else if (c <= 8e-74) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.3e-14:
		tmp = 1.0 / (c / b)
	elif c <= 8e-74:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.3e-14)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= 8e-74)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.3e-14)
		tmp = 1.0 / (c / b);
	elseif (c <= 8e-74)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.3e-14], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8e-74], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.3000000000000001e-14

   1. Initial program 55.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   7. Applied rewrites 71.6%

      \[\leadsto \frac{1}{\color{blue}{\frac{c}{b}}} \]

   if -5.3000000000000001e-14 < c < 7.99999999999999966e-74

   1. Initial program 73.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.3

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\frac{a}{-d}} \]

   if 7.99999999999999966e-74 < c

   1. Initial program 56.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.5

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

   5. Applied rewrites 70.5%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.3e-14) (/ b c) (if (<= c 8e-74) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.3e-14) {
		tmp = b / c;
	} else if (c <= 8e-74) {
		tmp = -a / d;
	} else {
		tmp = b / 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 <= (-5.3d-14)) then
        tmp = b / c
    else if (c <= 8d-74) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.3e-14) {
		tmp = b / c;
	} else if (c <= 8e-74) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.3e-14:
		tmp = b / c
	elif c <= 8e-74:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.3e-14)
		tmp = Float64(b / c);
	elseif (c <= 8e-74)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.3e-14)
		tmp = b / c;
	elseif (c <= 8e-74)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.3e-14], N[(b / c), $MachinePrecision], If[LessEqual[c, 8e-74], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.3000000000000001e-14 or 7.99999999999999966e-74 < c

   1. Initial program 55.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.0

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

   5. Applied rewrites 71.0%

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

   if -5.3000000000000001e-14 < c < 7.99999999999999966e-74

   1. Initial program 73.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{-1 \cdot d}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a}{-1 \cdot d}} \]

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.3

         \[\leadsto \frac{a}{\color{blue}{-d}} \]

   5. Applied rewrites 65.3%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-74}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return b / 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 = b / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 50.3

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

5. Applied rewrites 50.3%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (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(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (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[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024295 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))