Complex division, real part

Percentage Accurate: 61.9% → 82.4%

Time: 10.5s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.9% 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: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.06 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.2e+23)
     (* (/ d (hypot d c)) (/ b (hypot d c)))
     (if (<= d -8e-44)
       t_0
       (if (<= d 6.8e-139)
         (/ (+ a (/ (* b d) c)) c)
         (if (<= d 2.06e+77) t_0 (/ (+ b (* c (/ a d))) 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 <= -1.2e+23) {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	} else if (d <= -8e-44) {
		tmp = t_0;
	} else if (d <= 6.8e-139) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 2.06e+77) {
		tmp = t_0;
	} else {
		tmp = (b + (c * (a / d))) / 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 <= -1.2e+23) {
		tmp = (d / Math.hypot(d, c)) * (b / Math.hypot(d, c));
	} else if (d <= -8e-44) {
		tmp = t_0;
	} else if (d <= 6.8e-139) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 2.06e+77) {
		tmp = t_0;
	} else {
		tmp = (b + (c * (a / d))) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.2e+23:
		tmp = (d / math.hypot(d, c)) * (b / math.hypot(d, c))
	elif d <= -8e-44:
		tmp = t_0
	elif d <= 6.8e-139:
		tmp = (a + ((b * d) / c)) / c
	elif d <= 2.06e+77:
		tmp = t_0
	else:
		tmp = (b + (c * (a / d))) / 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 <= -1.2e+23)
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	elseif (d <= -8e-44)
		tmp = t_0;
	elseif (d <= 6.8e-139)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif (d <= 2.06e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / 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 <= -1.2e+23)
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	elseif (d <= -8e-44)
		tmp = t_0;
	elseif (d <= 6.8e-139)
		tmp = (a + ((b * d) / c)) / c;
	elseif (d <= 2.06e+77)
		tmp = t_0;
	else
		tmp = (b + (c * (a / d))) / 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, -1.2e+23], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -8e-44], t$95$0, If[LessEqual[d, 6.8e-139], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.06e+77], t$95$0, N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2e23

   1. Initial program 38.8%

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

   3. Taylor expanded in a around 0 34.6%

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

      1. *-commutative 34.6%

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

   5. Simplified 34.6%

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

      1. *-un-lft-identity 34.6%

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

      2. add-sqr-sqrt 34.6%

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

      3. hypot-undefine 34.6%

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

      4. hypot-undefine 34.6%

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

      5. times-frac 46.2%

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

   7. Applied egg-rr 46.2%

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

      1. associate-*l/ 46.3%

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

      2. *-lft-identity 46.3%

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

      3. associate-/l* 81.3%

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

      4. associate-*l/ 81.3%

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

      5. hypot-undefine 42.1%

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

      6. unpow2 42.1%

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

      7. unpow2 42.1%

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

      8. +-commutative 42.1%

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

      9. unpow2 42.1%

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

      10. unpow2 42.1%

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

      11. hypot-define 81.3%

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

      12. hypot-undefine 42.1%

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

      13. unpow2 42.1%

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

      14. unpow2 42.1%

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

      15. +-commutative 42.1%

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

      16. unpow2 42.1%

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

      17. unpow2 42.1%

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

      18. hypot-define 81.3%

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

   9. Simplified 81.3%

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

   if -1.2e23 < d < -7.99999999999999962e-44 or 6.79999999999999998e-139 < d < 2.06e77

   1. Initial program 80.3%

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

   if -7.99999999999999962e-44 < d < 6.79999999999999998e-139

   1. Initial program 64.5%

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

   3. Taylor expanded in c around inf 90.4%

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

      1. *-commutative 90.4%

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

   5. Simplified 90.4%

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

   if 2.06e77 < d

   1. Initial program 40.8%

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

   3. Taylor expanded in d around inf 73.4%

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

      1. associate-/l* 82.5%

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

   5. Simplified 82.5%

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

      1. clear-num 82.5%

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

      2. un-div-inv 82.5%

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

   7. Applied egg-rr 82.5%

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

      1. associate-/r/ 84.9%

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

   9. Simplified 84.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq -8 \cdot 10^{-44}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;\frac{b + \frac{a}{\frac{d}{c}}}{d}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+238}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 (- INFINITY))
     (/ (+ b (/ a (/ d c))) d)
     (if (<= t_0 4e+238)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (/ (+ a (/ b (/ c d))) c)))))

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 (t_0 <= -((double) INFINITY)) {
		tmp = (b + (a / (d / c))) / d;
	} else if (t_0 <= 4e+238) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a + (b / (c / d))) / c;
	}
	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 (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(b + Float64(a / Float64(d / c))) / d);
	elseif (t_0 <= 4e+238)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[t$95$0, 4e+238], 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[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 54.4%

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

   3. Taylor expanded in d around inf 83.4%

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

      1. associate-/l* 87.5%

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

   5. Simplified 87.5%

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

      1. clear-num 87.5%

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

      2. un-div-inv 87.6%

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

   7. Applied egg-rr 87.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 4.0000000000000002e238

   1. Initial program 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.7%

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

      4. hypot-define 78.7%

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

      5. fma-define 78.7%

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

      6. hypot-define 97.3%

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

   4. Applied egg-rr 97.3%

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

   if 4.0000000000000002e238 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.4%

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

      1. *-un-lft-identity 14.4%

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

      2. add-sqr-sqrt 14.4%

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

      3. times-frac 14.4%

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

      4. hypot-define 14.4%

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

      5. fma-define 14.4%

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

      6. hypot-define 22.1%

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

   4. Applied egg-rr 22.1%

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

   5. Taylor expanded in c around inf 52.4%

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

      1. associate-/l* 60.9%

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

   7. Simplified 60.9%

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

      1. clear-num 61.0%

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

      2. un-div-inv 61.0%

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

   9. Applied egg-rr 61.0%

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

Alternative 3: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-140}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* c (/ a d))) d)))
   (if (<= d -1.3e-43)
     t_0
     (if (<= d 2.05e-140)
       (/ (+ a (/ (* b d) c)) c)
       (if (<= d 1.85e+77) (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -1.3e-43) {
		tmp = t_0;
	} else if (d <= 2.05e-140) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 1.85e+77) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b + (c * (a / d))) / d
    if (d <= (-1.3d-43)) then
        tmp = t_0
    else if (d <= 2.05d-140) then
        tmp = (a + ((b * d) / c)) / c
    else if (d <= 1.85d+77) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b + (c * (a / d))) / d;
	double tmp;
	if (d <= -1.3e-43) {
		tmp = t_0;
	} else if (d <= 2.05e-140) {
		tmp = (a + ((b * d) / c)) / c;
	} else if (d <= 1.85e+77) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (c * (a / d))) / d
	tmp = 0
	if d <= -1.3e-43:
		tmp = t_0
	elif d <= 2.05e-140:
		tmp = (a + ((b * d) / c)) / c
	elif d <= 1.85e+77:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(c * Float64(a / d))) / d)
	tmp = 0.0
	if (d <= -1.3e-43)
		tmp = t_0;
	elseif (d <= 2.05e-140)
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	elseif (d <= 1.85e+77)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (c * (a / d))) / d;
	tmp = 0.0;
	if (d <= -1.3e-43)
		tmp = t_0;
	elseif (d <= 2.05e-140)
		tmp = (a + ((b * d) / c)) / c;
	elseif (d <= 1.85e+77)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.3e-43], t$95$0, If[LessEqual[d, 2.05e-140], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.85e+77], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.3e-43 or 1.84999999999999997e77 < d

   1. Initial program 45.2%

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

   3. Taylor expanded in d around inf 74.3%

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

      1. associate-/l* 78.3%

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

   5. Simplified 78.3%

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

      1. clear-num 78.1%

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

      2. un-div-inv 78.1%

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

   7. Applied egg-rr 78.1%

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

      1. associate-/r/ 79.9%

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

   9. Simplified 79.9%

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

   if -1.3e-43 < d < 2.0500000000000001e-140

   1. Initial program 64.5%

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

   3. Taylor expanded in c around inf 90.4%

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

      1. *-commutative 90.4%

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

   5. Simplified 90.4%

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

   if 2.0500000000000001e-140 < d < 1.84999999999999997e77

   1. Initial program 75.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{-140}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+77}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-43} \lor \neg \left(d \leq 5.5 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.15e-43) (not (<= d 5.5e+56)))
   (/ (+ b (* c (/ a d))) d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e-43) || !(d <= 5.5e+56)) {
		tmp = (b + (c * (a / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.15d-43)) .or. (.not. (d <= 5.5d+56))) then
        tmp = (b + (c * (a / d))) / d
    else
        tmp = (a + ((b * d) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e-43) || !(d <= 5.5e+56)) {
		tmp = (b + (c * (a / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.15e-43) or not (d <= 5.5e+56):
		tmp = (b + (c * (a / d))) / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.15e-43) || !(d <= 5.5e+56))
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.15e-43) || ~((d <= 5.5e+56)))
		tmp = (b + (c * (a / d))) / d;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.15e-43], N[Not[LessEqual[d, 5.5e+56]], $MachinePrecision]], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.1499999999999999e-43 or 5.5000000000000002e56 < d

   1. Initial program 46.5%

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

   3. Taylor expanded in d around inf 73.7%

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

      1. associate-/l* 77.6%

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

   5. Simplified 77.6%

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

      1. clear-num 77.4%

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

      2. un-div-inv 77.4%

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

   7. Applied egg-rr 77.4%

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

      1. associate-/r/ 79.1%

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

   9. Simplified 79.1%

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

   if -1.1499999999999999e-43 < d < 5.5000000000000002e56

   1. Initial program 66.6%

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

   3. Taylor expanded in c around inf 83.4%

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

      1. *-commutative 83.4%

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

   5. Simplified 83.4%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{-43} \lor \neg \left(d \leq 5.5 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{b + c \cdot \frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-44} \lor \neg \left(d \leq 2.05 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.2e-44) (not (<= d 2.05e-14)))
   (/ (+ b (* a (/ c d))) d)
   (/ (+ a (/ (* b d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e-44) || !(d <= 2.05e-14)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.2d-44)) .or. (.not. (d <= 2.05d-14))) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + ((b * d) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e-44) || !(d <= 2.05e-14)) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + ((b * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.2e-44) or not (d <= 2.05e-14):
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + ((b * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.2e-44) || !(d <= 2.05e-14))
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(Float64(b * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.2e-44) || ~((d <= 2.05e-14)))
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + ((b * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.2e-44], N[Not[LessEqual[d, 2.05e-14]], $MachinePrecision]], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.19999999999999984e-44 or 2.0500000000000001e-14 < d

   1. Initial program 48.9%

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

   3. Taylor expanded in d around inf 72.4%

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

      1. associate-/l* 76.1%

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

   5. Simplified 76.1%

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

   if -8.19999999999999984e-44 < d < 2.0500000000000001e-14

   1. Initial program 65.2%

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

   3. Taylor expanded in c around inf 85.5%

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

      1. *-commutative 85.5%

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

   5. Simplified 85.5%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{-44} \lor \neg \left(d \leq 2.05 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+50} \lor \neg \left(d \leq 5.4 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.2e+50) (not (<= d 5.4e+57)))
   (/ b d)
   (/ (+ a (* b (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+50) || !(d <= 5.4e+57)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-7.2d+50)) .or. (.not. (d <= 5.4d+57))) then
        tmp = b / d
    else
        tmp = (a + (b * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+50) || !(d <= 5.4e+57)) {
		tmp = b / d;
	} else {
		tmp = (a + (b * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.2e+50) or not (d <= 5.4e+57):
		tmp = b / d
	else:
		tmp = (a + (b * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.2e+50) || !(d <= 5.4e+57))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.2e+50) || ~((d <= 5.4e+57)))
		tmp = b / d;
	else
		tmp = (a + (b * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.2e+50], N[Not[LessEqual[d, 5.4e+57]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.19999999999999972e50 or 5.3999999999999997e57 < d

   1. Initial program 42.5%

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

   3. Taylor expanded in c around 0 70.3%

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

   if -7.19999999999999972e50 < d < 5.3999999999999997e57

   1. Initial program 67.6%

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

      1. *-un-lft-identity 67.6%

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

      2. add-sqr-sqrt 67.6%

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

      3. times-frac 67.6%

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

      4. hypot-define 67.7%

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

      5. fma-define 67.7%

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

      6. hypot-define 80.9%

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

   4. Applied egg-rr 80.9%

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

   5. Taylor expanded in c around inf 78.5%

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

      1. associate-/l* 78.5%

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

   7. Simplified 78.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+50} \lor \neg \left(d \leq 5.4 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.85 \cdot 10^{+102} \lor \neg \left(c \leq 2.4 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.85e+102) (not (<= c 2.4e+43))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.85e+102) || !(c <= 2.4e+43)) {
		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 <= (-3.85d+102)) .or. (.not. (c <= 2.4d+43))) 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 <= -3.85e+102) || !(c <= 2.4e+43)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.85e+102) or not (c <= 2.4e+43):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.85e+102) || !(c <= 2.4e+43))
		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 <= -3.85e+102) || ~((c <= 2.4e+43)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.85e+102], N[Not[LessEqual[c, 2.4e+43]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.85000000000000007e102 or 2.40000000000000023e43 < c

   1. Initial program 38.5%

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

   3. Taylor expanded in c around inf 74.1%

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

   if -3.85000000000000007e102 < c < 2.40000000000000023e43

   1. Initial program 67.4%

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

   3. Taylor expanded in c around 0 61.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.85 \cdot 10^{+102} \lor \neg \left(c \leq 2.4 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.6% 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 56.4%

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

3. Taylor expanded in c around inf 40.7%

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

Developer Target 1: 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 2024165 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

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

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