Complex division, real part

Percentage Accurate: 61.8% → 84.8%

Time: 13.3s

Alternatives: 14

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 74.6%

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

      1. *-un-lft-identity 74.6%

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

      2. add-sqr-sqrt 74.5%

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

      3. times-frac 74.6%

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

      4. hypot-def 74.6%

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

      5. fma-def 74.6%

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

      6. hypot-def 94.7%

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

   3. Applied egg-rr 94.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c around 0 43.1%

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

      1. associate-/l* 46.2%

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

   4. Simplified 46.2%

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

      1. *-un-lft-identity 46.2%

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

      2. metadata-eval 46.2%

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

      3. unpow2 46.2%

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

      4. associate-*l/ 59.4%

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

      5. times-frac 65.6%

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

      6. metadata-eval 65.6%

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

      7. clear-num 65.7%

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

   6. Applied egg-rr 65.7%

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

4. Final simplification 89.6%

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

Alternative 2: 81.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\frac{\mathsf{hypot}\left(c, d\right)}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))))
   (if (<= d -1.6e+159)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -3.9e-106)
       (* (fma a c (* b d)) (/ 1.0 (pow (hypot c d) 2.0)))
       (if (<= d 1.8e-89)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 3.8e+68)
           (/ (+ (* a c) (* b d)) (/ (hypot c d) t_0))
           (* t_0 (+ b (* c (/ a d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double tmp;
	if (d <= -1.6e+159) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -3.9e-106) {
		tmp = fma(a, c, (b * d)) * (1.0 / pow(hypot(c, d), 2.0));
	} else if (d <= 1.8e-89) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 3.8e+68) {
		tmp = ((a * c) + (b * d)) / (hypot(c, d) / t_0);
	} else {
		tmp = t_0 * (b + (c * (a / d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	tmp = 0.0
	if (d <= -1.6e+159)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -3.9e-106)
		tmp = Float64(fma(a, c, Float64(b * d)) * Float64(1.0 / (hypot(c, d) ^ 2.0)));
	elseif (d <= 1.8e-89)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 3.8e+68)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(hypot(c, d) / t_0));
	else
		tmp = Float64(t_0 * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6e+159], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.9e-106], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-89], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.8e+68], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.59999999999999992e159

   1. Initial program 34.1%

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

   2. Taylor expanded in c around 0 81.2%

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

      1. associate-/l* 81.7%

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

   4. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. metadata-eval 81.7%

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

      3. unpow2 81.7%

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

      4. associate-*l/ 86.9%

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

      5. times-frac 93.4%

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

      6. metadata-eval 93.4%

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

      7. clear-num 93.5%

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

   6. Applied egg-rr 93.5%

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

   if -1.59999999999999992e159 < d < -3.9000000000000001e-106

   1. Initial program 77.3%

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

      1. div-inv 77.4%

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

      2. fma-def 77.4%

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

      3. add-sqr-sqrt 77.4%

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

      4. pow2 77.4%

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

      5. hypot-def 77.4%

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

   3. Applied egg-rr 77.4%

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

   if -3.9000000000000001e-106 < d < 1.80000000000000003e-89

   1. Initial program 66.9%

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

   2. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 78.7%

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

      2. associate-/r/ 77.6%

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

   4. Simplified 77.6%

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

      1. associate-*l/ 81.7%

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

      2. unpow2 81.7%

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

      3. associate-/r* 90.1%

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

   6. Applied egg-rr 90.1%

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

   if 1.80000000000000003e-89 < d < 3.8000000000000001e68

   1. Initial program 81.2%

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

      1. flip3-+ 60.5%

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

      2. clear-num 60.5%

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

      3. *-un-lft-identity 60.5%

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

      4. associate-/l* 60.6%

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

      5. flip3-+ 81.1%

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

      6. associate-/l* 81.2%

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

      7. *-un-lft-identity 81.2%

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

      8. add-sqr-sqrt 81.1%

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

      9. associate-/l* 81.4%

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

      10. hypot-def 81.4%

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

   3. Applied egg-rr 81.4%

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

   if 3.8000000000000001e68 < d

   1. Initial program 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. add-sqr-sqrt 41.4%

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

      3. times-frac 41.2%

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

      4. hypot-def 41.2%

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

      5. fma-def 41.2%

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

      6. hypot-def 60.0%

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

   3. Applied egg-rr 60.0%

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

   4. Taylor expanded in c around 0 82.7%

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

      1. associate-/l* 86.5%

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

      2. associate-/r/ 88.2%

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

   6. Simplified 88.2%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{a}{d} \cdot c\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{1}{\mathsf{hypot}\left(c, d\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]

Alternative 3: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{t_0}{\frac{\mathsf{hypot}\left(c, d\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))) (t_1 (/ 1.0 (hypot c d))))
   (if (<= d -1.6e+159)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -6.2e-106)
       (/ t_0 (+ (* c c) (* d d)))
       (if (<= d 2.8e-91)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 5e+62)
           (/ t_0 (/ (hypot c d) t_1))
           (* t_1 (+ b (* c (/ a d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = 1.0 / hypot(c, d);
	double tmp;
	if (d <= -1.6e+159) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -6.2e-106) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (d <= 2.8e-91) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 5e+62) {
		tmp = t_0 / (hypot(c, d) / t_1);
	} else {
		tmp = t_1 * (b + (c * (a / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = 1.0 / Math.hypot(c, d);
	double tmp;
	if (d <= -1.6e+159) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -6.2e-106) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (d <= 2.8e-91) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 5e+62) {
		tmp = t_0 / (Math.hypot(c, d) / t_1);
	} else {
		tmp = t_1 * (b + (c * (a / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	t_1 = 1.0 / math.hypot(c, d)
	tmp = 0
	if d <= -1.6e+159:
		tmp = (b / d) + ((c / d) * (a / d))
	elif d <= -6.2e-106:
		tmp = t_0 / ((c * c) + (d * d))
	elif d <= 2.8e-91:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 5e+62:
		tmp = t_0 / (math.hypot(c, d) / t_1)
	else:
		tmp = t_1 * (b + (c * (a / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	t_1 = Float64(1.0 / hypot(c, d))
	tmp = 0.0
	if (d <= -1.6e+159)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -6.2e-106)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.8e-91)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 5e+62)
		tmp = Float64(t_0 / Float64(hypot(c, d) / t_1));
	else
		tmp = Float64(t_1 * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	t_1 = 1.0 / hypot(c, d);
	tmp = 0.0;
	if (d <= -1.6e+159)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (d <= -6.2e-106)
		tmp = t_0 / ((c * c) + (d * d));
	elseif (d <= 2.8e-91)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 5e+62)
		tmp = t_0 / (hypot(c, d) / t_1);
	else
		tmp = t_1 * (b + (c * (a / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6e+159], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.2e-106], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e-91], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5e+62], N[(t$95$0 / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.59999999999999992e159

   1. Initial program 34.1%

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

   2. Taylor expanded in c around 0 81.2%

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

      1. associate-/l* 81.7%

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

   4. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. metadata-eval 81.7%

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

      3. unpow2 81.7%

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

      4. associate-*l/ 86.9%

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

      5. times-frac 93.4%

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

      6. metadata-eval 93.4%

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

      7. clear-num 93.5%

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

   6. Applied egg-rr 93.5%

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

   if -1.59999999999999992e159 < d < -6.19999999999999971e-106

   1. Initial program 77.3%

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

   if -6.19999999999999971e-106 < d < 2.8e-91

   1. Initial program 66.9%

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

   2. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 78.7%

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

      2. associate-/r/ 77.6%

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

   4. Simplified 77.6%

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

      1. associate-*l/ 81.7%

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

      2. unpow2 81.7%

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

      3. associate-/r* 90.1%

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

   6. Applied egg-rr 90.1%

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

   if 2.8e-91 < d < 5.00000000000000029e62

   1. Initial program 81.2%

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

      1. flip3-+ 60.5%

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

      2. clear-num 60.5%

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

      3. *-un-lft-identity 60.5%

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

      4. associate-/l* 60.6%

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

      5. flip3-+ 81.1%

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

      6. associate-/l* 81.2%

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

      7. *-un-lft-identity 81.2%

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

      8. add-sqr-sqrt 81.1%

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

      9. associate-/l* 81.4%

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

      10. hypot-def 81.4%

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

   3. Applied egg-rr 81.4%

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

   if 5.00000000000000029e62 < d

   1. Initial program 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. add-sqr-sqrt 41.4%

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

      3. times-frac 41.2%

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

      4. hypot-def 41.2%

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

      5. fma-def 41.2%

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

      6. hypot-def 60.0%

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

   3. Applied egg-rr 60.0%

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

   4. Taylor expanded in c around 0 82.7%

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

      1. associate-/l* 86.5%

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

      2. associate-/r/ 88.2%

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

   6. Simplified 88.2%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{a}{d} \cdot c\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -6.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+62}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\frac{\mathsf{hypot}\left(c, d\right)}{\frac{1}{\mathsf{hypot}\left(c, d\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]

Alternative 4: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.6e+159)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (if (<= d -3.2e-108)
       t_0
       (if (<= d 2.05e-90)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 3.05e+69)
           t_0
           (* (/ 1.0 (hypot c d)) (+ b (* c (/ a d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.6e+159) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -3.2e-108) {
		tmp = t_0;
	} else if (d <= 2.05e-90) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 3.05e+69) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (c * (a / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.6e+159) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= -3.2e-108) {
		tmp = t_0;
	} else if (d <= 2.05e-90) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 3.05e+69) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (c * (a / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.6e+159:
		tmp = (b / d) + ((c / d) * (a / d))
	elif d <= -3.2e-108:
		tmp = t_0
	elif d <= 2.05e-90:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 3.05e+69:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (c * (a / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.6e+159)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= -3.2e-108)
		tmp = t_0;
	elseif (d <= 2.05e-90)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 3.05e+69)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.6e+159)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (d <= -3.2e-108)
		tmp = t_0;
	elseif (d <= 2.05e-90)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 3.05e+69)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b + (c * (a / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6e+159], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -3.2e-108], t$95$0, If[LessEqual[d, 2.05e-90], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.05e+69], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.59999999999999992e159

   1. Initial program 34.1%

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

   2. Taylor expanded in c around 0 81.2%

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

      1. associate-/l* 81.7%

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

   4. Simplified 81.7%

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

      1. *-un-lft-identity 81.7%

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

      2. metadata-eval 81.7%

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

      3. unpow2 81.7%

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

      4. associate-*l/ 86.9%

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

      5. times-frac 93.4%

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

      6. metadata-eval 93.4%

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

      7. clear-num 93.5%

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

   6. Applied egg-rr 93.5%

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

   if -1.59999999999999992e159 < d < -3.2e-108 or 2.05000000000000017e-90 < d < 3.05e69

   1. Initial program 78.8%

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

   if -3.2e-108 < d < 2.05000000000000017e-90

   1. Initial program 66.9%

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

   2. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 78.7%

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

      2. associate-/r/ 77.6%

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

   4. Simplified 77.6%

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

      1. associate-*l/ 81.7%

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

      2. unpow2 81.7%

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

      3. associate-/r* 90.1%

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

   6. Applied egg-rr 90.1%

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

   if 3.05e69 < d

   1. Initial program 40.3%

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

      1. *-un-lft-identity 40.3%

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

      2. add-sqr-sqrt 40.3%

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

      3. times-frac 40.2%

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

      4. hypot-def 40.2%

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

      5. fma-def 40.2%

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

      6. hypot-def 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in c around 0 82.4%

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

      1. associate-/l* 86.3%

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

      2. associate-/r/ 88.0%

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

   6. Simplified 88.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-90}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.05 \cdot 10^{+69}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]

Alternative 5: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -1.6e+159)
     t_1
     (if (<= d -5.6e-100)
       t_0
       (if (<= d 8.5e-84)
         (+ (/ a c) (/ (/ (* b d) c) c))
         (if (<= d 2e+71) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.6e+159) {
		tmp = t_1;
	} else if (d <= -5.6e-100) {
		tmp = t_0;
	} else if (d <= 8.5e-84) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 2e+71) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + ((c / d) * (a / d))
    if (d <= (-1.6d+159)) then
        tmp = t_1
    else if (d <= (-5.6d-100)) then
        tmp = t_0
    else if (d <= 8.5d-84) then
        tmp = (a / c) + (((b * d) / c) / c)
    else if (d <= 2d+71) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -1.6e+159) {
		tmp = t_1;
	} else if (d <= -5.6e-100) {
		tmp = t_0;
	} else if (d <= 8.5e-84) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 2e+71) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -1.6e+159:
		tmp = t_1
	elif d <= -5.6e-100:
		tmp = t_0
	elif d <= 8.5e-84:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 2e+71:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -1.6e+159)
		tmp = t_1;
	elseif (d <= -5.6e-100)
		tmp = t_0;
	elseif (d <= 8.5e-84)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 2e+71)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -1.6e+159)
		tmp = t_1;
	elseif (d <= -5.6e-100)
		tmp = t_0;
	elseif (d <= 8.5e-84)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 2e+71)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6e+159], t$95$1, If[LessEqual[d, -5.6e-100], t$95$0, If[LessEqual[d, 8.5e-84], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e+71], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.59999999999999992e159 or 2.0000000000000001e71 < d

   1. Initial program 38.2%

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

   2. Taylor expanded in c around 0 78.8%

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

      1. associate-/l* 78.1%

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

   4. Simplified 78.1%

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

      1. *-un-lft-identity 78.1%

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

      2. metadata-eval 78.1%

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

      3. unpow2 78.1%

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

      4. associate-*l/ 82.1%

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

      5. times-frac 88.8%

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

      6. metadata-eval 88.8%

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

      7. clear-num 88.8%

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

   6. Applied egg-rr 88.8%

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

   if -1.59999999999999992e159 < d < -5.59999999999999991e-100 or 8.4999999999999994e-84 < d < 2.0000000000000001e71

   1. Initial program 78.8%

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

   if -5.59999999999999991e-100 < d < 8.4999999999999994e-84

   1. Initial program 66.9%

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

   2. Taylor expanded in c around inf 81.7%

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

      1. associate-/l* 78.7%

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

      2. associate-/r/ 77.6%

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

   4. Simplified 77.6%

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

      1. associate-*l/ 81.7%

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

      2. unpow2 81.7%

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

      3. associate-/r* 90.1%

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

   6. Applied egg-rr 90.1%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -5.6 \cdot 10^{-100}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 8.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+71}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 6: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -2.75 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 8.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -2.75e-23)
     t_0
     (if (<= d 3.7e-22)
       (+ (/ a c) (/ (/ (* b d) c) c))
       (if (<= d 8.3e+46)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 1.02e+65) (* (/ -1.0 c) (- (- a) (/ b (/ c d)))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -2.75e-23) {
		tmp = t_0;
	} else if (d <= 3.7e-22) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 8.3e+46) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 1.02e+65) {
		tmp = (-1.0 / c) * (-a - (b / (c / 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 = (b / d) + ((c / d) * (a / d))
    if (d <= (-2.75d-23)) then
        tmp = t_0
    else if (d <= 3.7d-22) then
        tmp = (a / c) + (((b * d) / c) / c)
    else if (d <= 8.3d+46) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 1.02d+65) then
        tmp = ((-1.0d0) / c) * (-a - (b / (c / 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 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -2.75e-23) {
		tmp = t_0;
	} else if (d <= 3.7e-22) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 8.3e+46) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 1.02e+65) {
		tmp = (-1.0 / c) * (-a - (b / (c / d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -2.75e-23:
		tmp = t_0
	elif d <= 3.7e-22:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 8.3e+46:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 1.02e+65:
		tmp = (-1.0 / c) * (-a - (b / (c / d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -2.75e-23)
		tmp = t_0;
	elseif (d <= 3.7e-22)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 8.3e+46)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 1.02e+65)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(-a) - Float64(b / Float64(c / d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -2.75e-23)
		tmp = t_0;
	elseif (d <= 3.7e-22)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 8.3e+46)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 1.02e+65)
		tmp = (-1.0 / c) * (-a - (b / (c / d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.75e-23], t$95$0, If[LessEqual[d, 3.7e-22], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.3e+46], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.02e+65], N[(N[(-1.0 / c), $MachinePrecision] * N[((-a) - N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.7500000000000001e-23 or 1.02000000000000005e65 < d

   1. Initial program 52.9%

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

   2. Taylor expanded in c around 0 74.1%

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

      1. associate-/l* 72.9%

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

   4. Simplified 72.9%

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

      1. *-un-lft-identity 72.9%

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

      2. metadata-eval 72.9%

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

      3. unpow2 72.9%

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

      4. associate-*l/ 75.6%

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

      5. times-frac 80.7%

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

      6. metadata-eval 80.7%

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

      7. clear-num 80.7%

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

   6. Applied egg-rr 80.7%

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

   if -2.7500000000000001e-23 < d < 3.7e-22

   1. Initial program 67.8%

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

   2. Taylor expanded in c around inf 76.3%

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

      1. associate-/l* 73.9%

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

      2. associate-/r/ 73.1%

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

   4. Simplified 73.1%

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

      1. associate-*l/ 76.3%

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

      2. unpow2 76.3%

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

      3. associate-/r* 83.7%

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

   6. Applied egg-rr 83.7%

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

   if 3.7e-22 < d < 8.29999999999999951e46

   1. Initial program 92.9%

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

   2. Taylor expanded in a around 0 80.5%

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

   if 8.29999999999999951e46 < d < 1.02000000000000005e65

   1. Initial program 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. add-sqr-sqrt 37.8%

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

      3. times-frac 37.8%

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

      4. hypot-def 37.8%

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

      5. fma-def 37.8%

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

      6. hypot-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in c around -inf 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-/l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in c around -inf 100.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.75 \cdot 10^{-23}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-22}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 8.3 \cdot 10^{+46}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.02 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\left(-a\right) - \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 7: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (/ (* b d) c) c)))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -3.2e-23)
     t_1
     (if (<= d 5.8e-23)
       t_0
       (if (<= d 7.5e+49)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 4.9e+61) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (((b * d) / c) / c);
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -3.2e-23) {
		tmp = t_1;
	} else if (d <= 5.8e-23) {
		tmp = t_0;
	} else if (d <= 7.5e+49) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 4.9e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (a / c) + (((b * d) / c) / c)
    t_1 = (b / d) + ((c / d) * (a / d))
    if (d <= (-3.2d-23)) then
        tmp = t_1
    else if (d <= 5.8d-23) then
        tmp = t_0
    else if (d <= 7.5d+49) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 4.9d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (((b * d) / c) / c);
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -3.2e-23) {
		tmp = t_1;
	} else if (d <= 5.8e-23) {
		tmp = t_0;
	} else if (d <= 7.5e+49) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 4.9e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + (((b * d) / c) / c)
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -3.2e-23:
		tmp = t_1
	elif d <= 5.8e-23:
		tmp = t_0
	elif d <= 7.5e+49:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 4.9e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -3.2e-23)
		tmp = t_1;
	elseif (d <= 5.8e-23)
		tmp = t_0;
	elseif (d <= 7.5e+49)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 4.9e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + (((b * d) / c) / c);
	t_1 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -3.2e-23)
		tmp = t_1;
	elseif (d <= 5.8e-23)
		tmp = t_0;
	elseif (d <= 7.5e+49)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 4.9e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.2e-23], t$95$1, If[LessEqual[d, 5.8e-23], t$95$0, If[LessEqual[d, 7.5e+49], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.9e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.19999999999999976e-23 or 4.90000000000000025e61 < d

   1. Initial program 52.9%

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

   2. Taylor expanded in c around 0 74.1%

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

      1. associate-/l* 72.9%

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

   4. Simplified 72.9%

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

      1. *-un-lft-identity 72.9%

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

      2. metadata-eval 72.9%

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

      3. unpow2 72.9%

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

      4. associate-*l/ 75.6%

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

      5. times-frac 80.7%

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

      6. metadata-eval 80.7%

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

      7. clear-num 80.7%

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

   6. Applied egg-rr 80.7%

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

   if -3.19999999999999976e-23 < d < 5.8000000000000003e-23 or 7.4999999999999995e49 < d < 4.90000000000000025e61

   1. Initial program 67.0%

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

   2. Taylor expanded in c around inf 76.1%

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

      1. associate-/l* 73.8%

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

      2. associate-/r/ 73.0%

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

   4. Simplified 73.0%

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

      1. associate-*l/ 76.1%

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

      2. unpow2 76.1%

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

      3. associate-/r* 84.1%

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

   6. Applied egg-rr 84.1%

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

   if 5.8000000000000003e-23 < d < 7.4999999999999995e49

   1. Initial program 92.9%

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

   2. Taylor expanded in a around 0 80.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 8: 75.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{c + \frac{d}{\frac{c}{d}}}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d} + \frac{1}{d} \cdot \frac{a}{\frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.7e+39)
   (/ a (+ c (/ d (/ c d))))
   (if (<= c 6.4e+53)
     (+ (/ b d) (* (/ 1.0 d) (/ a (/ d c))))
     (+ (/ a c) (* d (* (/ 1.0 c) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e+39) {
		tmp = a / (c + (d / (c / d)));
	} else if (c <= 6.4e+53) {
		tmp = (b / d) + ((1.0 / d) * (a / (d / c)));
	} else {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.7d+39)) then
        tmp = a / (c + (d / (c / d)))
    else if (c <= 6.4d+53) then
        tmp = (b / d) + ((1.0d0 / d) * (a / (d / c)))
    else
        tmp = (a / c) + (d * ((1.0d0 / c) * (b / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e+39) {
		tmp = a / (c + (d / (c / d)));
	} else if (c <= 6.4e+53) {
		tmp = (b / d) + ((1.0 / d) * (a / (d / c)));
	} else {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.7e+39:
		tmp = a / (c + (d / (c / d)))
	elif c <= 6.4e+53:
		tmp = (b / d) + ((1.0 / d) * (a / (d / c)))
	else:
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.7e+39)
		tmp = Float64(a / Float64(c + Float64(d / Float64(c / d))));
	elseif (c <= 6.4e+53)
		tmp = Float64(Float64(b / d) + Float64(Float64(1.0 / d) * Float64(a / Float64(d / c))));
	else
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(1.0 / c) * Float64(b / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.7e+39)
		tmp = a / (c + (d / (c / d)));
	elseif (c <= 6.4e+53)
		tmp = (b / d) + ((1.0 / d) * (a / (d / c)));
	else
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.7e+39], N[(a / N[(c + N[(d / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.4e+53], N[(N[(b / d), $MachinePrecision] + N[(N[(1.0 / d), $MachinePrecision] * N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.70000000000000003e39

   1. Initial program 46.8%

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

   2. Taylor expanded in a around inf 42.2%

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

      1. associate-/l* 44.6%

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

      2. +-commutative 44.6%

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

      3. unpow2 44.6%

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

      4. fma-udef 44.6%

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

   4. Simplified 44.6%

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

   5. Taylor expanded in d around 0 65.3%

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

      1. div-inv 65.3%

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

      2. unpow2 65.3%

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

      3. associate-*l* 74.6%

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

      4. div-inv 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*r/ 65.3%

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

      2. associate-/l* 74.6%

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

   9. Applied egg-rr 74.6%

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

   if -2.70000000000000003e39 < c < 6.4e53

   1. Initial program 70.7%

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

   2. Taylor expanded in c around 0 70.5%

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

      1. associate-/l* 68.7%

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

   4. Simplified 68.7%

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

      1. *-un-lft-identity 68.7%

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

      2. metadata-eval 68.7%

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

      3. unpow2 68.7%

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

      4. associate-*r/ 75.5%

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

      5. times-frac 78.1%

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

      6. metadata-eval 78.1%

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

   6. Applied egg-rr 78.1%

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

   if 6.4e53 < c

   1. Initial program 52.4%

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

   2. Taylor expanded in c around inf 79.0%

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

      1. associate-/l* 77.7%

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

      2. associate-/r/ 82.7%

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

   4. Simplified 82.7%

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

      1. *-un-lft-identity 82.7%

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

      2. unpow2 82.7%

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

      3. times-frac 87.1%

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

   6. Applied egg-rr 87.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+39}:\\ \;\;\;\;\frac{a}{c + \frac{d}{\frac{c}{d}}}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d} + \frac{1}{d} \cdot \frac{a}{\frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \end{array} \]

Alternative 9: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-13} \lor \neg \left(d \leq 155000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.5e-13) (not (<= d 155000000000.0)))
   (/ b d)
   (+ (/ a c) (/ (/ (* b d) c) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e-13) || !(d <= 155000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.5d-13)) .or. (.not. (d <= 155000000000.0d0))) then
        tmp = b / d
    else
        tmp = (a / c) + (((b * d) / c) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.5e-13) || !(d <= 155000000000.0)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.5e-13) or not (d <= 155000000000.0):
		tmp = b / d
	else:
		tmp = (a / c) + (((b * d) / c) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.5e-13) || !(d <= 155000000000.0))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.5e-13) || ~((d <= 155000000000.0)))
		tmp = b / d;
	else
		tmp = (a / c) + (((b * d) / c) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.5e-13], N[Not[LessEqual[d, 155000000000.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.5000000000000001e-13 or 1.55e11 < d

   1. Initial program 53.3%

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

   2. Taylor expanded in c around 0 68.6%

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

   if -8.5000000000000001e-13 < d < 1.55e11

   1. Initial program 70.4%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. associate-/l* 71.4%

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

      2. associate-/r/ 70.6%

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

   4. Simplified 70.6%

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

      1. associate-*l/ 73.5%

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

      2. unpow2 73.5%

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

      3. associate-/r* 80.6%

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

   6. Applied egg-rr 80.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{-13} \lor \neg \left(d \leq 155000000000\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]

Alternative 10: 77.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-24} \lor \neg \left(d \leq 850\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.2e-24) (not (<= d 850.0)))
   (+ (/ b d) (* (/ c d) (/ a d)))
   (+ (/ a c) (/ (/ (* b d) c) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e-24) || !(d <= 850.0)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-7.2d-24)) .or. (.not. (d <= 850.0d0))) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + (((b * d) / c) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e-24) || !(d <= 850.0)) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.2e-24) or not (d <= 850.0):
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + (((b * d) / c) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.2e-24) || !(d <= 850.0))
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.2e-24) || ~((d <= 850.0)))
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + (((b * d) / c) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.2e-24], N[Not[LessEqual[d, 850.0]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.2000000000000002e-24 or 850 < d

   1. Initial program 55.2%

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

   2. Taylor expanded in c around 0 72.4%

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

      1. associate-/l* 71.4%

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

   4. Simplified 71.4%

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

      1. *-un-lft-identity 71.4%

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

      2. metadata-eval 71.4%

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

      3. unpow2 71.4%

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

      4. associate-*l/ 73.8%

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

      5. times-frac 78.4%

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

      6. metadata-eval 78.4%

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

      7. clear-num 78.4%

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

   6. Applied egg-rr 78.4%

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

   if -7.2000000000000002e-24 < d < 850

   1. Initial program 69.1%

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

   2. Taylor expanded in c around inf 75.6%

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

      1. associate-/l* 73.4%

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

      2. associate-/r/ 72.6%

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

   4. Simplified 72.6%

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

      1. associate-*l/ 75.6%

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

      2. unpow2 75.6%

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

      3. associate-/r* 82.7%

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

   6. Applied egg-rr 82.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{-24} \lor \neg \left(d \leq 850\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]

Alternative 11: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-8} \lor \neg \left(c \leq 2.4 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3e-8) (not (<= c 2.4e-171)))
   (/ a (+ c (* d (/ d c))))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3e-8) || !(c <= 2.4e-171)) {
		tmp = a / (c + (d * (d / 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 <= (-3d-8)) .or. (.not. (c <= 2.4d-171))) then
        tmp = a / (c + (d * (d / 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 <= -3e-8) || !(c <= 2.4e-171)) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3e-8) or not (c <= 2.4e-171):
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3e-8) || !(c <= 2.4e-171))
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3e-8) || ~((c <= 2.4e-171)))
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3e-8], N[Not[LessEqual[c, 2.4e-171]], $MachinePrecision]], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.99999999999999973e-8 or 2.39999999999999987e-171 < c

   1. Initial program 55.9%

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

   2. Taylor expanded in a around inf 44.5%

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

      1. associate-/l* 46.2%

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

      2. +-commutative 46.2%

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

      3. unpow2 46.2%

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

      4. fma-udef 46.2%

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

   4. Simplified 46.2%

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

   5. Taylor expanded in d around 0 64.2%

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

      1. div-inv 64.2%

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

      2. unpow2 64.2%

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

      3. associate-*l* 69.2%

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

      4. div-inv 69.2%

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

   7. Applied egg-rr 69.2%

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

   if -2.99999999999999973e-8 < c < 2.39999999999999987e-171

   1. Initial program 71.0%

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

   2. Taylor expanded in c around 0 75.3%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-8} \lor \neg \left(c \leq 2.4 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 12: 67.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{c + \frac{d}{\frac{c}{d}}}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-168}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.45e-11)
   (/ a (+ c (/ d (/ c d))))
   (if (<= c 1.25e-168) (/ b d) (/ a (+ c (* d (/ d c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.45e-11) {
		tmp = a / (c + (d / (c / d)));
	} else if (c <= 1.25e-168) {
		tmp = b / d;
	} else {
		tmp = a / (c + (d * (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.45d-11)) then
        tmp = a / (c + (d / (c / d)))
    else if (c <= 1.25d-168) then
        tmp = b / d
    else
        tmp = a / (c + (d * (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.45e-11) {
		tmp = a / (c + (d / (c / d)));
	} else if (c <= 1.25e-168) {
		tmp = b / d;
	} else {
		tmp = a / (c + (d * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.45e-11:
		tmp = a / (c + (d / (c / d)))
	elif c <= 1.25e-168:
		tmp = b / d
	else:
		tmp = a / (c + (d * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.45e-11)
		tmp = Float64(a / Float64(c + Float64(d / Float64(c / d))));
	elseif (c <= 1.25e-168)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.45e-11)
		tmp = a / (c + (d / (c / d)));
	elseif (c <= 1.25e-168)
		tmp = b / d;
	else
		tmp = a / (c + (d * (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.45e-11], N[(a / N[(c + N[(d / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.25e-168], N[(b / d), $MachinePrecision], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.45e-11

   1. Initial program 53.0%

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

   2. Taylor expanded in a around inf 44.0%

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

      1. associate-/l* 45.9%

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

      2. +-commutative 45.9%

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

      3. unpow2 45.9%

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

      4. fma-udef 45.9%

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

   4. Simplified 45.9%

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

   5. Taylor expanded in d around 0 62.3%

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

      1. div-inv 62.3%

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

      2. unpow2 62.3%

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

      3. associate-*l* 70.9%

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

      4. div-inv 70.9%

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

   7. Applied egg-rr 70.9%

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

      1. associate-*r/ 62.3%

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

      2. associate-/l* 70.9%

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

   9. Applied egg-rr 70.9%

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

   if -1.45e-11 < c < 1.25e-168

   1. Initial program 71.0%

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

   2. Taylor expanded in c around 0 75.3%

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

   if 1.25e-168 < c

   1. Initial program 58.2%

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

   2. Taylor expanded in a around inf 44.8%

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

      1. associate-/l* 46.5%

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

      2. +-commutative 46.5%

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

      3. unpow2 46.5%

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

      4. fma-udef 46.5%

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

   4. Simplified 46.5%

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

   5. Taylor expanded in d around 0 65.8%

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

      1. div-inv 65.7%

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

      2. unpow2 65.7%

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

      3. associate-*l* 67.8%

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

      4. div-inv 67.9%

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

   7. Applied egg-rr 67.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{c + \frac{d}{\frac{c}{d}}}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-168}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \end{array} \]

Alternative 13: 63.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+39} \lor \neg \left(c \leq 6 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.7e+39) (not (<= c 6e+53))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.7e+39) || !(c <= 6e+53)) {
		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 <= (-1.7d+39)) .or. (.not. (c <= 6d+53))) 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 <= -1.7e+39) || !(c <= 6e+53)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.7e+39) or not (c <= 6e+53):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.7e+39) || !(c <= 6e+53))
		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 <= -1.7e+39) || ~((c <= 6e+53)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.7e+39], N[Not[LessEqual[c, 6e+53]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.6999999999999999e39 or 5.99999999999999996e53 < c

   1. Initial program 49.6%

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

   2. Taylor expanded in c around inf 69.8%

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

   if -1.6999999999999999e39 < c < 5.99999999999999996e53

   1. Initial program 70.7%

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

   2. Taylor expanded in c around 0 64.5%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+39} \lor \neg \left(c \leq 6 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 14: 43.4% 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 41.8%

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

3. Final simplification 41.8%

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

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 2023301 
(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))))