Complex division, real part

Percentage Accurate: 61.5% → 84.9%

Time: 9.9s

Alternatives: 11

Speedup: 2.1×

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: 84.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d \cdot b + c \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.3 \cdot 10^{-141}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (+ (* d b) (* c a)) (hypot c d)))))
   (if (<= c -1.8e+150)
     (* (/ c (hypot c d)) (/ a (hypot c d)))
     (if (<= c -5.3e-141)
       t_0
       (if (<= c 1.75e-80)
         (* (/ 1.0 d) (+ b (/ a (/ d c))))
         (if (<= c 6.6e+69) t_0 (/ (+ a (/ b (/ c d))) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (((d * b) + (c * a)) / hypot(c, d));
	double tmp;
	if (c <= -1.8e+150) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (c <= -5.3e-141) {
		tmp = t_0;
	} else if (c <= 1.75e-80) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 6.6e+69) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / Math.hypot(c, d)) * (((d * b) + (c * a)) / Math.hypot(c, d));
	double tmp;
	if (c <= -1.8e+150) {
		tmp = (c / Math.hypot(c, d)) * (a / Math.hypot(c, d));
	} else if (c <= -5.3e-141) {
		tmp = t_0;
	} else if (c <= 1.75e-80) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 6.6e+69) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / math.hypot(c, d)) * (((d * b) + (c * a)) / math.hypot(c, d))
	tmp = 0
	if c <= -1.8e+150:
		tmp = (c / math.hypot(c, d)) * (a / math.hypot(c, d))
	elif c <= -5.3e-141:
		tmp = t_0
	elif c <= 1.75e-80:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	elif c <= 6.6e+69:
		tmp = t_0
	else:
		tmp = (a + (b / (c / d))) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(d * b) + Float64(c * a)) / hypot(c, d)))
	tmp = 0.0
	if (c <= -1.8e+150)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (c <= -5.3e-141)
		tmp = t_0;
	elseif (c <= 1.75e-80)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (c <= 6.6e+69)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / hypot(c, d)) * (((d * b) + (c * a)) / hypot(c, d));
	tmp = 0.0;
	if (c <= -1.8e+150)
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	elseif (c <= -5.3e-141)
		tmp = t_0;
	elseif (c <= 1.75e-80)
		tmp = (1.0 / d) * (b + (a / (d / c)));
	elseif (c <= 6.6e+69)
		tmp = t_0;
	else
		tmp = (a + (b / (c / d))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.8e+150], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.3e-141], t$95$0, If[LessEqual[c, 1.75e-80], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.6e+69], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.79999999999999993e150

   1. Initial program 31.2%

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

   2. Taylor expanded in a around inf 31.2%

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

      1. *-commutative 31.2%

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

   4. Simplified 31.2%

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

      1. add-sqr-sqrt 31.2%

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

      2. hypot-udef 31.2%

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

      3. hypot-udef 31.2%

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

      4. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -1.79999999999999993e150 < c < -5.30000000000000007e-141 or 1.75000000000000007e-80 < c < 6.5999999999999997e69

   1. Initial program 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. +-commutative 80.6%

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

      3. fma-udef 80.6%

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

      4. add-sqr-sqrt 80.6%

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

      5. times-frac 80.6%

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

      6. fma-udef 80.6%

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

      7. +-commutative 80.6%

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

      8. hypot-def 80.6%

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

      9. fma-def 80.6%

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

      10. fma-udef 80.6%

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

      11. +-commutative 80.6%

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

      12. hypot-def 87.2%

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

   3. Applied egg-rr 87.2%

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

      1. fma-def 87.2%

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

      2. +-commutative 87.2%

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

   5. Applied egg-rr 87.2%

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

   if -5.30000000000000007e-141 < c < 1.75000000000000007e-80

   1. Initial program 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. +-commutative 63.5%

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

      3. fma-udef 63.5%

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

      4. add-sqr-sqrt 63.5%

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

      5. times-frac 63.5%

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

      6. fma-udef 63.5%

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

      7. +-commutative 63.5%

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

      8. hypot-def 63.5%

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

      9. fma-def 63.5%

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

      10. fma-udef 63.5%

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

      11. +-commutative 63.5%

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

      12. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in c around 0 61.0%

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

      1. associate-/l* 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in c around 0 94.9%

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

   if 6.5999999999999997e69 < c

   1. Initial program 46.2%

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

      1. *-un-lft-identity 46.2%

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

      2. +-commutative 46.2%

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

      3. fma-udef 46.2%

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

      4. add-sqr-sqrt 46.2%

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

      5. times-frac 46.2%

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

      6. fma-udef 46.2%

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

      7. +-commutative 46.2%

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

      8. hypot-def 46.2%

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

      9. fma-def 46.2%

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

      10. fma-udef 46.2%

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

      11. +-commutative 46.2%

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

      12. hypot-def 56.4%

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

   3. Applied egg-rr 56.4%

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

   4. Taylor expanded in c around inf 78.2%

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

      1. associate-*l/ 78.3%

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

      2. *-un-lft-identity 78.3%

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

      3. associate-/l* 88.6%

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

   6. Applied egg-rr 88.6%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.3 \cdot 10^{-141}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d \cdot b + c \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 6.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d \cdot b + c \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot b + c \cdot a\\ \mathbf{if}\;c \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 250000:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* d b) (* c a))))
   (if (<= c -2e+141)
     (* (/ c (hypot c d)) (/ a (hypot c d)))
     (if (<= c -9.8e-141)
       (/ t_0 (fma d d (* c c)))
       (if (<= c 1.3e-77)
         (* (/ 1.0 d) (+ b (/ a (/ d c))))
         (if (<= c 250000.0)
           (/ t_0 (+ (* c c) (* d d)))
           (if (<= c 4.2e+21)
             (fma (/ (/ a d) d) c (/ b d))
             (/ (+ a (/ b (/ c d))) (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (d * b) + (c * a);
	double tmp;
	if (c <= -2e+141) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (c <= -9.8e-141) {
		tmp = t_0 / fma(d, d, (c * c));
	} else if (c <= 1.3e-77) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 250000.0) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 4.2e+21) {
		tmp = fma(((a / d) / d), c, (b / d));
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(d * b) + Float64(c * a))
	tmp = 0.0
	if (c <= -2e+141)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (c <= -9.8e-141)
		tmp = Float64(t_0 / fma(d, d, Float64(c * c)));
	elseif (c <= 1.3e-77)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (c <= 250000.0)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 4.2e+21)
		tmp = fma(Float64(Float64(a / d) / d), c, Float64(b / d));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2e+141], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -9.8e-141], N[(t$95$0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-77], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 250000.0], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.2e+21], N[(N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] * c + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -2.00000000000000003e141

   1. Initial program 33.1%

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

   2. Taylor expanded in a around inf 33.1%

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

      1. *-commutative 33.1%

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

   4. Simplified 33.1%

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

      1. add-sqr-sqrt 33.1%

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

      2. hypot-udef 33.1%

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

      3. hypot-udef 33.1%

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

      4. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -2.00000000000000003e141 < c < -9.80000000000000012e-141

   1. Initial program 83.5%

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

      1. fma-def 83.5%

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

      2. +-commutative 83.5%

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

      3. fma-def 83.6%

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

   3. Simplified 83.6%

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

      1. fma-def 87.1%

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

      2. +-commutative 87.1%

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

   5. Applied egg-rr 83.6%

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

   if -9.80000000000000012e-141 < c < 1.3000000000000001e-77

   1. Initial program 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. +-commutative 63.5%

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

      3. fma-udef 63.5%

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

      4. add-sqr-sqrt 63.5%

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

      5. times-frac 63.5%

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

      6. fma-udef 63.5%

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

      7. +-commutative 63.5%

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

      8. hypot-def 63.5%

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

      9. fma-def 63.5%

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

      10. fma-udef 63.5%

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

      11. +-commutative 63.5%

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

      12. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in c around 0 61.0%

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

      1. associate-/l* 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in c around 0 94.9%

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

   if 1.3000000000000001e-77 < c < 2.5e5

   1. Initial program 79.5%

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

   if 2.5e5 < c < 4.2e21

   1. Initial program 27.0%

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

   2. Taylor expanded in c around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. associate-/l* 76.1%

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

      3. associate-/r/ 76.1%

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

      4. fma-def 76.1%

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

   4. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. pow2 76.1%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if 4.2e21 < c

   1. Initial program 53.3%

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

      1. *-un-lft-identity 53.3%

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

      2. +-commutative 53.3%

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

      3. fma-udef 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. times-frac 53.3%

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

      6. fma-udef 53.3%

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

      7. +-commutative 53.3%

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

      8. hypot-def 53.3%

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

      9. fma-def 53.3%

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

      10. fma-udef 53.3%

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

      11. +-commutative 53.3%

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

      12. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

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

   4. Taylor expanded in c around inf 79.5%

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

      1. associate-*l/ 79.6%

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

      2. *-un-lft-identity 79.6%

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

      3. associate-/l* 88.0%

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

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 250000:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;c \leq -1.3 \cdot 10^{-140}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 205000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* d b) (* c a)) (+ (* c c) (* d d)))))
   (if (<= c -6.8e+67)
     (+ (/ a c) (/ b (/ (pow c 2.0) d)))
     (if (<= c -1.3e-140)
       t_0
       (if (<= c 1.6e-78)
         (* (/ 1.0 d) (+ b (/ a (/ d c))))
         (if (<= c 205000.0)
           t_0
           (if (<= c 1.7e+22)
             (fma (/ (/ a d) d) c (/ b d))
             (/ (+ a (/ b (/ c d))) (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((d * b) + (c * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -6.8e+67) {
		tmp = (a / c) + (b / (pow(c, 2.0) / d));
	} else if (c <= -1.3e-140) {
		tmp = t_0;
	} else if (c <= 1.6e-78) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 205000.0) {
		tmp = t_0;
	} else if (c <= 1.7e+22) {
		tmp = fma(((a / d) / d), c, (b / d));
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -6.8e+67)
		tmp = Float64(Float64(a / c) + Float64(b / Float64((c ^ 2.0) / d)));
	elseif (c <= -1.3e-140)
		tmp = t_0;
	elseif (c <= 1.6e-78)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (c <= 205000.0)
		tmp = t_0;
	elseif (c <= 1.7e+22)
		tmp = fma(Float64(Float64(a / d) / d), c, Float64(b / d));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	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[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.8e+67], N[(N[(a / c), $MachinePrecision] + N[(b / N[(N[Power[c, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.3e-140], t$95$0, If[LessEqual[c, 1.6e-78], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 205000.0], t$95$0, If[LessEqual[c, 1.7e+22], N[(N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] * c + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -6.8000000000000003e67

   1. Initial program 47.2%

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

   2. Taylor expanded in c around inf 84.7%

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

      1. associate-/l* 86.6%

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

   4. Simplified 86.6%

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

   if -6.8000000000000003e67 < c < -1.2999999999999999e-140 or 1.6e-78 < c < 205000

   1. Initial program 82.9%

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

   if -1.2999999999999999e-140 < c < 1.6e-78

   1. Initial program 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. +-commutative 63.5%

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

      3. fma-udef 63.5%

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

      4. add-sqr-sqrt 63.5%

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

      5. times-frac 63.5%

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

      6. fma-udef 63.5%

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

      7. +-commutative 63.5%

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

      8. hypot-def 63.5%

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

      9. fma-def 63.5%

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

      10. fma-udef 63.5%

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

      11. +-commutative 63.5%

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

      12. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in c around 0 61.0%

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

      1. associate-/l* 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in c around 0 94.9%

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

   if 205000 < c < 1.7e22

   1. Initial program 27.0%

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

   2. Taylor expanded in c around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. associate-/l* 76.1%

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

      3. associate-/r/ 76.1%

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

      4. fma-def 76.1%

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

   4. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. pow2 76.1%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if 1.7e22 < c

   1. Initial program 53.3%

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

      1. *-un-lft-identity 53.3%

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

      2. +-commutative 53.3%

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

      3. fma-udef 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. times-frac 53.3%

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

      6. fma-udef 53.3%

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

      7. +-commutative 53.3%

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

      8. hypot-def 53.3%

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

      9. fma-def 53.3%

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

      10. fma-udef 53.3%

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

      11. +-commutative 53.3%

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

      12. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

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

   4. Taylor expanded in c around inf 79.5%

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

      1. associate-*l/ 79.6%

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

      2. *-un-lft-identity 79.6%

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

      3. associate-/l* 88.0%

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

   6. Applied egg-rr 88.0%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;c \leq -1.3 \cdot 10^{-140}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 205000:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4: 80.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := d \cdot b + c \cdot a\\ \mathbf{if}\;c \leq -4.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 250000:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* d b) (* c a))))
   (if (<= c -4.2e+67)
     (+ (/ a c) (/ b (/ (pow c 2.0) d)))
     (if (<= c -5.5e-139)
       (/ t_0 (fma d d (* c c)))
       (if (<= c 4.8e-80)
         (* (/ 1.0 d) (+ b (/ a (/ d c))))
         (if (<= c 250000.0)
           (/ t_0 (+ (* c c) (* d d)))
           (if (<= c 1.9e+22)
             (fma (/ (/ a d) d) c (/ b d))
             (/ (+ a (/ b (/ c d))) (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (d * b) + (c * a);
	double tmp;
	if (c <= -4.2e+67) {
		tmp = (a / c) + (b / (pow(c, 2.0) / d));
	} else if (c <= -5.5e-139) {
		tmp = t_0 / fma(d, d, (c * c));
	} else if (c <= 4.8e-80) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (c <= 250000.0) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1.9e+22) {
		tmp = fma(((a / d) / d), c, (b / d));
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(d * b) + Float64(c * a))
	tmp = 0.0
	if (c <= -4.2e+67)
		tmp = Float64(Float64(a / c) + Float64(b / Float64((c ^ 2.0) / d)));
	elseif (c <= -5.5e-139)
		tmp = Float64(t_0 / fma(d, d, Float64(c * c)));
	elseif (c <= 4.8e-80)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (c <= 250000.0)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.9e+22)
		tmp = fma(Float64(Float64(a / d) / d), c, Float64(b / d));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.2e+67], N[(N[(a / c), $MachinePrecision] + N[(b / N[(N[Power[c, 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.5e-139], N[(t$95$0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e-80], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 250000.0], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.9e+22], N[(N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] * c + N[(b / d), $MachinePrecision]), $MachinePrecision], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if c < -4.2000000000000003e67

   1. Initial program 47.2%

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

   2. Taylor expanded in c around inf 84.7%

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

      1. associate-/l* 86.6%

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

   4. Simplified 86.6%

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

   if -4.2000000000000003e67 < c < -5.4999999999999997e-139

   1. Initial program 84.8%

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

      1. fma-def 84.8%

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

      2. +-commutative 84.8%

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

      3. fma-def 84.9%

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

   3. Simplified 84.9%

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

      1. fma-def 89.6%

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

      2. +-commutative 89.6%

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

   5. Applied egg-rr 84.9%

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

   if -5.4999999999999997e-139 < c < 4.7999999999999998e-80

   1. Initial program 63.5%

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

      1. *-un-lft-identity 63.5%

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

      2. +-commutative 63.5%

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

      3. fma-udef 63.5%

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

      4. add-sqr-sqrt 63.5%

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

      5. times-frac 63.5%

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

      6. fma-udef 63.5%

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

      7. +-commutative 63.5%

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

      8. hypot-def 63.5%

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

      9. fma-def 63.5%

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

      10. fma-udef 63.5%

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

      11. +-commutative 63.5%

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

      12. hypot-def 77.8%

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

   3. Applied egg-rr 77.8%

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

   4. Taylor expanded in c around 0 61.0%

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

      1. associate-/l* 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in c around 0 94.9%

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

   if 4.7999999999999998e-80 < c < 2.5e5

   1. Initial program 79.5%

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

   if 2.5e5 < c < 1.9000000000000002e22

   1. Initial program 27.0%

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

   2. Taylor expanded in c around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. associate-/l* 76.1%

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

      3. associate-/r/ 76.1%

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

      4. fma-def 76.1%

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

   4. Simplified 76.1%

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

      1. *-un-lft-identity 76.1%

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

      2. pow2 76.1%

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

      3. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   8. Simplified 100.0%

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

   if 1.9000000000000002e22 < c

   1. Initial program 53.3%

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

      1. *-un-lft-identity 53.3%

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

      2. +-commutative 53.3%

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

      3. fma-udef 53.3%

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

      4. add-sqr-sqrt 53.3%

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

      5. times-frac 53.3%

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

      6. fma-udef 53.3%

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

      7. +-commutative 53.3%

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

      8. hypot-def 53.3%

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

      9. fma-def 53.3%

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

      10. fma-udef 53.3%

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

      11. +-commutative 53.3%

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

      12. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

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

   4. Taylor expanded in c around inf 79.5%

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

      1. associate-*l/ 79.6%

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

      2. *-un-lft-identity 79.6%

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

      3. associate-/l* 88.0%

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

   6. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{\frac{{c}^{2}}{d}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 250000:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 5: 80.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.3e+46)
   (* (/ 1.0 d) (+ b (/ a (/ d c))))
   (if (<= d 4e-125)
     (* (/ 1.0 c) (+ a (/ (* d b) c)))
     (if (<= d 7.2e+106)
       (/ (+ (* d b) (* c a)) (+ (* c c) (* d d)))
       (fma (/ (/ a d) d) c (/ b d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.3e+46) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else if (d <= 4e-125) {
		tmp = (1.0 / c) * (a + ((d * b) / c));
	} else if (d <= 7.2e+106) {
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d));
	} else {
		tmp = fma(((a / d) / d), c, (b / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.3e+46)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	elseif (d <= 4e-125)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(d * b) / c)));
	elseif (d <= 7.2e+106)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = fma(Float64(Float64(a / d) / d), c, Float64(b / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.3e+46], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4e-125], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+106], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision] * c + N[(b / d), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.3000000000000001e46

   1. Initial program 39.0%

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

      1. *-un-lft-identity 39.0%

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

      2. +-commutative 39.0%

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

      3. fma-udef 39.0%

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

      4. add-sqr-sqrt 39.0%

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

      5. times-frac 39.0%

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

      6. fma-udef 39.0%

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

      7. +-commutative 39.0%

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

      8. hypot-def 39.0%

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

      9. fma-def 39.0%

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

      10. fma-udef 39.0%

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

      11. +-commutative 39.0%

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

      12. hypot-def 55.4%

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

   3. Applied egg-rr 55.4%

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

   4. Taylor expanded in c around 0 26.3%

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

      1. associate-/l* 26.5%

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

   6. Simplified 26.5%

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

   7. Taylor expanded in c around 0 86.5%

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

   if -2.3000000000000001e46 < d < 4.00000000000000005e-125

   1. Initial program 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. +-commutative 73.0%

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

      3. fma-udef 73.0%

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

      4. add-sqr-sqrt 73.0%

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

      5. times-frac 73.1%

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

      6. fma-udef 73.1%

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

      7. +-commutative 73.1%

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

      8. hypot-def 73.1%

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

      9. fma-def 73.1%

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

      10. fma-udef 73.1%

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

      11. +-commutative 73.1%

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

      12. hypot-def 83.6%

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

   3. Applied egg-rr 83.6%

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

   4. Taylor expanded in c around inf 50.9%

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

   5. Taylor expanded in c around inf 84.5%

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

   if 4.00000000000000005e-125 < d < 7.2000000000000002e106

   1. Initial program 76.2%

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

   if 7.2000000000000002e106 < d

   1. Initial program 24.6%

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

   2. Taylor expanded in c around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-/l* 72.6%

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

      3. associate-/r/ 75.7%

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

      4. fma-def 75.7%

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

   4. Simplified 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. pow2 75.7%

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

      3. times-frac 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. associate-*l/ 87.6%

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

      2. *-lft-identity 87.6%

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

   8. Simplified 87.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-125}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{a}{d}}{d}, c, \frac{b}{d}\right)\\ \end{array} \]

Alternative 6: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{if}\;d \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+108}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (/ a (/ d c))))))
   (if (<= d -3.5e+47)
     t_0
     (if (<= d 2e-126)
       (* (/ 1.0 c) (+ a (/ (* d b) c)))
       (if (<= d 7e+108) (/ (+ (* d b) (* c a)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (d <= -3.5e+47) {
		tmp = t_0;
	} else if (d <= 2e-126) {
		tmp = (1.0 / c) * (a + ((d * b) / c));
	} else if (d <= 7e+108) {
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	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 = (1.0d0 / d) * (b + (a / (d / c)))
    if (d <= (-3.5d+47)) then
        tmp = t_0
    else if (d <= 2d-126) then
        tmp = (1.0d0 / c) * (a + ((d * b) / c))
    else if (d <= 7d+108) then
        tmp = ((d * b) + (c * a)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (d <= -3.5e+47) {
		tmp = t_0;
	} else if (d <= 2e-126) {
		tmp = (1.0 / c) * (a + ((d * b) / c));
	} else if (d <= 7e+108) {
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a / (d / c)))
	tmp = 0
	if d <= -3.5e+47:
		tmp = t_0
	elif d <= 2e-126:
		tmp = (1.0 / c) * (a + ((d * b) / c))
	elif d <= 7e+108:
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))))
	tmp = 0.0
	if (d <= -3.5e+47)
		tmp = t_0;
	elseif (d <= 2e-126)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(d * b) / c)));
	elseif (d <= 7e+108)
		tmp = Float64(Float64(Float64(d * b) + Float64(c * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a / (d / c)));
	tmp = 0.0;
	if (d <= -3.5e+47)
		tmp = t_0;
	elseif (d <= 2e-126)
		tmp = (1.0 / c) * (a + ((d * b) / c));
	elseif (d <= 7e+108)
		tmp = ((d * b) + (c * a)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.5e+47], t$95$0, If[LessEqual[d, 2e-126], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e+108], N[(N[(N[(d * b), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.50000000000000015e47 or 7.0000000000000005e108 < d

   1. Initial program 33.3%

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

      1. *-un-lft-identity 33.3%

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

      2. +-commutative 33.3%

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

      3. fma-udef 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. times-frac 33.3%

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

      6. fma-udef 33.3%

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

      7. +-commutative 33.3%

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

      8. hypot-def 33.3%

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

      9. fma-def 33.3%

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

      10. fma-udef 33.3%

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

      11. +-commutative 33.3%

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

      12. hypot-def 50.7%

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

   3. Applied egg-rr 50.7%

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

   4. Taylor expanded in c around 0 47.7%

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

      1. associate-/l* 50.5%

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

   6. Simplified 50.5%

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

   7. Taylor expanded in c around 0 86.9%

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

   if -3.50000000000000015e47 < d < 1.9999999999999999e-126

   1. Initial program 73.0%

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

      1. *-un-lft-identity 73.0%

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

      2. +-commutative 73.0%

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

      3. fma-udef 73.0%

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

      4. add-sqr-sqrt 73.0%

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

      5. times-frac 73.1%

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

      6. fma-udef 73.1%

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

      7. +-commutative 73.1%

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

      8. hypot-def 73.1%

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

      9. fma-def 73.1%

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

      10. fma-udef 73.1%

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

      11. +-commutative 73.1%

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

      12. hypot-def 83.6%

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

   3. Applied egg-rr 83.6%

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

   4. Taylor expanded in c around inf 50.9%

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

   5. Taylor expanded in c around inf 84.5%

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

   if 1.9999999999999999e-126 < d < 7.0000000000000005e108

   1. Initial program 76.2%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+108}:\\ \;\;\;\;\frac{d \cdot b + c \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]

Alternative 7: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+47} \lor \neg \left(d \leq 1.9 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.2e+47) (not (<= d 1.9e+103)))
   (/ b d)
   (* (/ 1.0 c) (+ a (/ (* d b) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.2e+47) || !(d <= 1.9e+103)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + ((d * 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 <= (-2.2d+47)) .or. (.not. (d <= 1.9d+103))) then
        tmp = b / d
    else
        tmp = (1.0d0 / c) * (a + ((d * 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 <= -2.2e+47) || !(d <= 1.9e+103)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + ((d * b) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.2e+47) or not (d <= 1.9e+103):
		tmp = b / d
	else:
		tmp = (1.0 / c) * (a + ((d * b) / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.2e+47) || !(d <= 1.9e+103))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(d * b) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.2e+47) || ~((d <= 1.9e+103)))
		tmp = b / d;
	else
		tmp = (1.0 / c) * (a + ((d * b) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.2e+47], N[Not[LessEqual[d, 1.9e+103]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1999999999999999e47 or 1.8999999999999998e103 < d

   1. Initial program 35.8%

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

   2. Taylor expanded in c around 0 72.5%

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

   if -2.1999999999999999e47 < d < 1.8999999999999998e103

   1. Initial program 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. +-commutative 73.6%

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

      3. fma-udef 73.6%

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

      4. add-sqr-sqrt 73.6%

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

      5. times-frac 73.7%

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

      6. fma-udef 73.6%

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

      7. +-commutative 73.6%

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

      8. hypot-def 73.7%

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

      9. fma-def 73.7%

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

      10. fma-udef 73.7%

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

      11. +-commutative 73.7%

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

      12. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in c around inf 50.2%

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

   5. Taylor expanded in c around inf 78.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+47} \lor \neg \left(d \leq 1.9 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \end{array} \]

Alternative 8: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+47} \lor \neg \left(d \leq 1.75 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.4e+47) (not (<= d 1.75e+62)))
   (* (/ 1.0 d) (+ b (/ a (/ d c))))
   (* (/ 1.0 c) (+ a (/ (* d b) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.4e+47) || !(d <= 1.75e+62)) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else {
		tmp = (1.0 / c) * (a + ((d * 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 <= (-4.4d+47)) .or. (.not. (d <= 1.75d+62))) then
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    else
        tmp = (1.0d0 / c) * (a + ((d * 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 <= -4.4e+47) || !(d <= 1.75e+62)) {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	} else {
		tmp = (1.0 / c) * (a + ((d * b) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.4e+47) or not (d <= 1.75e+62):
		tmp = (1.0 / d) * (b + (a / (d / c)))
	else:
		tmp = (1.0 / c) * (a + ((d * b) / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.4e+47) || !(d <= 1.75e+62))
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(Float64(d * b) / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.4e+47) || ~((d <= 1.75e+62)))
		tmp = (1.0 / d) * (b + (a / (d / c)));
	else
		tmp = (1.0 / c) * (a + ((d * b) / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.4e+47], N[Not[LessEqual[d, 1.75e+62]], $MachinePrecision]], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(N[(d * b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.3999999999999999e47 or 1.74999999999999992e62 < d

   1. Initial program 36.5%

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

      1. *-un-lft-identity 36.5%

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

      2. +-commutative 36.5%

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

      3. fma-udef 36.5%

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

      4. add-sqr-sqrt 36.5%

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

      5. times-frac 36.4%

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

      6. fma-udef 36.4%

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

      7. +-commutative 36.4%

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

      8. hypot-def 36.4%

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

      9. fma-def 36.4%

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

      10. fma-udef 36.4%

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

      11. +-commutative 36.4%

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

      12. hypot-def 53.0%

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

   3. Applied egg-rr 53.0%

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

   4. Taylor expanded in c around 0 48.3%

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

      1. associate-/l* 50.8%

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

   6. Simplified 50.8%

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

   7. Taylor expanded in c around 0 83.0%

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

   if -4.3999999999999999e47 < d < 1.74999999999999992e62

   1. Initial program 74.8%

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

      1. *-un-lft-identity 74.8%

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

      2. +-commutative 74.8%

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

      3. fma-udef 74.8%

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

      4. add-sqr-sqrt 74.8%

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

      5. times-frac 74.9%

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

      6. fma-udef 74.9%

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

      7. +-commutative 74.9%

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

      8. hypot-def 74.9%

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

      9. fma-def 74.9%

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

      10. fma-udef 74.9%

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

      11. +-commutative 74.9%

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

      12. hypot-def 84.6%

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

   3. Applied egg-rr 84.6%

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

   4. Taylor expanded in c around inf 51.5%

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

   5. Taylor expanded in c around inf 80.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+47} \lor \neg \left(d \leq 1.75 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d \cdot b}{c}\right)\\ \end{array} \]

Alternative 9: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.65 \cdot 10^{+67} \lor \neg \left(c \leq 0.016\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.65e+67) (not (<= c 0.016))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.65e+67) || !(c <= 0.016)) {
		tmp = a / 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 ((c <= (-2.65d+67)) .or. (.not. (c <= 0.016d0))) then
        tmp = a / 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 ((c <= -2.65e+67) || !(c <= 0.016)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.65e+67) or not (c <= 0.016):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.65e+67) || !(c <= 0.016))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.65e+67) || ~((c <= 0.016)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.65e+67], N[Not[LessEqual[c, 0.016]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.65e67 or 0.016 < c

   1. Initial program 51.9%

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

   2. Taylor expanded in c around inf 75.4%

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

   if -2.65e67 < c < 0.016

   1. Initial program 71.3%

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

   2. Taylor expanded in c around 0 60.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.65 \cdot 10^{+67} \lor \neg \left(c \leq 0.016\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 10: 42.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+240}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -4.4e+240) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.4e+240) {
		tmp = a / 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 <= (-4.4d+240)) then
        tmp = a / 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 <= -4.4e+240) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.4e+240:
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.4e+240)
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4.4e+240)
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.4e+240], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.4000000000000003e240

   1. Initial program 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. +-commutative 48.7%

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

      3. fma-udef 48.7%

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

      4. add-sqr-sqrt 48.7%

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

      5. times-frac 48.7%

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

      6. fma-udef 48.7%

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

      7. +-commutative 48.7%

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

      8. hypot-def 48.7%

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

      9. fma-def 48.7%

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

      10. fma-udef 48.7%

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

      11. +-commutative 48.7%

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

      12. hypot-def 54.4%

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

   3. Applied egg-rr 54.4%

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

   4. Taylor expanded in c around inf 31.9%

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

   5. Taylor expanded in c around 0 32.4%

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

   if -4.4000000000000003e240 < d

   1. Initial program 62.3%

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

   2. Taylor expanded in c around inf 51.6%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{+240}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 11: 41.4% 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 61.5%

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

2. Taylor expanded in c around inf 48.8%

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

3. Final simplification 48.8%

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

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 2023309 
(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))))