Complex division, real part

Percentage Accurate: 61.4% → 90.1%

Time: 11.3s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.42 \cdot 10^{-62}:\\ \;\;\;\;t\_0 \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma d (/ b c) a) (hypot c d))))
   (if (<= c -1.42e-62)
     (* t_0 (/ c (hypot c d)))
     (if (<= c 7.2e-58) (/ (+ b (* a (/ c d))) d) (/ t_0 (/ (hypot c d) c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, (b / c), a) / hypot(c, d);
	double tmp;
	if (c <= -1.42e-62) {
		tmp = t_0 * (c / hypot(c, d));
	} else if (c <= 7.2e-58) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = t_0 / (hypot(c, d) / c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(d, Float64(b / c), a) / hypot(c, d))
	tmp = 0.0
	if (c <= -1.42e-62)
		tmp = Float64(t_0 * Float64(c / hypot(c, d)));
	elseif (c <= 7.2e-58)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(t_0 / Float64(hypot(c, d) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.42e-62], N[(t$95$0 * N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e-58], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(t$95$0 / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.42e-62

   1. Initial program 64.1%

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

      1. fma-define 64.1%

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

      2. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in c around inf 63.0%

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

      1. *-commutative 63.0%

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

   7. Simplified 63.0%

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

      1. *-commutative 63.0%

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

      2. add-sqr-sqrt 62.9%

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

      3. times-frac 63.5%

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

      4. +-commutative 63.5%

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

      5. associate-/l* 63.8%

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

      6. fma-define 63.8%

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

      7. fma-undefine 63.8%

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

      8. hypot-define 63.8%

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

      9. fma-undefine 63.8%

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

      10. hypot-define 95.1%

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

   9. Applied egg-rr 95.1%

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

   if -1.42e-62 < c < 7.20000000000000019e-58

   1. Initial program 72.7%

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

      1. fma-define 72.7%

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

      2. fma-define 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in d around inf 87.3%

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

      1. associate-/l* 87.9%

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

   7. Simplified 87.9%

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

   if 7.20000000000000019e-58 < c

   1. Initial program 57.6%

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

      1. fma-define 57.6%

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

      2. fma-define 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in c around inf 57.5%

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

      1. *-commutative 57.5%

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

   7. Simplified 57.5%

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

      1. *-commutative 57.5%

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

      2. add-sqr-sqrt 57.5%

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

      3. times-frac 60.5%

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

      4. +-commutative 60.5%

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

      5. associate-/l* 62.2%

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

      6. fma-define 62.2%

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

      7. fma-undefine 62.2%

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

      8. hypot-define 62.2%

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

      9. fma-undefine 62.2%

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

      10. hypot-define 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. clear-num 95.8%

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

       2. un-div-inv 95.8%

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

   11. Applied egg-rr 95.8%

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

Alternative 2: 83.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{a + \left(\left(b \cdot \frac{d}{c} - b \cdot {\left(\frac{d}{c}\right)}^{3}\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}\right)}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+121}:\\ \;\;\;\;c \cdot \frac{\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.2e+142)
   (/
    (+ a (- (- (* b (/ d c)) (* b (pow (/ d c) 3.0))) (* a (pow (/ d c) 2.0))))
    c)
   (if (<= c -1.7e-97)
     (/ (fma a c (* d b)) (fma c c (* d d)))
     (if (<= c 7.2e-58)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c 1.55e+121)
         (* c (/ (/ (fma d (/ b c) a) (hypot c d)) (hypot c d)))
         (/ (+ a (* d (/ b c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e+142) {
		tmp = (a + (((b * (d / c)) - (b * pow((d / c), 3.0))) - (a * pow((d / c), 2.0)))) / c;
	} else if (c <= -1.7e-97) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else if (c <= 7.2e-58) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 1.55e+121) {
		tmp = c * ((fma(d, (b / c), a) / hypot(c, d)) / hypot(c, d));
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.2e+142)
		tmp = Float64(Float64(a + Float64(Float64(Float64(b * Float64(d / c)) - Float64(b * (Float64(d / c) ^ 3.0))) - Float64(a * (Float64(d / c) ^ 2.0)))) / c);
	elseif (c <= -1.7e-97)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	elseif (c <= 7.2e-58)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 1.55e+121)
		tmp = Float64(c * Float64(Float64(fma(d, Float64(b / c), a) / hypot(c, d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.2e+142], N[(N[(a + N[(N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] - N[(b * N[Power[N[(d / c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[Power[N[(d / c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.7e-97], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.2e-58], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.55e+121], N[(c * N[(N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -8.19999999999999963e142

   1. Initial program 37.4%

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

      1. fma-define 37.4%

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

      2. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in c around inf 37.4%

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

      1. *-commutative 37.4%

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

   7. Simplified 37.4%

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

   8. Taylor expanded in c around inf 55.6%

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

   10. Simplified 88.6%

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

   if -8.19999999999999963e142 < c < -1.6999999999999999e-97

   1. Initial program 85.9%

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

      1. fma-define 85.9%

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

      2. fma-define 85.9%

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

   3. Simplified 85.9%

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

   if -1.6999999999999999e-97 < c < 7.20000000000000019e-58

   1. Initial program 71.8%

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

      1. fma-define 71.8%

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

      2. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in d around inf 86.9%

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

      1. associate-/l* 87.5%

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

   7. Simplified 87.5%

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

   if 7.20000000000000019e-58 < c < 1.55000000000000004e121

   1. Initial program 83.1%

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

      1. fma-define 83.1%

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

      2. fma-define 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in c around inf 82.8%

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

      1. *-commutative 82.8%

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

   7. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. add-sqr-sqrt 82.8%

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

      3. times-frac 85.4%

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

      4. +-commutative 85.4%

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

      5. associate-/l* 88.4%

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

      6. fma-define 88.4%

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

      7. fma-undefine 88.4%

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

      8. hypot-define 88.4%

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

      9. fma-undefine 88.4%

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

      10. hypot-define 94.0%

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

   9. Applied egg-rr 94.0%

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

       1. clear-num 93.9%

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

       2. un-div-inv 94.0%

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

   11. Applied egg-rr 94.0%

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

       1. associate-/r/ 91.4%

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

   13. Applied egg-rr 91.4%

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

   if 1.55000000000000004e121 < c

   1. Initial program 35.5%

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

      1. fma-define 35.5%

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

      2. fma-define 35.5%

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

   3. Simplified 35.5%

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

   5. Taylor expanded in c around inf 73.1%

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

      1. +-commutative 73.1%

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

      2. associate-/l* 80.7%

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

      3. fma-define 80.7%

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

   7. Simplified 80.7%

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

      1. fma-undefine 80.7%

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

      2. associate-*r/ 73.1%

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

      3. *-commutative 73.1%

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

      4. associate-/l* 83.1%

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

   9. Applied egg-rr 83.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{a + \left(\left(b \cdot \frac{d}{c} - b \cdot {\left(\frac{d}{c}\right)}^{3}\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}\right)}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{+121}:\\ \;\;\;\;c \cdot \frac{\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.85 \cdot 10^{-63} \lor \neg \left(c \leq 1.9 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.85e-63) (not (<= c 1.9e-56)))
   (* (/ (fma d (/ b c) a) (hypot c d)) (/ c (hypot c d)))
   (/ (+ b (* a (/ c d))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.85e-63) || !(c <= 1.9e-56)) {
		tmp = (fma(d, (b / c), a) / hypot(c, d)) * (c / hypot(c, d));
	} else {
		tmp = (b + (a * (c / d))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.85e-63) || !(c <= 1.9e-56))
		tmp = Float64(Float64(fma(d, Float64(b / c), a) / hypot(c, d)) * Float64(c / hypot(c, d)));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.85e-63], N[Not[LessEqual[c, 1.9e-56]], $MachinePrecision]], N[(N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.85000000000000026e-63 or 1.9000000000000001e-56 < c

   1. Initial program 61.1%

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

      1. fma-define 61.1%

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

      2. fma-define 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in c around inf 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

      1. *-commutative 60.4%

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

      2. add-sqr-sqrt 60.4%

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

      3. times-frac 62.1%

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

      4. +-commutative 62.1%

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

      5. associate-/l* 63.1%

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

      6. fma-define 63.1%

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

      7. fma-undefine 63.1%

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

      8. hypot-define 63.1%

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

      9. fma-undefine 63.1%

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

      10. hypot-define 95.4%

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

   9. Applied egg-rr 95.4%

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

   if -2.85000000000000026e-63 < c < 1.9000000000000001e-56

   1. Initial program 72.7%

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

      1. fma-define 72.7%

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

      2. fma-define 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in d around inf 87.3%

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

      1. associate-/l* 87.9%

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

   7. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{b + a \cdot \frac{c}{d}}{d}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.85 \cdot 10^{-63} \lor \neg \left(c \leq 1.9 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+142}:\\ \;\;\;\;\frac{a + \left(\left(b \cdot \frac{d}{c} - b \cdot {\left(\frac{d}{c}\right)}^{3}\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}\right)}{c}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9e+142)
   (/
    (+ a (- (- (* b (/ d c)) (* b (pow (/ d c) 3.0))) (* a (pow (/ d c) 2.0))))
    c)
   (if (<= c -1.6e-97)
     (/ (fma a c (* d b)) (fma c c (* d d)))
     (if (<= c 6.3e-142)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c 5e+100)
         (/ (+ (* d b) (* c a)) (+ (* d d) (* c c)))
         (/ (+ a (* d (/ b c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9e+142) {
		tmp = (a + (((b * (d / c)) - (b * pow((d / c), 3.0))) - (a * pow((d / c), 2.0)))) / c;
	} else if (c <= -1.6e-97) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else if (c <= 6.3e-142) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 5e+100) {
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9e+142)
		tmp = Float64(Float64(a + Float64(Float64(Float64(b * Float64(d / c)) - Float64(b * (Float64(d / c) ^ 3.0))) - Float64(a * (Float64(d / c) ^ 2.0)))) / c);
	elseif (c <= -1.6e-97)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	elseif (c <= 6.3e-142)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 5e+100)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9e+142], N[(N[(a + N[(N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] - N[(b * N[Power[N[(d / c), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[Power[N[(d / c), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.6e-97], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.3e-142], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5e+100], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -8.9999999999999998e142

   1. Initial program 37.4%

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

      1. fma-define 37.4%

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

      2. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in c around inf 37.4%

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

      1. *-commutative 37.4%

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

   7. Simplified 37.4%

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

   8. Taylor expanded in c around inf 55.6%

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

   10. Simplified 88.6%

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

   if -8.9999999999999998e142 < c < -1.5999999999999999e-97

   1. Initial program 85.9%

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

      1. fma-define 85.9%

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

      2. fma-define 85.9%

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

   3. Simplified 85.9%

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

   if -1.5999999999999999e-97 < c < 6.2999999999999998e-142

   1. Initial program 69.3%

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

      1. fma-define 69.3%

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

      2. fma-define 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 88.6%

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

   7. Simplified 88.6%

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

   if 6.2999999999999998e-142 < c < 4.9999999999999999e100

   1. Initial program 85.5%

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

   if 4.9999999999999999e100 < c

   1. Initial program 38.4%

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

      1. fma-define 38.4%

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

      2. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-/l* 78.6%

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

      3. fma-define 78.6%

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

   7. Simplified 78.6%

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

      1. fma-undefine 78.6%

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

      2. associate-*r/ 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-/l* 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+142}:\\ \;\;\;\;\frac{a + \left(\left(b \cdot \frac{d}{c} - b \cdot {\left(\frac{d}{c}\right)}^{3}\right) - a \cdot {\left(\frac{d}{c}\right)}^{2}\right)}{c}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-142}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-140}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.2e+142)
   (/ (fma d (/ b c) a) (- (hypot c d)))
   (if (<= c -1.9e-97)
     (/ (fma a c (* d b)) (fma c c (* d d)))
     (if (<= c 1.35e-140)
       (/ (+ b (* a (/ c d))) d)
       (if (<= c 4.4e+99)
         (/ (+ (* d b) (* c a)) (+ (* d d) (* c c)))
         (/ (+ a (* d (/ b c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e+142) {
		tmp = fma(d, (b / c), a) / -hypot(c, d);
	} else if (c <= -1.9e-97) {
		tmp = fma(a, c, (d * b)) / fma(c, c, (d * d));
	} else if (c <= 1.35e-140) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 4.4e+99) {
		tmp = ((d * b) + (c * a)) / ((d * d) + (c * c));
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.2e+142)
		tmp = Float64(fma(d, Float64(b / c), a) / Float64(-hypot(c, d)));
	elseif (c <= -1.9e-97)
		tmp = Float64(fma(a, c, Float64(d * b)) / fma(c, c, Float64(d * d)));
	elseif (c <= 1.35e-140)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 4.4e+99)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.2e+142], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])), $MachinePrecision], If[LessEqual[c, -1.9e-97], N[(N[(a * c + N[(d * b), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.35e-140], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.4e+99], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -8.19999999999999963e142

   1. Initial program 37.4%

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

      1. fma-define 37.4%

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

      2. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in c around inf 37.4%

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

      1. *-commutative 37.4%

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

   7. Simplified 37.4%

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

      1. *-commutative 37.4%

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

      2. add-sqr-sqrt 37.4%

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

      3. times-frac 38.6%

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

      4. +-commutative 38.6%

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

      5. associate-/l* 39.2%

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

      6. fma-define 39.2%

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

      7. fma-undefine 39.2%

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

      8. hypot-define 39.2%

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

      9. fma-undefine 39.2%

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

      10. hypot-define 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in c around -inf 88.2%

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

   if -8.19999999999999963e142 < c < -1.9e-97

   1. Initial program 85.9%

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

      1. fma-define 85.9%

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

      2. fma-define 85.9%

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

   3. Simplified 85.9%

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

   if -1.9e-97 < c < 1.35e-140

   1. Initial program 69.3%

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

      1. fma-define 69.3%

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

      2. fma-define 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 88.6%

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

   7. Simplified 88.6%

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

   if 1.35e-140 < c < 4.39999999999999956e99

   1. Initial program 85.5%

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

   if 4.39999999999999956e99 < c

   1. Initial program 38.4%

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

      1. fma-define 38.4%

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

      2. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-/l* 78.6%

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

      3. fma-define 78.6%

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

   7. Simplified 78.6%

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

      1. fma-undefine 78.6%

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

      2. associate-*r/ 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-/l* 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, d \cdot b\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{-140}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -8.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{-142}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* d d) (* c c)))))
   (if (<= c -8.6e+142)
     (/ (fma d (/ b c) a) (- (hypot c d)))
     (if (<= c -1.6e-97)
       t_0
       (if (<= c 5.9e-142)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 2.4e+99) t_0 (/ (+ a (* d (/ b c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -8.6e+142) {
		tmp = fma(d, (b / c), a) / -hypot(c, d);
	} else if (c <= -1.6e-97) {
		tmp = t_0;
	} else if (c <= 5.9e-142) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 2.4e+99) {
		tmp = t_0;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -8.6e+142)
		tmp = Float64(fma(d, Float64(b / c), a) / Float64(-hypot(c, d)));
	elseif (c <= -1.6e-97)
		tmp = t_0;
	elseif (c <= 5.9e-142)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 2.4e+99)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.6e+142], N[(N[(d * N[(b / c), $MachinePrecision] + a), $MachinePrecision] / (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])), $MachinePrecision], If[LessEqual[c, -1.6e-97], t$95$0, If[LessEqual[c, 5.9e-142], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.4e+99], t$95$0, N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -8.60000000000000025e142

   1. Initial program 37.4%

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

      1. fma-define 37.4%

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

      2. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in c around inf 37.4%

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

      1. *-commutative 37.4%

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

   7. Simplified 37.4%

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

      1. *-commutative 37.4%

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

      2. add-sqr-sqrt 37.4%

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

      3. times-frac 38.6%

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

      4. +-commutative 38.6%

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

      5. associate-/l* 39.2%

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

      6. fma-define 39.2%

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

      7. fma-undefine 39.2%

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

      8. hypot-define 39.2%

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

      9. fma-undefine 39.2%

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

      10. hypot-define 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in c around -inf 88.2%

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

   if -8.60000000000000025e142 < c < -1.5999999999999999e-97 or 5.89999999999999966e-142 < c < 2.4000000000000001e99

   1. Initial program 85.7%

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

   if -1.5999999999999999e-97 < c < 5.89999999999999966e-142

   1. Initial program 69.3%

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

      1. fma-define 69.3%

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

      2. fma-define 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 88.6%

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

   7. Simplified 88.6%

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

   if 2.4000000000000001e99 < c

   1. Initial program 38.4%

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

      1. fma-define 38.4%

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

      2. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-/l* 78.6%

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

      3. fma-define 78.6%

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

   7. Simplified 78.6%

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

      1. fma-undefine 78.6%

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

      2. associate-*r/ 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-/l* 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{b}{c}, a\right)}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{-142}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -1.95 \cdot 10^{+143}:\\ \;\;\;\;\frac{a + \frac{b}{c \cdot \frac{1}{d}}}{c}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+99}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* d d) (* c c)))))
   (if (<= c -1.95e+143)
     (/ (+ a (/ b (* c (/ 1.0 d)))) c)
     (if (<= c -1.6e-97)
       t_0
       (if (<= c 3.2e-141)
         (/ (+ b (* a (/ c d))) d)
         (if (<= c 8e+99) t_0 (/ (+ a (* d (/ b c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -1.95e+143) {
		tmp = (a + (b / (c * (1.0 / d)))) / c;
	} else if (c <= -1.6e-97) {
		tmp = t_0;
	} else if (c <= 3.2e-141) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 8e+99) {
		tmp = t_0;
	} else {
		tmp = (a + (d * (b / c))) / 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) :: t_0
    real(8) :: tmp
    t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c))
    if (c <= (-1.95d+143)) then
        tmp = (a + (b / (c * (1.0d0 / d)))) / c
    else if (c <= (-1.6d-97)) then
        tmp = t_0
    else if (c <= 3.2d-141) then
        tmp = (b + (a * (c / d))) / d
    else if (c <= 8d+99) then
        tmp = t_0
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -1.95e+143) {
		tmp = (a + (b / (c * (1.0 / d)))) / c;
	} else if (c <= -1.6e-97) {
		tmp = t_0;
	} else if (c <= 3.2e-141) {
		tmp = (b + (a * (c / d))) / d;
	} else if (c <= 8e+99) {
		tmp = t_0;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c))
	tmp = 0
	if c <= -1.95e+143:
		tmp = (a + (b / (c * (1.0 / d)))) / c
	elif c <= -1.6e-97:
		tmp = t_0
	elif c <= 3.2e-141:
		tmp = (b + (a * (c / d))) / d
	elif c <= 8e+99:
		tmp = t_0
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -1.95e+143)
		tmp = Float64(Float64(a + Float64(b / Float64(c * Float64(1.0 / d)))) / c);
	elseif (c <= -1.6e-97)
		tmp = t_0;
	elseif (c <= 3.2e-141)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (c <= 8e+99)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((d * b) + (c * a)) / ((d * d) + (c * c));
	tmp = 0.0;
	if (c <= -1.95e+143)
		tmp = (a + (b / (c * (1.0 / d)))) / c;
	elseif (c <= -1.6e-97)
		tmp = t_0;
	elseif (c <= 3.2e-141)
		tmp = (b + (a * (c / d))) / d;
	elseif (c <= 8e+99)
		tmp = t_0;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.95e+143], N[(N[(a + N[(b / N[(c * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.6e-97], t$95$0, If[LessEqual[c, 3.2e-141], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8e+99], t$95$0, N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.9499999999999999e143

   1. Initial program 37.4%

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

      1. fma-define 37.4%

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

      2. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in c around inf 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

      1. *-commutative 81.1%

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

      2. associate-*r/ 88.1%

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

      3. *-commutative 88.1%

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

   9. Applied egg-rr 88.1%

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

       1. *-commutative 88.1%

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

       2. clear-num 88.0%

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

       3. un-div-inv 88.1%

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

   11. Applied egg-rr 88.1%

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

       1. div-inv 88.1%

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

   13. Applied egg-rr 88.1%

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

   if -1.9499999999999999e143 < c < -1.5999999999999999e-97 or 3.2000000000000001e-141 < c < 7.9999999999999997e99

   1. Initial program 85.7%

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

   if -1.5999999999999999e-97 < c < 3.2000000000000001e-141

   1. Initial program 69.3%

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

      1. fma-define 69.3%

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

      2. fma-define 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in d around inf 87.8%

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

      1. associate-/l* 88.6%

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

   7. Simplified 88.6%

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

   if 7.9999999999999997e99 < c

   1. Initial program 38.4%

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

      1. fma-define 38.4%

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

      2. fma-define 38.4%

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

   3. Simplified 38.4%

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

   5. Taylor expanded in c around inf 71.7%

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

      1. +-commutative 71.7%

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

      2. associate-/l* 78.6%

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

      3. fma-define 78.6%

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

   7. Simplified 78.6%

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

      1. fma-undefine 78.6%

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

      2. associate-*r/ 71.7%

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

      3. *-commutative 71.7%

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

      4. associate-/l* 80.8%

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

   9. Applied egg-rr 80.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.95 \cdot 10^{+143}:\\ \;\;\;\;\frac{a + \frac{b}{c \cdot \frac{1}{d}}}{c}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-97}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-141}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+99}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+23} \lor \neg \left(d \leq 1.95 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.22e+23) (not (<= d 1.95e-50)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.22e+23) || !(d <= 1.95e-50)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (d / c))) / 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 ((d <= (-1.22d+23)) .or. (.not. (d <= 1.95d-50))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.22e+23) || !(d <= 1.95e-50)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.22e+23) or not (d <= 1.95e-50):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.22e+23) || !(d <= 1.95e-50))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.22e+23) || ~((d <= 1.95e-50)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.22e+23], N[Not[LessEqual[d, 1.95e-50]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.22e23 or 1.9500000000000001e-50 < d

   1. Initial program 57.5%

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

      1. fma-define 57.6%

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

      2. fma-define 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in d around inf 76.0%

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

      1. associate-/l* 78.3%

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

   7. Simplified 78.3%

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

   if -1.22e23 < d < 1.9500000000000001e-50

   1. Initial program 75.4%

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

      1. fma-define 75.4%

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

      2. fma-define 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in c around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-*r/ 83.9%

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

      3. *-commutative 83.9%

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

   9. Applied egg-rr 83.9%

      \[\leadsto \frac{a + \color{blue}{\frac{d}{c} \cdot b}}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+23} \lor \neg \left(d \leq 1.95 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+24} \lor \neg \left(d \leq 13500000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.2e+24) (not (<= d 13500000.0)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.2e+24) || !(d <= 13500000.0)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / 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 ((d <= (-4.2d+24)) .or. (.not. (d <= 13500000.0d0))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.2e+24) || !(d <= 13500000.0)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.2e+24) or not (d <= 13500000.0):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.2e+24) || !(d <= 13500000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.2e+24) || ~((d <= 13500000.0)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.2e+24], N[Not[LessEqual[d, 13500000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.2000000000000003e24 or 1.35e7 < d

   1. Initial program 53.2%

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

      1. fma-define 53.3%

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

      2. fma-define 53.3%

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

   3. Simplified 53.3%

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

   5. Taylor expanded in c around 0 67.8%

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

   if -4.2000000000000003e24 < d < 1.35e7

   1. Initial program 77.8%

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

      1. fma-define 77.8%

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

      2. fma-define 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in c around inf 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*r/ 81.0%

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

      3. *-commutative 81.0%

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

   9. Applied egg-rr 81.0%

      \[\leadsto \frac{a + \color{blue}{\frac{d}{c} \cdot b}}{c} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+24} \lor \neg \left(d \leq 13500000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.45e+23)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 2.9e-50) (/ (+ a (* d (/ b c))) c) (/ (+ b (/ c (/ d a))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.45e+23) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.9e-50) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (c / (d / a))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.45d+23)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 2.9d-50) then
        tmp = (a + (d * (b / c))) / c
    else
        tmp = (b + (c / (d / a))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.45e+23) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 2.9e-50) {
		tmp = (a + (d * (b / c))) / c;
	} else {
		tmp = (b + (c / (d / a))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.45e+23:
		tmp = (b + (a * (c / d))) / d
	elif d <= 2.9e-50:
		tmp = (a + (d * (b / c))) / c
	else:
		tmp = (b + (c / (d / a))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.45e+23)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 2.9e-50)
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.45e+23)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 2.9e-50)
		tmp = (a + (d * (b / c))) / c;
	else
		tmp = (b + (c / (d / a))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.45e+23], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.9e-50], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.45000000000000006e23

   1. Initial program 49.5%

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

      1. fma-define 49.5%

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

      2. fma-define 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in d around inf 76.4%

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

      1. associate-/l* 78.2%

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

   7. Simplified 78.2%

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

   if -1.45000000000000006e23 < d < 2.90000000000000008e-50

   1. Initial program 75.4%

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

      1. fma-define 75.4%

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

      2. fma-define 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in c around inf 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-/l* 83.9%

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

      3. fma-define 83.9%

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

   7. Simplified 83.9%

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

      1. fma-undefine 83.9%

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

      2. associate-*r/ 83.6%

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

      3. *-commutative 83.6%

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

      4. associate-/l* 84.8%

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

   9. Applied egg-rr 84.8%

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

   if 2.90000000000000008e-50 < d

   1. Initial program 63.7%

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

      1. fma-define 63.8%

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

      2. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in d around inf 75.7%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

      3. *-commutative 39.8%

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

      4. associate-/l* 46.2%

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

   7. Applied egg-rr 46.2%

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

      1. unpow2 46.2%

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

      2. add-sqr-sqrt 79.7%

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

      3. clear-num 79.5%

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

      4. un-div-inv 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+23}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-50}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.3e+26)
   (/ (+ b (* a (/ c d))) d)
   (if (<= d 1.75e-48) (/ (+ a (* b (/ d c))) c) (/ (+ b (/ c (/ d a))) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.3e+26) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.75e-48) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c / (d / a))) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-2.3d+26)) then
        tmp = (b + (a * (c / d))) / d
    else if (d <= 1.75d-48) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = (b + (c / (d / a))) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.3e+26) {
		tmp = (b + (a * (c / d))) / d;
	} else if (d <= 1.75e-48) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = (b + (c / (d / a))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.3e+26:
		tmp = (b + (a * (c / d))) / d
	elif d <= 1.75e-48:
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = (b + (c / (d / a))) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.3e+26)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	elseif (d <= 1.75e-48)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.3e+26)
		tmp = (b + (a * (c / d))) / d;
	elseif (d <= 1.75e-48)
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = (b + (c / (d / a))) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.3e+26], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.75e-48], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3000000000000001e26

   1. Initial program 49.5%

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

      1. fma-define 49.5%

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

      2. fma-define 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in d around inf 76.4%

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

      1. associate-/l* 78.2%

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

   7. Simplified 78.2%

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

   if -2.3000000000000001e26 < d < 1.74999999999999996e-48

   1. Initial program 75.4%

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

      1. fma-define 75.4%

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

      2. fma-define 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in c around inf 83.6%

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

      1. *-commutative 83.6%

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

   7. Simplified 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-*r/ 83.9%

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

      3. *-commutative 83.9%

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

   9. Applied egg-rr 83.9%

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

   if 1.74999999999999996e-48 < d

   1. Initial program 63.7%

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

      1. fma-define 63.8%

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

      2. fma-define 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in d around inf 75.7%

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

      1. add-sqr-sqrt 39.8%

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

      2. pow2 39.8%

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

      3. *-commutative 39.8%

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

      4. associate-/l* 46.2%

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

   7. Applied egg-rr 46.2%

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

      1. unpow2 46.2%

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

      2. add-sqr-sqrt 79.7%

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

      3. clear-num 79.5%

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

      4. un-div-inv 79.6%

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

   9. Applied egg-rr 79.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{c}{\frac{d}{a}}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-27} \lor \neg \left(d \leq 1850\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.4e-27) (not (<= d 1850.0))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.4e-27) || !(d <= 1850.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.4d-27)) .or. (.not. (d <= 1850.0d0))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.4e-27) || !(d <= 1850.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.4e-27) or not (d <= 1850.0):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.4e-27) || !(d <= 1850.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.4e-27) || ~((d <= 1850.0)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.4e-27], N[Not[LessEqual[d, 1850.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.39999999999999978e-27 or 1850 < d

   1. Initial program 55.5%

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

      1. fma-define 55.5%

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

      2. fma-define 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in c around 0 65.0%

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

   if -5.39999999999999978e-27 < d < 1850

   1. Initial program 77.9%

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

      1. fma-define 77.9%

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

      2. fma-define 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in c around inf 65.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{-27} \lor \neg \left(d \leq 1850\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

   1. fma-define 65.6%

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

   2. fma-define 65.6%

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

3. Simplified 65.6%

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

5. Taylor expanded in c around inf 38.0%

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

Developer Target 1: 99.1% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (abs(d) < abs(c)) then
        tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024133 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

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

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