Complex division, real part

Percentage Accurate: 61.5% → 85.0%

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (* (/ d (hypot c d)) (/ b (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= math.inf:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= Inf)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < +inf.0

   1. Initial program 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. associate-*r/ 78.1%

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

      3. fma-define 78.1%

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

      4. add-sqr-sqrt 78.1%

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

      5. times-frac 78.0%

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

      6. fma-define 78.0%

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

      7. hypot-define 78.1%

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

      8. fma-define 78.0%

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

      9. fma-define 78.0%

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

      10. hypot-define 96.3%

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

   4. Applied egg-rr 96.3%

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

      1. fma-define 96.3%

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

      2. +-commutative 96.3%

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

   6. Applied egg-rr 96.3%

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

   if +inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   3. Taylor expanded in a around 0 1.3%

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

      1. *-commutative 1.3%

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

      2. add-sqr-sqrt 1.3%

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

      3. hypot-undefine 1.3%

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

      4. hypot-undefine 1.3%

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

      5. times-frac 54.7%

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

   5. Applied egg-rr 54.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a \cdot c + b \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -9.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -9.5e+151)
     (* (/ d (hypot c d)) (/ b (hypot c d)))
     (if (<= d -6e-49)
       t_0
       (if (<= d 7.2e-139)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.4e+92)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (* a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -9.5e+151) {
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	} else if (d <= -6e-49) {
		tmp = t_0;
	} else if (d <= 7.2e-139) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.4e+92) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -9.5e+151) {
		tmp = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	} else if (d <= -6e-49) {
		tmp = t_0;
	} else if (d <= 7.2e-139) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.4e+92) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -9.5e+151:
		tmp = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	elif d <= -6e-49:
		tmp = t_0
	elif d <= 7.2e-139:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	elif d <= 2.4e+92:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -9.5e+151)
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	elseif (d <= -6e-49)
		tmp = t_0;
	elseif (d <= 7.2e-139)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.4e+92)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -9.5e+151)
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	elseif (d <= -6e-49)
		tmp = t_0;
	elseif (d <= 7.2e-139)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	elseif (d <= 2.4e+92)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.5e+151], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-49], t$95$0, If[LessEqual[d, 7.2e-139], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.4e+92], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -9.5000000000000001e151

   1. Initial program 32.0%

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

   3. Taylor expanded in a around 0 32.4%

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

      1. *-commutative 32.4%

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

      2. add-sqr-sqrt 32.4%

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

      3. hypot-undefine 32.4%

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

      4. hypot-undefine 32.4%

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

      5. times-frac 83.0%

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

   5. Applied egg-rr 83.0%

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

   if -9.5000000000000001e151 < d < -6e-49 or 7.20000000000000007e-139 < d < 2.40000000000000005e92

   1. Initial program 82.7%

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

   if -6e-49 < d < 7.20000000000000007e-139

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

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

   if 2.40000000000000005e92 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. associate-*r/ 39.5%

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

      3. fma-define 39.5%

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

      4. add-sqr-sqrt 39.5%

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

      5. times-frac 39.5%

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

      6. fma-define 39.5%

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

      7. hypot-define 39.5%

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

      8. fma-define 39.5%

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

      9. fma-define 39.5%

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

      10. hypot-define 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 88.4%

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

      1. associate-/l* 91.9%

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

   7. Simplified 91.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + a \cdot \frac{c}{d}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-49}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{1}{d} \cdot \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -3.6e+80)
     (fma c (* (/ 1.0 d) (/ a d)) (/ b d))
     (if (<= d -5.7e-49)
       t_0
       (if (<= d 1.4e-140)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 1.7e+92)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (* a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -3.6e+80) {
		tmp = fma(c, ((1.0 / d) * (a / d)), (b / d));
	} else if (d <= -5.7e-49) {
		tmp = t_0;
	} else if (d <= 1.4e-140) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 1.7e+92) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -3.6e+80)
		tmp = fma(c, Float64(Float64(1.0 / d) * Float64(a / d)), Float64(b / d));
	elseif (d <= -5.7e-49)
		tmp = t_0;
	elseif (d <= 1.4e-140)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 1.7e+92)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+80], N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.7e-49], t$95$0, If[LessEqual[d, 1.4e-140], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e+92], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.59999999999999995e80

   1. Initial program 45.5%

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

   3. Taylor expanded in c around 0 66.7%

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

      1. +-commutative 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-/l* 71.8%

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

      4. fma-define 71.8%

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

   5. Simplified 71.8%

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

      1. *-un-lft-identity 71.8%

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

      2. pow2 71.8%

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

      3. times-frac 78.3%

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

   7. Applied egg-rr 78.3%

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

   if -3.59999999999999995e80 < d < -5.7000000000000003e-49 or 1.4000000000000001e-140 < d < 1.6999999999999999e92

   1. Initial program 81.6%

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

   if -5.7000000000000003e-49 < d < 1.4000000000000001e-140

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

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

   if 1.6999999999999999e92 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. associate-*r/ 39.5%

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

      3. fma-define 39.5%

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

      4. add-sqr-sqrt 39.5%

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

      5. times-frac 39.5%

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

      6. fma-define 39.5%

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

      7. hypot-define 39.5%

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

      8. fma-define 39.5%

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

      9. fma-define 39.5%

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

      10. hypot-define 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 88.4%

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

      1. associate-/l* 91.9%

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

   7. Simplified 91.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + a \cdot \frac{c}{d}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{1}{d} \cdot \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -7.2 \cdot 10^{+82}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ b (* a (/ c d)))))
   (if (<= d -7.2e+82)
     (* t_1 (/ -1.0 (hypot c d)))
     (if (<= d -6.2e-49)
       t_0
       (if (<= d 4.6e-138)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.15e+92) t_0 (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -7.2e+82) {
		tmp = t_1 * (-1.0 / hypot(c, d));
	} else if (d <= -6.2e-49) {
		tmp = t_0;
	} else if (d <= 4.6e-138) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.15e+92) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = b + (a * (c / d));
	double tmp;
	if (d <= -7.2e+82) {
		tmp = t_1 * (-1.0 / Math.hypot(c, d));
	} else if (d <= -6.2e-49) {
		tmp = t_0;
	} else if (d <= 4.6e-138) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.15e+92) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = b + (a * (c / d))
	tmp = 0
	if d <= -7.2e+82:
		tmp = t_1 * (-1.0 / math.hypot(c, d))
	elif d <= -6.2e-49:
		tmp = t_0
	elif d <= 4.6e-138:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	elif d <= 2.15e+92:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -7.2e+82)
		tmp = Float64(t_1 * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -6.2e-49)
		tmp = t_0;
	elseif (d <= 4.6e-138)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.15e+92)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = b + (a * (c / d));
	tmp = 0.0;
	if (d <= -7.2e+82)
		tmp = t_1 * (-1.0 / hypot(c, d));
	elseif (d <= -6.2e-49)
		tmp = t_0;
	elseif (d <= 4.6e-138)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	elseif (d <= 2.15e+92)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.2e+82], N[(t$95$1 * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.2e-49], t$95$0, If[LessEqual[d, 4.6e-138], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e+92], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.20000000000000028e82

   1. Initial program 45.5%

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

      1. *-un-lft-identity 45.5%

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

      2. associate-*r/ 45.5%

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

      3. fma-define 45.5%

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

      4. add-sqr-sqrt 45.5%

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

      5. times-frac 45.5%

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

      6. fma-define 45.5%

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

      7. hypot-define 45.5%

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

      8. fma-define 45.5%

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

      9. fma-define 45.5%

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

      10. hypot-define 63.7%

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

   4. Applied egg-rr 63.7%

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

   5. Taylor expanded in d around -inf 73.6%

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

      1. distribute-lft-out 73.6%

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

      2. associate-/l* 83.1%

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

   7. Simplified 83.1%

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

   if -7.20000000000000028e82 < d < -6.2e-49 or 4.5999999999999998e-138 < d < 2.1499999999999999e92

   1. Initial program 81.6%

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

   if -6.2e-49 < d < 4.5999999999999998e-138

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

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

   if 2.1499999999999999e92 < d

   1. Initial program 39.5%

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

      1. *-un-lft-identity 39.5%

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

      2. associate-*r/ 39.5%

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

      3. fma-define 39.5%

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

      4. add-sqr-sqrt 39.5%

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

      5. times-frac 39.5%

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

      6. fma-define 39.5%

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

      7. hypot-define 39.5%

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

      8. fma-define 39.5%

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

      9. fma-define 39.5%

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

      10. hypot-define 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 88.4%

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

      1. associate-/l* 91.9%

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

   7. Simplified 91.9%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + a \cdot \frac{c}{d}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+82}:\\ \;\;\;\;\left(b + a \cdot \frac{c}{d}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \mathsf{fma}\left(c, \frac{1}{d} \cdot \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (fma c (* (/ 1.0 d) (/ a d)) (/ b d))))
   (if (<= d -3.8e+84)
     t_1
     (if (<= d -6.4e-49)
       t_0
       (if (<= d 3.8e-139)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.15e+92) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = fma(c, ((1.0 / d) * (a / d)), (b / d));
	double tmp;
	if (d <= -3.8e+84) {
		tmp = t_1;
	} else if (d <= -6.4e-49) {
		tmp = t_0;
	} else if (d <= 3.8e-139) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.15e+92) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = fma(c, Float64(Float64(1.0 / d) * Float64(a / d)), Float64(b / d))
	tmp = 0.0
	if (d <= -3.8e+84)
		tmp = t_1;
	elseif (d <= -6.4e-49)
		tmp = t_0;
	elseif (d <= 3.8e-139)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.15e+92)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(N[(1.0 / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.8e+84], t$95$1, If[LessEqual[d, -6.4e-49], t$95$0, If[LessEqual[d, 3.8e-139], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e+92], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.8000000000000001e84 or 2.1499999999999999e92 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-/l* 77.4%

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

      4. fma-define 77.4%

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

   5. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 84.2%

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

   7. Applied egg-rr 84.2%

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

   if -3.8000000000000001e84 < d < -6.40000000000000005e-49 or 3.80000000000000008e-139 < d < 2.1499999999999999e92

   1. Initial program 81.6%

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

   if -6.40000000000000005e-49 < d < 3.80000000000000008e-139

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{c} \cdot \frac{b \cdot d}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{1}{d} \cdot \frac{a}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -6.4 \cdot 10^{-49}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{1}{d} \cdot \frac{a}{d}, \frac{b}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \mathsf{fma}\left(c, \frac{\frac{a}{d}}{d}, \frac{b}{d}\right)\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.12 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (fma c (/ (/ a d) d) (/ b d))))
   (if (<= d -4.4e+79)
     t_1
     (if (<= d -1.75e-48)
       t_0
       (if (<= d 3.7e-138)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.12e+92) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = fma(c, ((a / d) / d), (b / d));
	double tmp;
	if (d <= -4.4e+79) {
		tmp = t_1;
	} else if (d <= -1.75e-48) {
		tmp = t_0;
	} else if (d <= 3.7e-138) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.12e+92) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = fma(c, Float64(Float64(a / d) / d), Float64(b / d))
	tmp = 0.0
	if (d <= -4.4e+79)
		tmp = t_1;
	elseif (d <= -1.75e-48)
		tmp = t_0;
	elseif (d <= 3.7e-138)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.12e+92)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] + N[(b / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.4e+79], t$95$1, If[LessEqual[d, -1.75e-48], t$95$0, If[LessEqual[d, 3.7e-138], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.12e+92], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.3999999999999998e79 or 2.11999999999999999e92 < d

   1. Initial program 42.2%

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

   3. Taylor expanded in c around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-/l* 77.4%

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

      4. fma-define 77.4%

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

   5. Simplified 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. pow2 77.4%

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

      3. times-frac 84.2%

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

   7. Applied egg-rr 84.2%

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

      1. associate-*l/ 84.2%

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

      2. *-un-lft-identity 84.2%

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

   9. Applied egg-rr 84.2%

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

   if -4.3999999999999998e79 < d < -1.74999999999999996e-48 or 3.69999999999999991e-138 < d < 2.11999999999999999e92

   1. Initial program 81.6%

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

   if -1.74999999999999996e-48 < d < 3.69999999999999991e-138

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{c} \cdot \frac{b \cdot d}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{\frac{a}{d}}{d}, \frac{b}{d}\right)\\ \mathbf{elif}\;d \leq -1.75 \cdot 10^{-48}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.12 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{\frac{a}{d}}{d}, \frac{b}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;b \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-48}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.32e+154)
     (* b (/ -1.0 (hypot c d)))
     (if (<= d -1.9e-48)
       t_0
       (if (<= d 2.6e-138)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.45e+92) t_0 (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.32e+154) {
		tmp = b * (-1.0 / hypot(c, d));
	} else if (d <= -1.9e-48) {
		tmp = t_0;
	} else if (d <= 2.6e-138) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.45e+92) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.32e+154) {
		tmp = b * (-1.0 / Math.hypot(c, d));
	} else if (d <= -1.9e-48) {
		tmp = t_0;
	} else if (d <= 2.6e-138) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.45e+92) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.32e+154:
		tmp = b * (-1.0 / math.hypot(c, d))
	elif d <= -1.9e-48:
		tmp = t_0
	elif d <= 2.6e-138:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	elif d <= 2.45e+92:
		tmp = t_0
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.32e+154)
		tmp = Float64(b * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -1.9e-48)
		tmp = t_0;
	elseif (d <= 2.6e-138)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.45e+92)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.32e+154)
		tmp = b * (-1.0 / hypot(c, d));
	elseif (d <= -1.9e-48)
		tmp = t_0;
	elseif (d <= 2.6e-138)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	elseif (d <= 2.45e+92)
		tmp = t_0;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.32e+154], N[(b * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.9e-48], t$95$0, If[LessEqual[d, 2.6e-138], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.45e+92], t$95$0, N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.31999999999999998e154

   1. Initial program 32.0%

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

      1. *-un-lft-identity 32.0%

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

      2. associate-*r/ 32.0%

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

      3. fma-define 32.0%

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

      4. add-sqr-sqrt 32.0%

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

      5. times-frac 32.0%

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

      6. fma-define 32.0%

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

      7. hypot-define 32.0%

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

      8. fma-define 32.0%

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

      9. fma-define 32.0%

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

      10. hypot-define 55.7%

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

   4. Applied egg-rr 55.7%

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

   5. Taylor expanded in d around -inf 68.2%

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

      1. mul-1-neg 68.2%

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

   7. Simplified 68.2%

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

   if -1.31999999999999998e154 < d < -1.90000000000000001e-48 or 2.6e-138 < d < 2.4500000000000001e92

   1. Initial program 82.7%

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

   if -1.90000000000000001e-48 < d < 2.6e-138

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

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

   if 2.4500000000000001e92 < d

   1. Initial program 39.5%

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

   3. Taylor expanded in c around 0 78.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;b \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.9 \cdot 10^{-48}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -2.6e+153)
     (/ b d)
     (if (<= d -5.7e-49)
       t_0
       (if (<= d 5.5e-137)
         (+ (/ a c) (* (/ 1.0 c) (/ (* b d) c)))
         (if (<= d 2.55e+92) t_0 (/ b d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.6e+153) {
		tmp = b / d;
	} else if (d <= -5.7e-49) {
		tmp = t_0;
	} else if (d <= 5.5e-137) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.55e+92) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (d <= (-2.6d+153)) then
        tmp = b / d
    else if (d <= (-5.7d-49)) then
        tmp = t_0
    else if (d <= 5.5d-137) then
        tmp = (a / c) + ((1.0d0 / c) * ((b * d) / c))
    else if (d <= 2.55d+92) then
        tmp = t_0
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.6e+153) {
		tmp = b / d;
	} else if (d <= -5.7e-49) {
		tmp = t_0;
	} else if (d <= 5.5e-137) {
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	} else if (d <= 2.55e+92) {
		tmp = t_0;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -2.6e+153:
		tmp = b / d
	elif d <= -5.7e-49:
		tmp = t_0
	elif d <= 5.5e-137:
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c))
	elif d <= 2.55e+92:
		tmp = t_0
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.6e+153)
		tmp = Float64(b / d);
	elseif (d <= -5.7e-49)
		tmp = t_0;
	elseif (d <= 5.5e-137)
		tmp = Float64(Float64(a / c) + Float64(Float64(1.0 / c) * Float64(Float64(b * d) / c)));
	elseif (d <= 2.55e+92)
		tmp = t_0;
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -2.6e+153)
		tmp = b / d;
	elseif (d <= -5.7e-49)
		tmp = t_0;
	elseif (d <= 5.5e-137)
		tmp = (a / c) + ((1.0 / c) * ((b * d) / c));
	elseif (d <= 2.55e+92)
		tmp = t_0;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.6e+153], N[(b / d), $MachinePrecision], If[LessEqual[d, -5.7e-49], t$95$0, If[LessEqual[d, 5.5e-137], N[(N[(a / c), $MachinePrecision] + N[(N[(1.0 / c), $MachinePrecision] * N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.55e+92], t$95$0, N[(b / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.5999999999999999e153 or 2.5500000000000001e92 < d

   1. Initial program 36.6%

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

   3. Taylor expanded in c around 0 73.5%

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

   if -2.5999999999999999e153 < d < -5.7000000000000003e-49 or 5.5000000000000003e-137 < d < 2.5500000000000001e92

   1. Initial program 82.7%

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

   if -5.7000000000000003e-49 < d < 5.5000000000000003e-137

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 91.9%

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

   5. Applied egg-rr 91.9%

      \[\leadsto \frac{a}{c} + \color{blue}{\frac{1}{c} \cdot \frac{b \cdot d}{c}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c} \cdot \frac{b \cdot d}{c}\\ \mathbf{elif}\;d \leq 2.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 135:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.5e+85)
   (/ b d)
   (if (<= d 135.0)
     (+ (/ a c) (/ (/ d (/ c b)) c))
     (if (<= d 3.2e+51)
       (/ (* b d) (+ (* c c) (* d d)))
       (if (<= d 3.1e+92) (+ (/ a c) (* (/ d c) (/ b c))) (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.5e+85) {
		tmp = b / d;
	} else if (d <= 135.0) {
		tmp = (a / c) + ((d / (c / b)) / c);
	} else if (d <= 3.2e+51) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 3.1e+92) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.5d+85)) then
        tmp = b / d
    else if (d <= 135.0d0) then
        tmp = (a / c) + ((d / (c / b)) / c)
    else if (d <= 3.2d+51) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 3.1d+92) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.5e+85) {
		tmp = b / d;
	} else if (d <= 135.0) {
		tmp = (a / c) + ((d / (c / b)) / c);
	} else if (d <= 3.2e+51) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 3.1e+92) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.5e+85:
		tmp = b / d
	elif d <= 135.0:
		tmp = (a / c) + ((d / (c / b)) / c)
	elif d <= 3.2e+51:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 3.1e+92:
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.5e+85)
		tmp = Float64(b / d);
	elseif (d <= 135.0)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / Float64(c / b)) / c));
	elseif (d <= 3.2e+51)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 3.1e+92)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.5e+85)
		tmp = b / d;
	elseif (d <= 135.0)
		tmp = (a / c) + ((d / (c / b)) / c);
	elseif (d <= 3.2e+51)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 3.1e+92)
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.5e+85], N[(b / d), $MachinePrecision], If[LessEqual[d, 135.0], N[(N[(a / c), $MachinePrecision] + N[(N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.2e+51], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.1e+92], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.5e85 or 3.1000000000000002e92 < d

   1. Initial program 43.1%

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

   3. Taylor expanded in c around 0 73.3%

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

   if -1.5e85 < d < 135

   1. Initial program 75.4%

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

   3. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. pow2 74.7%

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

      3. times-frac 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*l/ 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

   if 135 < d < 3.2000000000000002e51

   1. Initial program 92.8%

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

   3. Taylor expanded in a around 0 61.2%

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

   if 3.2000000000000002e51 < d < 3.1000000000000002e92

   1. Initial program 35.2%

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

   3. Taylor expanded in c around inf 48.8%

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

      1. *-commutative 48.8%

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

      2. pow2 48.8%

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

      3. times-frac 81.5%

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

   5. Applied egg-rr 81.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+85}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 135:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+85} \lor \neg \left(d \leq 1350\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.8e+85) (not (<= d 1350.0)))
   (/ b d)
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+85) || !(d <= 1350.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4.8d+85)) .or. (.not. (d <= 1350.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+85) || !(d <= 1350.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.8e+85) or not (d <= 1350.0):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.8e+85) || !(d <= 1350.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.8e+85) || ~((d <= 1350.0)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.8e+85], N[Not[LessEqual[d, 1350.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.79999999999999993e85 or 1350 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0 68.2%

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

   if -4.79999999999999993e85 < d < 1350

   1. Initial program 75.4%

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

   3. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. pow2 74.7%

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

      3. times-frac 81.7%

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

   5. Applied egg-rr 81.7%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+85} \lor \neg \left(d \leq 1350\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+84} \lor \neg \left(d \leq 65000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.5e+84) (not (<= d 65000.0)))
   (/ b d)
   (+ (/ a c) (/ (/ d (/ c b)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.5e+84) || !(d <= 65000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / (c / b)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-7.5d+84)) .or. (.not. (d <= 65000.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / (c / b)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.5e+84) || !(d <= 65000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / (c / b)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.5e+84) or not (d <= 65000.0):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / (c / b)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.5e+84) || !(d <= 65000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / Float64(c / b)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.5e+84) || ~((d <= 65000.0)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / (c / b)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.5e+84], N[Not[LessEqual[d, 65000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.5000000000000001e84 or 65000 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0 68.2%

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

   if -7.5000000000000001e84 < d < 65000

   1. Initial program 75.4%

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

   3. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. pow2 74.7%

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

      3. times-frac 81.7%

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

   5. Applied egg-rr 81.7%

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

      1. associate-*l/ 82.4%

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

      2. clear-num 82.4%

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

      3. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+84} \lor \neg \left(d \leq 65000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{d}{\frac{c}{b}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+84} \lor \neg \left(d \leq 0.108\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8e+84) (not (<= d 0.108))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8e+84) || !(d <= 0.108)) {
		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 <= (-8d+84)) .or. (.not. (d <= 0.108d0))) 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 <= -8e+84) || !(d <= 0.108)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8e+84) or not (d <= 0.108):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8e+84) || !(d <= 0.108))
		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 <= -8e+84) || ~((d <= 0.108)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8e+84], N[Not[LessEqual[d, 0.108]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.00000000000000046e84 or 0.107999999999999999 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0 68.2%

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

   if -8.00000000000000046e84 < d < 0.107999999999999999

   1. Initial program 75.4%

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

   3. Taylor expanded in c around inf 68.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+84} \lor \neg \left(d \leq 0.108\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.0% 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 63.8%

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

3. Taylor expanded in c around inf 46.0%

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

4. Final simplification 46.0%

   \[\leadsto \frac{a}{c} \]
5. Add Preprocessing

Developer target: 99.3% 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 2024041 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (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))))