Complex division, real part

Percentage Accurate: 61.7% → 84.8%

Time: 11.6s

Alternatives: 11

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: 61.7% 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

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) INFINITY)
   (/ (/ (fma a c (* b d)) (hypot c d)) (hypot c d))
   (* (/ c (hypot c d)) (/ a (hypot c 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 = (fma(a, c, (b * d)) / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (c / hypot(c, d)) * (a / hypot(c, 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(fma(a, c, Float64(b * d)) / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, 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[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. associate-*r/ 80.4%

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

      3. fma-define 80.4%

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

      4. add-sqr-sqrt 80.4%

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

      5. times-frac 80.3%

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

      6. fma-define 80.3%

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

      7. hypot-define 80.4%

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

      8. fma-define 80.4%

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

      9. fma-define 80.4%

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

      10. hypot-define 94.9%

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

   4. Applied egg-rr 94.9%

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

      1. associate-*l/ 95.0%

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

      2. *-un-lft-identity 95.0%

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

   6. Applied egg-rr 95.0%

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

   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. Add Preprocessing

   3. Taylor expanded in a around inf 1.1%

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

      1. *-commutative 1.1%

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

      2. rem-square-sqrt 1.1%

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

      3. unpow2 1.1%

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

      4. unpow2 1.1%

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

      5. hypot-undefine 1.1%

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

      6. unpow2 1.1%

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

      7. unpow2 1.1%

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

      8. hypot-undefine 1.1%

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

      9. unpow2 1.1%

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

      10. hypot-undefine 1.1%

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

      11. unpow2 1.1%

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

      12. unpow2 1.1%

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

      13. +-commutative 1.1%

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

      14. unpow2 1.1%

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

      15. unpow2 1.1%

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

      16. hypot-define 1.1%

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

   5. Simplified 1.1%

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

      1. hypot-undefine 1.1%

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

      2. +-commutative 1.1%

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

      3. hypot-undefine 1.1%

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

      4. pow2 1.1%

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

      5. times-frac 62.4%

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

   7. Applied egg-rr 62.4%

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

4. Final simplification 87.9%

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

Alternative 2: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -170:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -170.0)
   (/ (- (- b) (* c (/ a d))) (hypot c d))
   (if (<= d 1.05e-24)
     (+ (/ a c) (/ (/ (* b d) c) c))
     (if (<= d 2.85e+76)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (* (/ 1.0 (hypot c d)) (+ b (* a (/ c d))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -170.0) {
		tmp = (-b - (c * (a / d))) / hypot(c, d);
	} else if (d <= 1.05e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 2.85e+76) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -170.0) {
		tmp = (-b - (c * (a / d))) / Math.hypot(c, d);
	} else if (d <= 1.05e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 2.85e+76) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -170.0:
		tmp = (-b - (c * (a / d))) / math.hypot(c, d)
	elif d <= 1.05e-24:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 2.85e+76:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -170.0)
		tmp = Float64(Float64(Float64(-b) - Float64(c * Float64(a / d))) / hypot(c, d));
	elseif (d <= 1.05e-24)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 2.85e+76)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -170.0)
		tmp = (-b - (c * (a / d))) / hypot(c, d);
	elseif (d <= 1.05e-24)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 2.85e+76)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (1.0 / hypot(c, d)) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -170.0], N[(N[((-b) - N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e-24], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.85e+76], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -170

   1. Initial program 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-*r/ 50.2%

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

      3. fma-define 50.2%

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

      4. add-sqr-sqrt 50.2%

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

      5. times-frac 50.1%

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

      6. fma-define 50.1%

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

      7. hypot-define 50.1%

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

      8. fma-define 50.1%

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

      9. fma-define 50.1%

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

      10. hypot-define 63.8%

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

   4. Applied egg-rr 63.8%

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

   5. Taylor expanded in d around -inf 71.0%

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

      1. frac-2neg 71.0%

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

      2. metadata-eval 71.0%

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

      3. associate-*l/ 71.1%

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

   7. Applied egg-rr 77.6%

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

   if -170 < d < 1.05e-24

   1. Initial program 69.3%

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

   3. Taylor expanded in c around inf 80.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 80.2%

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

      3. associate-/r* 86.7%

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

   7. Applied egg-rr 86.7%

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

   if 1.05e-24 < d < 2.85000000000000002e76

   1. Initial program 94.4%

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

   if 2.85000000000000002e76 < d

   1. Initial program 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. associate-*r/ 53.8%

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

      3. fma-define 53.8%

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

      4. add-sqr-sqrt 53.8%

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

      5. times-frac 53.8%

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

      6. fma-define 53.8%

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

      7. hypot-define 53.8%

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

      8. fma-define 53.8%

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

      9. fma-define 53.8%

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

      10. hypot-define 68.0%

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

   4. Applied egg-rr 68.0%

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

   5. Taylor expanded in c around 0 78.8%

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

      1. associate-/l* 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -170:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.85 \cdot 10^{+76}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -26000:\\ \;\;\;\;\frac{\left(-b\right) - t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+91}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + t\_0 \cdot \frac{1}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* c (/ a d))))
   (if (<= d -26000.0)
     (/ (- (- b) t_0) (hypot c d))
     (if (<= d 1e-24)
       (+ (/ a c) (/ (/ (* b d) c) c))
       (if (<= d 7e+91)
         (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
         (+ (/ b d) (* t_0 (/ 1.0 d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c * (a / d);
	double tmp;
	if (d <= -26000.0) {
		tmp = (-b - t_0) / hypot(c, d);
	} else if (d <= 1e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 7e+91) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b / d) + (t_0 * (1.0 / d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = c * (a / d);
	double tmp;
	if (d <= -26000.0) {
		tmp = (-b - t_0) / Math.hypot(c, d);
	} else if (d <= 1e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 7e+91) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b / d) + (t_0 * (1.0 / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = c * (a / d)
	tmp = 0
	if d <= -26000.0:
		tmp = (-b - t_0) / math.hypot(c, d)
	elif d <= 1e-24:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 7e+91:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (b / d) + (t_0 * (1.0 / d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(c * Float64(a / d))
	tmp = 0.0
	if (d <= -26000.0)
		tmp = Float64(Float64(Float64(-b) - t_0) / hypot(c, d));
	elseif (d <= 1e-24)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 7e+91)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b / d) + Float64(t_0 * Float64(1.0 / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = c * (a / d);
	tmp = 0.0;
	if (d <= -26000.0)
		tmp = (-b - t_0) / hypot(c, d);
	elseif (d <= 1e-24)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 7e+91)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (b / d) + (t_0 * (1.0 / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -26000.0], N[(N[((-b) - t$95$0), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1e-24], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7e+91], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(t$95$0 * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -26000

   1. Initial program 50.2%

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

      1. *-un-lft-identity 50.2%

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

      2. associate-*r/ 50.2%

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

      3. fma-define 50.2%

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

      4. add-sqr-sqrt 50.2%

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

      5. times-frac 50.1%

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

      6. fma-define 50.1%

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

      7. hypot-define 50.1%

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

      8. fma-define 50.1%

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

      9. fma-define 50.1%

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

      10. hypot-define 63.8%

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

   4. Applied egg-rr 63.8%

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

   5. Taylor expanded in d around -inf 71.0%

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

      1. frac-2neg 71.0%

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

      2. metadata-eval 71.0%

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

      3. associate-*l/ 71.1%

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

   7. Applied egg-rr 77.6%

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

   if -26000 < d < 9.99999999999999924e-25

   1. Initial program 69.3%

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

   3. Taylor expanded in c around inf 80.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 80.2%

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

      3. associate-/r* 86.7%

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

   7. Applied egg-rr 86.7%

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

   if 9.99999999999999924e-25 < d < 7.00000000000000001e91

   1. Initial program 94.9%

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

   if 7.00000000000000001e91 < d

   1. Initial program 52.0%

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

   3. Taylor expanded in c around 0 75.8%

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

      1. *-un-lft-identity 75.8%

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

      2. unpow2 75.8%

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

      3. times-frac 77.8%

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

      4. *-commutative 77.8%

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

      5. associate-/l* 85.7%

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

   5. Applied egg-rr 85.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -26000:\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+91}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{if}\;d \leq -1150:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (* (* c (/ a d)) (/ 1.0 d)))))
   (if (<= d -1150.0)
     t_0
     (if (<= d 4e-24)
       (+ (/ a c) (/ (/ (* b d) c) c))
       (if (<= d 7.3e+90) (/ (+ (* 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 / d) + ((c * (a / d)) * (1.0 / d));
	double tmp;
	if (d <= -1150.0) {
		tmp = t_0;
	} else if (d <= 4e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 7.3e+90) {
		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 / d) + ((c * (a / d)) * (1.0d0 / d))
    if (d <= (-1150.0d0)) then
        tmp = t_0
    else if (d <= 4d-24) then
        tmp = (a / c) + (((b * d) / c) / c)
    else if (d <= 7.3d+90) 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 / d) + ((c * (a / d)) * (1.0 / d));
	double tmp;
	if (d <= -1150.0) {
		tmp = t_0;
	} else if (d <= 4e-24) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else if (d <= 7.3e+90) {
		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 / d) + ((c * (a / d)) * (1.0 / d))
	tmp = 0
	if d <= -1150.0:
		tmp = t_0
	elif d <= 4e-24:
		tmp = (a / c) + (((b * d) / c) / c)
	elif d <= 7.3e+90:
		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 / d) + Float64(Float64(c * Float64(a / d)) * Float64(1.0 / d)))
	tmp = 0.0
	if (d <= -1150.0)
		tmp = t_0;
	elseif (d <= 4e-24)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	elseif (d <= 7.3e+90)
		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 / d) + ((c * (a / d)) * (1.0 / d));
	tmp = 0.0;
	if (d <= -1150.0)
		tmp = t_0;
	elseif (d <= 4e-24)
		tmp = (a / c) + (((b * d) / c) / c);
	elseif (d <= 7.3e+90)
		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 / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1150.0], t$95$0, If[LessEqual[d, 4e-24], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.3e+90], 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 < -1150 or 7.29999999999999995e90 < d

   1. Initial program 50.9%

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

   3. Taylor expanded in c around 0 71.9%

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

      1. *-un-lft-identity 71.9%

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

      2. unpow2 71.9%

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

      3. times-frac 73.8%

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

      4. *-commutative 73.8%

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

      5. associate-/l* 80.8%

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

   5. Applied egg-rr 80.8%

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

   if -1150 < d < 3.99999999999999969e-24

   1. Initial program 69.3%

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

   3. Taylor expanded in c around inf 80.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 80.2%

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

      3. associate-/r* 86.7%

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

   7. Applied egg-rr 86.7%

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

   if 3.99999999999999969e-24 < d < 7.29999999999999995e90

   1. Initial program 94.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1150:\\ \;\;\;\;\frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -145000 \lor \neg \left(d \leq 0.0145\right):\\ \;\;\;\;\frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -145000.0) (not (<= d 0.0145)))
   (+ (/ b d) (* (* c (/ a d)) (/ 1.0 d)))
   (+ (/ a c) (/ (/ (* b d) c) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -145000.0) || !(d <= 0.0145)) {
		tmp = (b / d) + ((c * (a / d)) * (1.0 / 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 <= (-145000.0d0)) .or. (.not. (d <= 0.0145d0))) then
        tmp = (b / d) + ((c * (a / d)) * (1.0d0 / 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 <= -145000.0) || !(d <= 0.0145)) {
		tmp = (b / d) + ((c * (a / d)) * (1.0 / d));
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -145000.0) or not (d <= 0.0145):
		tmp = (b / d) + ((c * (a / d)) * (1.0 / d))
	else:
		tmp = (a / c) + (((b * d) / c) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -145000.0) || !(d <= 0.0145))
		tmp = Float64(Float64(b / d) + Float64(Float64(c * Float64(a / d)) * Float64(1.0 / 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 <= -145000.0) || ~((d <= 0.0145)))
		tmp = (b / d) + ((c * (a / d)) * (1.0 / d));
	else
		tmp = (a / c) + (((b * d) / c) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -145000.0], N[Not[LessEqual[d, 0.0145]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / 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 < -145000 or 0.0145000000000000007 < d

   1. Initial program 56.0%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-/l* 79.9%

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

   5. Applied egg-rr 79.9%

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

   if -145000 < d < 0.0145000000000000007

   1. Initial program 70.3%

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

   3. Taylor expanded in c around inf 79.8%

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

      1. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

      1. pow2 76.8%

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

      2. associate-*r/ 79.8%

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

      3. associate-/r* 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -145000 \lor \neg \left(d \leq 0.0145\right):\\ \;\;\;\;\frac{b}{d} + \left(c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -82000 \lor \neg \left(d \leq 1.9 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -82000.0) (not (<= d 1.9e+21)))
   (/ b d)
   (* (/ 1.0 c) (+ a (* b (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -82000.0) || !(d <= 1.9e+21)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + (b * (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 ((d <= (-82000.0d0)) .or. (.not. (d <= 1.9d+21))) then
        tmp = b / d
    else
        tmp = (1.0d0 / c) * (a + (b * (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -82000.0) || !(d <= 1.9e+21)) {
		tmp = b / d;
	} else {
		tmp = (1.0 / c) * (a + (b * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -82000.0) or not (d <= 1.9e+21):
		tmp = b / d
	else:
		tmp = (1.0 / c) * (a + (b * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -82000.0) || !(d <= 1.9e+21))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -82000.0) || ~((d <= 1.9e+21)))
		tmp = b / d;
	else
		tmp = (1.0 / c) * (a + (b * (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -82000.0], N[Not[LessEqual[d, 1.9e+21]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -82000 or 1.9e21 < d

   1. Initial program 55.0%

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

   3. Taylor expanded in c around 0 65.0%

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

   if -82000 < d < 1.9e21

   1. Initial program 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. associate-*r/ 70.9%

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

      3. fma-define 70.9%

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

      4. add-sqr-sqrt 70.9%

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

      5. times-frac 70.9%

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

      6. fma-define 70.9%

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

      7. hypot-define 71.0%

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

      8. fma-define 71.0%

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

      9. fma-define 70.9%

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

      10. hypot-define 81.8%

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

   4. Applied egg-rr 81.8%

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

   5. Taylor expanded in c around inf 46.8%

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

      1. associate-/l* 46.3%

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

   7. Simplified 46.3%

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

   8. Taylor expanded in c around inf 84.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -82000 \lor \neg \left(d \leq 1.9 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -40000 \lor \neg \left(d \leq 6.4\right):\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -40000.0) (not (<= d 6.4)))
   (* (+ b (* c (/ a d))) (/ 1.0 d))
   (* (/ 1.0 c) (+ a (* b (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -40000.0) || !(d <= 6.4)) {
		tmp = (b + (c * (a / d))) * (1.0 / d);
	} else {
		tmp = (1.0 / c) * (a + (b * (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 ((d <= (-40000.0d0)) .or. (.not. (d <= 6.4d0))) then
        tmp = (b + (c * (a / d))) * (1.0d0 / d)
    else
        tmp = (1.0d0 / c) * (a + (b * (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -40000.0) || !(d <= 6.4)) {
		tmp = (b + (c * (a / d))) * (1.0 / d);
	} else {
		tmp = (1.0 / c) * (a + (b * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -40000.0) or not (d <= 6.4):
		tmp = (b + (c * (a / d))) * (1.0 / d)
	else:
		tmp = (1.0 / c) * (a + (b * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -40000.0) || !(d <= 6.4))
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) * Float64(1.0 / d));
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -40000.0) || ~((d <= 6.4)))
		tmp = (b + (c * (a / d))) * (1.0 / d);
	else
		tmp = (1.0 / c) * (a + (b * (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -40000.0], N[Not[LessEqual[d, 6.4]], $MachinePrecision]], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4e4 or 6.4000000000000004 < d

   1. Initial program 56.0%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-/l* 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. +-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. div-inv 79.8%

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

      4. distribute-rgt-out 79.8%

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

   7. Applied egg-rr 79.8%

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

   if -4e4 < d < 6.4000000000000004

   1. Initial program 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. associate-*r/ 70.3%

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

      3. fma-define 70.2%

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

      4. add-sqr-sqrt 70.2%

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

      5. times-frac 70.2%

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

      6. fma-define 70.2%

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

      7. hypot-define 70.3%

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

      8. fma-define 70.3%

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

      9. fma-define 70.2%

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

      10. hypot-define 81.4%

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

   4. Applied egg-rr 81.4%

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

   5. Taylor expanded in c around inf 47.9%

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

      1. associate-/l* 47.4%

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

   7. Simplified 47.4%

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

   8. Taylor expanded in c around inf 85.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -40000 \lor \neg \left(d \leq 6.4\right):\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + b \cdot \frac{d}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -82000 \lor \neg \left(d \leq 380\right):\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -82000.0) (not (<= d 380.0)))
   (* (+ b (* c (/ a d))) (/ 1.0 d))
   (+ (/ a c) (/ (/ (* b d) c) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -82000.0) || !(d <= 380.0)) {
		tmp = (b + (c * (a / d))) * (1.0 / 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 <= (-82000.0d0)) .or. (.not. (d <= 380.0d0))) then
        tmp = (b + (c * (a / d))) * (1.0d0 / 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 <= -82000.0) || !(d <= 380.0)) {
		tmp = (b + (c * (a / d))) * (1.0 / d);
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -82000.0) or not (d <= 380.0):
		tmp = (b + (c * (a / d))) * (1.0 / d)
	else:
		tmp = (a / c) + (((b * d) / c) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -82000.0) || !(d <= 380.0))
		tmp = Float64(Float64(b + Float64(c * Float64(a / d))) * Float64(1.0 / 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 <= -82000.0) || ~((d <= 380.0)))
		tmp = (b + (c * (a / d))) * (1.0 / d);
	else
		tmp = (a / c) + (((b * d) / c) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -82000.0], N[Not[LessEqual[d, 380.0]], $MachinePrecision]], N[(N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / d), $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 < -82000 or 380 < d

   1. Initial program 56.0%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. *-un-lft-identity 72.0%

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

      2. unpow2 72.0%

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

      3. times-frac 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-/l* 79.9%

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

   5. Applied egg-rr 79.9%

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

      1. +-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. div-inv 79.8%

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

      4. distribute-rgt-out 79.8%

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

   7. Applied egg-rr 79.8%

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

   if -82000 < d < 380

   1. Initial program 70.3%

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

   3. Taylor expanded in c around inf 79.8%

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

      1. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

      1. pow2 76.8%

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

      2. associate-*r/ 79.8%

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

      3. associate-/r* 85.9%

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

   7. Applied egg-rr 85.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -82000 \lor \neg \left(d \leq 380\right):\\ \;\;\;\;\left(b + c \cdot \frac{a}{d}\right) \cdot \frac{1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -50000 \lor \neg \left(d \leq 4.3 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -50000.0) (not (<= d 4.3e+20))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -50000.0) || !(d <= 4.3e+20)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-50000.0d0)) .or. (.not. (d <= 4.3d+20))) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -50000.0) || !(d <= 4.3e+20)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -50000.0) or not (d <= 4.3e+20):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -50000.0) || !(d <= 4.3e+20))
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -50000.0) || ~((d <= 4.3e+20)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -50000.0], N[Not[LessEqual[d, 4.3e+20]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5e4 or 4.3e20 < d

   1. Initial program 55.0%

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

   3. Taylor expanded in c around 0 65.0%

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

   if -5e4 < d < 4.3e20

   1. Initial program 70.9%

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

   3. Taylor expanded in c around inf 64.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -50000 \lor \neg \left(d \leq 4.3 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 2e+218) (/ a c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 2e+218) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 2d+218) then
        tmp = a / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 2e+218) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 2e+218:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 2e+218)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 2e+218)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 2e+218], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.00000000000000017e218

   1. Initial program 63.4%

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

   3. Taylor expanded in c around inf 44.6%

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

   if 2.00000000000000017e218 < d

   1. Initial program 57.1%

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

      1. *-un-lft-identity 57.1%

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

      2. associate-*r/ 57.1%

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

      3. fma-define 57.1%

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

      4. add-sqr-sqrt 57.1%

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

      5. times-frac 57.1%

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

      6. fma-define 57.1%

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

      7. hypot-define 57.1%

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

      8. fma-define 57.1%

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

      9. fma-define 57.1%

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

      10. hypot-define 72.5%

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

   4. Applied egg-rr 72.5%

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

   5. Taylor expanded in d around -inf 57.3%

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

   6. Taylor expanded in c around -inf 39.4%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

3. Taylor expanded in c around inf 41.7%

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

4. Final simplification 41.7%

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

Developer target: 99.4% 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 2024044 
(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))))