Complex division, real part

Percentage Accurate: 61.5% → 84.8%

Time: 13.3s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 10^{+235}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+235)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (/ (- (- b) (* a (/ c d))) (- d)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+235) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (-b - (a * (c / d))) / -d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+235) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (-b - (a * (c / d))) / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+235:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (-b - (a * (c / d))) / -d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+235)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+235)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (-b - (a * (c / d))) / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+235], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $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))) < 1.0000000000000001e235

   1. Initial program 80.1%

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

      1. +-commutative 80.1%

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

      2. fma-udef 80.1%

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

      3. *-un-lft-identity 80.1%

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

      4. associate-*r/ 80.1%

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

      5. add-sqr-sqrt 80.1%

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

      6. times-frac 80.0%

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

      7. fma-udef 80.0%

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

      8. +-commutative 80.0%

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

      9. hypot-def 80.1%

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

      10. fma-def 80.0%

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

      11. fma-udef 80.0%

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

      12. +-commutative 80.0%

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

      13. hypot-def 97.9%

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

   4. Applied egg-rr 97.9%

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

      1. fma-def 98.0%

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

      2. +-commutative 98.0%

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

   6. Applied egg-rr 98.0%

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

   if 1.0000000000000001e235 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.7%

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

   3. Taylor expanded in c around 0 46.2%

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

      1. associate-/l* 51.9%

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

      2. associate-/r/ 50.3%

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

   5. Simplified 50.3%

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

      1. pow2 50.3%

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

      2. associate-*l/ 46.2%

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

      3. *-commutative 46.2%

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

      4. associate-/r* 51.3%

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

      5. *-commutative 51.3%

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

   7. Applied egg-rr 51.3%

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

      1. div-inv 51.3%

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

      2. associate-/l* 64.0%

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

      3. add-sqr-sqrt 31.7%

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

      4. sqrt-unprod 33.8%

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

      5. sqr-neg 33.8%

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

      6. sqrt-unprod 19.3%

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

      7. add-sqr-sqrt 46.3%

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

      8. distribute-frac-neg 46.3%

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

      9. associate-/l* 43.0%

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

      10. cancel-sign-sub-inv 43.0%

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

      11. div-inv 43.0%

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

      12. frac-2neg 43.0%

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

      13. frac-2neg 43.0%

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

      14. associate-/l* 46.3%

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

      15. distribute-frac-neg 46.3%

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

      16. sub-div 46.3%

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

   9. Applied egg-rr 64.1%

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

       1. associate-*l/ 52.9%

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

       2. associate-*r/ 65.7%

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

   11. Simplified 65.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+235}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a \cdot c + b \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.36 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.3e+81)
   (* (/ c (hypot c d)) (/ a (hypot c d)))
   (if (<= c -2.2e-105)
     (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
     (if (<= c 7.6e-35)
       (/ (- (- b) (* a (/ c d))) (- d))
       (if (<= c 1.36e+101)
         (/ (fma a c (* b d)) (fma d d (* c c)))
         (* (/ 1.0 (hypot c d)) (+ a (* d (/ b c)))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.3e+81) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (c <= -2.2e-105) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 7.6e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.36e+101) {
		tmp = fma(a, c, (b * d)) / fma(d, d, (c * c));
	} else {
		tmp = (1.0 / hypot(c, d)) * (a + (d * (b / c)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.3e+81)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (c <= -2.2e-105)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 7.6e-35)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 1.36e+101)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(d, d, Float64(c * c)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(a + Float64(d * Float64(b / c))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.3e+81], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.2e-105], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.6e-35], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.36e+101], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3.3e81

   1. Initial program 49.6%

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

   3. Taylor expanded in a around inf 46.3%

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

      1. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. add-sqr-sqrt 46.3%

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

      2. hypot-udef 46.3%

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

      3. hypot-udef 46.3%

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

      4. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

   if -3.3e81 < c < -2.20000000000000004e-105

   1. Initial program 89.6%

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

   if -2.20000000000000004e-105 < c < 7.6000000000000002e-35

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 7.6000000000000002e-35 < c < 1.35999999999999998e101

   1. Initial program 80.2%

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

      1. fma-def 80.2%

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

      2. +-commutative 80.2%

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

      3. fma-def 80.2%

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

   3. Simplified 80.2%

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

   if 1.35999999999999998e101 < c

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-udef 42.7%

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

      3. *-un-lft-identity 42.7%

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

      4. associate-*r/ 42.7%

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

      5. add-sqr-sqrt 42.7%

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

      6. times-frac 42.7%

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

      7. fma-udef 42.7%

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

      8. +-commutative 42.7%

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

      9. hypot-def 42.7%

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

      10. fma-def 42.7%

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

      11. fma-udef 42.7%

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

      12. +-commutative 42.7%

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

      13. hypot-def 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in c around inf 82.0%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.3 \cdot 10^{+81}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.36 \cdot 10^{+101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -8.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+103}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -8.8e+82)
     (* (/ c (hypot c d)) (/ a (hypot c d)))
     (if (<= c -1.15e-103)
       t_0
       (if (<= c 7e-35)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 2.8e+103)
           t_0
           (* (/ 1.0 (hypot c d)) (+ a (* d (/ b 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 (c <= -8.8e+82) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (c <= -1.15e-103) {
		tmp = t_0;
	} else if (c <= 7e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 2.8e+103) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (a + (d * (b / c)));
	}
	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 (c <= -8.8e+82) {
		tmp = (c / Math.hypot(c, d)) * (a / Math.hypot(c, d));
	} else if (c <= -1.15e-103) {
		tmp = t_0;
	} else if (c <= 7e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 2.8e+103) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (a + (d * (b / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -8.8e+82:
		tmp = (c / math.hypot(c, d)) * (a / math.hypot(c, d))
	elif c <= -1.15e-103:
		tmp = t_0
	elif c <= 7e-35:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 2.8e+103:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (a + (d * (b / 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 (c <= -8.8e+82)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (c <= -1.15e-103)
		tmp = t_0;
	elseif (c <= 7e-35)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 2.8e+103)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(a + Float64(d * Float64(b / c))));
	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 (c <= -8.8e+82)
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	elseif (c <= -1.15e-103)
		tmp = t_0;
	elseif (c <= 7e-35)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 2.8e+103)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (a + (d * (b / c)));
	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[c, -8.8e+82], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.15e-103], t$95$0, If[LessEqual[c, 7e-35], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 2.8e+103], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -8.8000000000000005e82

   1. Initial program 49.6%

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

   3. Taylor expanded in a around inf 46.3%

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

      1. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. add-sqr-sqrt 46.3%

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

      2. hypot-udef 46.3%

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

      3. hypot-udef 46.3%

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

      4. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

   if -8.8000000000000005e82 < c < -1.15e-103 or 6.99999999999999992e-35 < c < 2.80000000000000008e103

   1. Initial program 85.1%

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

   if -1.15e-103 < c < 6.99999999999999992e-35

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 2.80000000000000008e103 < c

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-udef 42.7%

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

      3. *-un-lft-identity 42.7%

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

      4. associate-*r/ 42.7%

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

      5. add-sqr-sqrt 42.7%

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

      6. times-frac 42.7%

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

      7. fma-udef 42.7%

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

      8. +-commutative 42.7%

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

      9. hypot-def 42.7%

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

      10. fma-def 42.7%

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

      11. fma-udef 42.7%

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

      12. +-commutative 42.7%

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

      13. hypot-def 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in c around inf 82.0%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.6e+83)
     (+ (/ a c) (* d (/ b (pow c 2.0))))
     (if (<= c -5.5e-105)
       t_0
       (if (<= c 1.25e-34)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 1.6e+104)
           t_0
           (* (/ 1.0 (hypot c d)) (+ a (* d (/ b 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 (c <= -1.6e+83) {
		tmp = (a / c) + (d * (b / pow(c, 2.0)));
	} else if (c <= -5.5e-105) {
		tmp = t_0;
	} else if (c <= 1.25e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.6e+104) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (a + (d * (b / c)));
	}
	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 (c <= -1.6e+83) {
		tmp = (a / c) + (d * (b / Math.pow(c, 2.0)));
	} else if (c <= -5.5e-105) {
		tmp = t_0;
	} else if (c <= 1.25e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.6e+104) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (a + (d * (b / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.6e+83:
		tmp = (a / c) + (d * (b / math.pow(c, 2.0)))
	elif c <= -5.5e-105:
		tmp = t_0
	elif c <= 1.25e-34:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 1.6e+104:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (a + (d * (b / 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 (c <= -1.6e+83)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(b / (c ^ 2.0))));
	elseif (c <= -5.5e-105)
		tmp = t_0;
	elseif (c <= 1.25e-34)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 1.6e+104)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(a + Float64(d * Float64(b / c))));
	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 (c <= -1.6e+83)
		tmp = (a / c) + (d * (b / (c ^ 2.0)));
	elseif (c <= -5.5e-105)
		tmp = t_0;
	elseif (c <= 1.25e-34)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 1.6e+104)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (a + (d * (b / c)));
	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[c, -1.6e+83], N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.5e-105], t$95$0, If[LessEqual[c, 1.25e-34], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.6e+104], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.5999999999999999e83

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. fma-udef 49.6%

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

      3. *-un-lft-identity 49.6%

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

      4. associate-*r/ 49.6%

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

      5. add-sqr-sqrt 49.6%

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

      6. times-frac 49.5%

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

      7. fma-udef 49.5%

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

      8. +-commutative 49.5%

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

      9. hypot-def 49.5%

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

      10. fma-def 49.5%

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

      11. fma-udef 49.5%

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

      12. +-commutative 49.5%

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

      13. hypot-def 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in c around inf 78.5%

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

      1. associate-/l* 78.9%

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

      2. associate-/r/ 82.4%

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

   7. Simplified 82.4%

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

   if -1.5999999999999999e83 < c < -5.50000000000000029e-105 or 1.2500000000000001e-34 < c < 1.6e104

   1. Initial program 85.1%

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

   if -5.50000000000000029e-105 < c < 1.2500000000000001e-34

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 1.6e104 < c

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-udef 42.7%

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

      3. *-un-lft-identity 42.7%

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

      4. associate-*r/ 42.7%

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

      5. add-sqr-sqrt 42.7%

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

      6. times-frac 42.7%

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

      7. fma-udef 42.7%

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

      8. +-commutative 42.7%

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

      9. hypot-def 42.7%

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

      10. fma-def 42.7%

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

      11. fma-udef 42.7%

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

      12. +-commutative 42.7%

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

      13. hypot-def 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in c around inf 82.0%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{+83}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := a + d \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;t\_1 \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ a (* d (/ b c)))))
   (if (<= c -7.5e+82)
     (* t_1 (/ -1.0 (hypot c d)))
     (if (<= c -3.1e-103)
       t_0
       (if (<= c 1e-34)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 1.66e+102) t_0 (* (/ 1.0 (hypot c d)) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = a + (d * (b / c));
	double tmp;
	if (c <= -7.5e+82) {
		tmp = t_1 * (-1.0 / hypot(c, d));
	} else if (c <= -3.1e-103) {
		tmp = t_0;
	} else if (c <= 1e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.66e+102) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = a + (d * (b / c));
	double tmp;
	if (c <= -7.5e+82) {
		tmp = t_1 * (-1.0 / Math.hypot(c, d));
	} else if (c <= -3.1e-103) {
		tmp = t_0;
	} else if (c <= 1e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.66e+102) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = a + (d * (b / c))
	tmp = 0
	if c <= -7.5e+82:
		tmp = t_1 * (-1.0 / math.hypot(c, d))
	elif c <= -3.1e-103:
		tmp = t_0
	elif c <= 1e-34:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 1.66e+102:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(a + Float64(d * Float64(b / c)))
	tmp = 0.0
	if (c <= -7.5e+82)
		tmp = Float64(t_1 * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -3.1e-103)
		tmp = t_0;
	elseif (c <= 1e-34)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 1.66e+102)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = a + (d * (b / c));
	tmp = 0.0;
	if (c <= -7.5e+82)
		tmp = t_1 * (-1.0 / hypot(c, d));
	elseif (c <= -3.1e-103)
		tmp = t_0;
	elseif (c <= 1e-34)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 1.66e+102)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.5e+82], N[(t$95$1 * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.1e-103], t$95$0, If[LessEqual[c, 1e-34], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.66e+102], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.4999999999999999e82

   1. Initial program 49.6%

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

      1. +-commutative 49.6%

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

      2. fma-udef 49.6%

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

      3. *-un-lft-identity 49.6%

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

      4. associate-*r/ 49.6%

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

      5. add-sqr-sqrt 49.6%

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

      6. times-frac 49.5%

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

      7. fma-udef 49.5%

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

      8. +-commutative 49.5%

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

      9. hypot-def 49.5%

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

      10. fma-def 49.5%

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

      11. fma-udef 49.5%

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

      12. +-commutative 49.5%

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

      13. hypot-def 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in c around -inf 78.7%

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

      1. mul-1-neg 78.7%

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

      2. unsub-neg 78.7%

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

      3. neg-mul-1 78.7%

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

      4. associate-/l* 86.0%

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

      5. associate-/r/ 86.0%

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

   7. Simplified 86.0%

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

   if -7.4999999999999999e82 < c < -3.1000000000000001e-103 or 9.99999999999999928e-35 < c < 1.66e102

   1. Initial program 85.1%

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

   if -3.1000000000000001e-103 < c < 9.99999999999999928e-35

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 1.66e102 < c

   1. Initial program 42.7%

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

      1. +-commutative 42.7%

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

      2. fma-udef 42.7%

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

      3. *-un-lft-identity 42.7%

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

      4. associate-*r/ 42.7%

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

      5. add-sqr-sqrt 42.7%

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

      6. times-frac 42.7%

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

      7. fma-udef 42.7%

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

      8. +-commutative 42.7%

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

      9. hypot-def 42.7%

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

      10. fma-def 42.7%

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

      11. fma-udef 42.7%

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

      12. +-commutative 42.7%

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

      13. hypot-def 71.1%

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

   4. Applied egg-rr 71.1%

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

   5. Taylor expanded in c around inf 82.0%

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

      1. associate-/l* 80.4%

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

      2. associate-/r/ 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+82}:\\ \;\;\;\;\left(a + d \cdot \frac{b}{c}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.66 \cdot 10^{+102}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a + d \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{if}\;c \leq -4.3 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ a c) (* d (/ b (pow c 2.0))))))
   (if (<= c -4.3e+82)
     t_1
     (if (<= c -5.5e-105)
       t_0
       (if (<= c 1.25e-34)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 4.4e+100) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (a / c) + (d * (b / pow(c, 2.0)));
	double tmp;
	if (c <= -4.3e+82) {
		tmp = t_1;
	} else if (c <= -5.5e-105) {
		tmp = t_0;
	} else if (c <= 1.25e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 4.4e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (a / c) + (d * (b / (c ** 2.0d0)))
    if (c <= (-4.3d+82)) then
        tmp = t_1
    else if (c <= (-5.5d-105)) then
        tmp = t_0
    else if (c <= 1.25d-34) then
        tmp = (-b - (a * (c / d))) / -d
    else if (c <= 4.4d+100) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (a / c) + (d * (b / Math.pow(c, 2.0)));
	double tmp;
	if (c <= -4.3e+82) {
		tmp = t_1;
	} else if (c <= -5.5e-105) {
		tmp = t_0;
	} else if (c <= 1.25e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 4.4e+100) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (a / c) + (d * (b / math.pow(c, 2.0)))
	tmp = 0
	if c <= -4.3e+82:
		tmp = t_1
	elif c <= -5.5e-105:
		tmp = t_0
	elif c <= 1.25e-34:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 4.4e+100:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(a / c) + Float64(d * Float64(b / (c ^ 2.0))))
	tmp = 0.0
	if (c <= -4.3e+82)
		tmp = t_1;
	elseif (c <= -5.5e-105)
		tmp = t_0;
	elseif (c <= 1.25e-34)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 4.4e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (a / c) + (d * (b / (c ^ 2.0)));
	tmp = 0.0;
	if (c <= -4.3e+82)
		tmp = t_1;
	elseif (c <= -5.5e-105)
		tmp = t_0;
	elseif (c <= 1.25e-34)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 4.4e+100)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.3e+82], t$95$1, If[LessEqual[c, -5.5e-105], t$95$0, If[LessEqual[c, 1.25e-34], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 4.4e+100], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.30000000000000015e82 or 4.4000000000000001e100 < c

   1. Initial program 46.9%

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

      1. +-commutative 46.9%

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

      2. fma-udef 46.9%

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

      3. *-un-lft-identity 46.9%

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

      4. associate-*r/ 46.9%

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

      5. add-sqr-sqrt 46.9%

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

      6. times-frac 46.9%

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

      7. fma-udef 46.9%

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

      8. +-commutative 46.9%

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

      9. hypot-def 46.9%

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

      10. fma-def 46.9%

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

      11. fma-udef 46.9%

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

      12. +-commutative 46.9%

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

      13. hypot-def 71.0%

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

   4. Applied egg-rr 71.0%

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

   5. Taylor expanded in c around inf 73.8%

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

      1. associate-/l* 73.4%

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

      2. associate-/r/ 76.2%

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

   7. Simplified 76.2%

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

   if -4.30000000000000015e82 < c < -5.50000000000000029e-105 or 1.2500000000000001e-34 < c < 4.4000000000000001e100

   1. Initial program 85.1%

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

   if -5.50000000000000029e-105 < c < 1.2500000000000001e-34

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+82}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{{c}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -4.5e+130)
     (/ a c)
     (if (<= c -3.9e-103)
       t_0
       (if (<= c 6.8e-35)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 1.45e+118) t_0 (/ a (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.5e+130) {
		tmp = a / c;
	} else if (c <= -3.9e-103) {
		tmp = t_0;
	} else if (c <= 6.8e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.45e+118) {
		tmp = t_0;
	} else {
		tmp = a / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.5e+130) {
		tmp = a / c;
	} else if (c <= -3.9e-103) {
		tmp = t_0;
	} else if (c <= 6.8e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.45e+118) {
		tmp = t_0;
	} else {
		tmp = a / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.5e+130:
		tmp = a / c
	elif c <= -3.9e-103:
		tmp = t_0
	elif c <= 6.8e-35:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 1.45e+118:
		tmp = t_0
	else:
		tmp = a / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -4.5e+130)
		tmp = Float64(a / c);
	elseif (c <= -3.9e-103)
		tmp = t_0;
	elseif (c <= 6.8e-35)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 1.45e+118)
		tmp = t_0;
	else
		tmp = Float64(a / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.5e+130)
		tmp = a / c;
	elseif (c <= -3.9e-103)
		tmp = t_0;
	elseif (c <= 6.8e-35)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 1.45e+118)
		tmp = t_0;
	else
		tmp = a / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.5e+130], N[(a / c), $MachinePrecision], If[LessEqual[c, -3.9e-103], t$95$0, If[LessEqual[c, 6.8e-35], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.45e+118], t$95$0, N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.50000000000000039e130

   1. Initial program 39.9%

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

   3. Taylor expanded in c around inf 81.9%

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

   if -4.50000000000000039e130 < c < -3.9000000000000002e-103 or 6.8000000000000005e-35 < c < 1.45000000000000008e118

   1. Initial program 84.6%

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

   if -3.9000000000000002e-103 < c < 6.8000000000000005e-35

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 1.45000000000000008e118 < c

   1. Initial program 36.4%

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

      1. +-commutative 36.4%

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

      2. fma-udef 36.4%

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

      3. *-un-lft-identity 36.4%

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

      4. associate-*r/ 36.4%

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

      5. add-sqr-sqrt 36.4%

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

      6. times-frac 36.4%

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

      7. fma-udef 36.4%

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

      8. +-commutative 36.4%

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

      9. hypot-def 36.4%

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

      10. fma-def 36.4%

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

      11. fma-udef 36.4%

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

      12. +-commutative 36.4%

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

      13. hypot-def 69.6%

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

   4. Applied egg-rr 69.6%

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

   5. Taylor expanded in c around -inf 23.2%

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

      1. mul-1-neg 23.2%

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

      2. unsub-neg 23.2%

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

      3. neg-mul-1 23.2%

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

      4. associate-/l* 23.3%

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

      5. associate-/r/ 23.2%

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

   7. Simplified 23.2%

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

      1. associate-*l/ 23.2%

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

      2. *-un-lft-identity 23.2%

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

      3. add-sqr-sqrt 11.5%

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

      4. sqrt-unprod 26.7%

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

      5. sqr-neg 26.7%

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

      6. sqrt-unprod 25.2%

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

      7. add-sqr-sqrt 61.8%

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

      8. *-commutative 61.8%

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

      9. clear-num 61.8%

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

      10. un-div-inv 61.8%

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

   9. Applied egg-rr 61.8%

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

   10. Taylor expanded in a around inf 62.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -3.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+118}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1 \cdot 10^{+126}:\\ \;\;\;\;a \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1e+126)
     (* a (/ -1.0 (hypot c d)))
     (if (<= c -6.2e-103)
       t_0
       (if (<= c 6.5e-35)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 5e+119) t_0 (/ a (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+126) {
		tmp = a * (-1.0 / hypot(c, d));
	} else if (c <= -6.2e-103) {
		tmp = t_0;
	} else if (c <= 6.5e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 5e+119) {
		tmp = t_0;
	} else {
		tmp = a / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+126) {
		tmp = a * (-1.0 / Math.hypot(c, d));
	} else if (c <= -6.2e-103) {
		tmp = t_0;
	} else if (c <= 6.5e-35) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 5e+119) {
		tmp = t_0;
	} else {
		tmp = a / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1e+126:
		tmp = a * (-1.0 / math.hypot(c, d))
	elif c <= -6.2e-103:
		tmp = t_0
	elif c <= 6.5e-35:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 5e+119:
		tmp = t_0
	else:
		tmp = a / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1e+126)
		tmp = Float64(a * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -6.2e-103)
		tmp = t_0;
	elseif (c <= 6.5e-35)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 5e+119)
		tmp = t_0;
	else
		tmp = Float64(a / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1e+126)
		tmp = a * (-1.0 / hypot(c, d));
	elseif (c <= -6.2e-103)
		tmp = t_0;
	elseif (c <= 6.5e-35)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 5e+119)
		tmp = t_0;
	else
		tmp = a / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1e+126], N[(a * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -6.2e-103], t$95$0, If[LessEqual[c, 6.5e-35], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 5e+119], t$95$0, N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.99999999999999925e125

   1. Initial program 39.9%

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

      1. +-commutative 39.9%

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

      2. fma-udef 39.9%

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

      3. *-un-lft-identity 39.9%

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

      4. associate-*r/ 39.9%

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

      5. add-sqr-sqrt 39.9%

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

      6. times-frac 39.9%

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

      7. fma-udef 39.9%

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

      8. +-commutative 39.9%

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

      9. hypot-def 39.9%

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

      10. fma-def 39.9%

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

      11. fma-udef 39.9%

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

      12. +-commutative 39.9%

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

      13. hypot-def 65.1%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in c around -inf 81.9%

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

      1. neg-mul-1 81.9%

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

   7. Simplified 81.9%

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

   if -9.99999999999999925e125 < c < -6.2000000000000003e-103 or 6.4999999999999999e-35 < c < 4.9999999999999999e119

   1. Initial program 84.6%

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

   if -6.2000000000000003e-103 < c < 6.4999999999999999e-35

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

   if 4.9999999999999999e119 < c

   1. Initial program 36.4%

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

      1. +-commutative 36.4%

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

      2. fma-udef 36.4%

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

      3. *-un-lft-identity 36.4%

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

      4. associate-*r/ 36.4%

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

      5. add-sqr-sqrt 36.4%

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

      6. times-frac 36.4%

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

      7. fma-udef 36.4%

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

      8. +-commutative 36.4%

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

      9. hypot-def 36.4%

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

      10. fma-def 36.4%

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

      11. fma-udef 36.4%

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

      12. +-commutative 36.4%

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

      13. hypot-def 69.6%

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

   4. Applied egg-rr 69.6%

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

   5. Taylor expanded in c around -inf 23.2%

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

      1. mul-1-neg 23.2%

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

      2. unsub-neg 23.2%

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

      3. neg-mul-1 23.2%

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

      4. associate-/l* 23.3%

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

      5. associate-/r/ 23.2%

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

   7. Simplified 23.2%

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

      1. associate-*l/ 23.2%

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

      2. *-un-lft-identity 23.2%

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

      3. add-sqr-sqrt 11.5%

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

      4. sqrt-unprod 26.7%

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

      5. sqr-neg 26.7%

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

      6. sqrt-unprod 25.2%

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

      7. add-sqr-sqrt 61.8%

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

      8. *-commutative 61.8%

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

      9. clear-num 61.8%

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

      10. un-div-inv 61.8%

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

   9. Applied egg-rr 61.8%

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

   10. Taylor expanded in a around inf 62.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+126}:\\ \;\;\;\;a \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+119}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.1e+129)
     (/ a c)
     (if (<= c -1.25e-103)
       t_0
       (if (<= c 1.7e-34)
         (/ (- (- b) (* a (/ c d))) (- d))
         (if (<= c 1.1e+125) t_0 (/ a 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 (c <= -1.1e+129) {
		tmp = a / c;
	} else if (c <= -1.25e-103) {
		tmp = t_0;
	} else if (c <= 1.7e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.1e+125) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-1.1d+129)) then
        tmp = a / c
    else if (c <= (-1.25d-103)) then
        tmp = t_0
    else if (c <= 1.7d-34) then
        tmp = (-b - (a * (c / d))) / -d
    else if (c <= 1.1d+125) then
        tmp = t_0
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.1e+129) {
		tmp = a / c;
	} else if (c <= -1.25e-103) {
		tmp = t_0;
	} else if (c <= 1.7e-34) {
		tmp = (-b - (a * (c / d))) / -d;
	} else if (c <= 1.1e+125) {
		tmp = t_0;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.1e+129:
		tmp = a / c
	elif c <= -1.25e-103:
		tmp = t_0
	elif c <= 1.7e-34:
		tmp = (-b - (a * (c / d))) / -d
	elif c <= 1.1e+125:
		tmp = t_0
	else:
		tmp = a / 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 (c <= -1.1e+129)
		tmp = Float64(a / c);
	elseif (c <= -1.25e-103)
		tmp = t_0;
	elseif (c <= 1.7e-34)
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	elseif (c <= 1.1e+125)
		tmp = t_0;
	else
		tmp = Float64(a / c);
	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 (c <= -1.1e+129)
		tmp = a / c;
	elseif (c <= -1.25e-103)
		tmp = t_0;
	elseif (c <= 1.7e-34)
		tmp = (-b - (a * (c / d))) / -d;
	elseif (c <= 1.1e+125)
		tmp = t_0;
	else
		tmp = a / c;
	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[c, -1.1e+129], N[(a / c), $MachinePrecision], If[LessEqual[c, -1.25e-103], t$95$0, If[LessEqual[c, 1.7e-34], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], If[LessEqual[c, 1.1e+125], t$95$0, N[(a / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1e129 or 1.09999999999999995e125 < 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 73.8%

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

   if -1.1e129 < c < -1.24999999999999992e-103 or 1.7e-34 < c < 1.09999999999999995e125

   1. Initial program 84.6%

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

   if -1.24999999999999992e-103 < c < 1.7e-34

   1. Initial program 62.9%

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

   3. Taylor expanded in c around 0 81.0%

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

      1. associate-/l* 79.7%

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

      2. associate-/r/ 77.7%

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

   5. Simplified 77.7%

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

      1. pow2 77.7%

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

      2. associate-*l/ 81.0%

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

      3. *-commutative 81.0%

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

      4. associate-/r* 86.1%

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

      5. *-commutative 86.1%

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

   7. Applied egg-rr 86.1%

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

      1. div-inv 86.1%

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

      2. associate-/l* 86.8%

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

      3. add-sqr-sqrt 37.0%

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

      4. sqrt-unprod 62.6%

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

      5. sqr-neg 62.6%

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

      6. sqrt-unprod 36.1%

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

      7. add-sqr-sqrt 63.8%

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

      8. distribute-frac-neg 63.8%

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

      9. associate-/l* 63.8%

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

      10. cancel-sign-sub-inv 63.8%

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

      11. div-inv 63.8%

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

      12. frac-2neg 63.8%

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

      13. frac-2neg 63.8%

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

      14. associate-/l* 63.8%

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

      15. distribute-frac-neg 63.8%

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

      16. sub-div 63.8%

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

   9. Applied egg-rr 84.9%

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

       1. associate-*l/ 87.2%

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

       2. associate-*r/ 88.0%

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

   11. Simplified 88.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-103}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+104}:\\ \;\;\;\;\frac{a \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5e+18)
   (/ a c)
   (if (<= c 7.6e-35)
     (+ (/ b d) (/ (/ (* a c) d) d))
     (if (<= c 1.9e+104) (/ (* a c) (+ (* c c) (* d d))) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5e+18) {
		tmp = a / c;
	} else if (c <= 7.6e-35) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.9e+104) {
		tmp = (a * c) / ((c * c) + (d * 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 (c <= (-5d+18)) then
        tmp = a / c
    else if (c <= 7.6d-35) then
        tmp = (b / d) + (((a * c) / d) / d)
    else if (c <= 1.9d+104) then
        tmp = (a * c) / ((c * c) + (d * 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 (c <= -5e+18) {
		tmp = a / c;
	} else if (c <= 7.6e-35) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 1.9e+104) {
		tmp = (a * c) / ((c * c) + (d * d));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5e+18:
		tmp = a / c
	elif c <= 7.6e-35:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 1.9e+104:
		tmp = (a * c) / ((c * c) + (d * d))
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5e+18)
		tmp = Float64(a / c);
	elseif (c <= 7.6e-35)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 1.9e+104)
		tmp = Float64(Float64(a * c) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5e+18)
		tmp = a / c;
	elseif (c <= 7.6e-35)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 1.9e+104)
		tmp = (a * c) / ((c * c) + (d * d));
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5e+18], N[(a / c), $MachinePrecision], If[LessEqual[c, 7.6e-35], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.9e+104], N[(N[(a * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5e18 or 1.89999999999999984e104 < c

   1. Initial program 55.9%

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

   3. Taylor expanded in c around inf 72.2%

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

   if -5e18 < c < 7.6000000000000002e-35

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0 76.8%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 74.1%

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

   5. Simplified 74.1%

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

      1. pow2 74.1%

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

      2. associate-*l/ 76.8%

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

      3. *-commutative 76.8%

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

      4. associate-/r* 81.9%

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

      5. *-commutative 81.9%

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

   7. Applied egg-rr 81.9%

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

   if 7.6000000000000002e-35 < c < 1.89999999999999984e104

   1. Initial program 80.2%

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

   3. Taylor expanded in a around inf 68.5%

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

      1. *-commutative 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+18}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+104}:\\ \;\;\;\;\frac{a \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{+18} \lor \neg \left(c \leq 1.9 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.3e+18) (not (<= c 1.9e-34)))
   (/ a c)
   (+ (/ b d) (/ (/ (* a c) d) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.3e+18) || !(c <= 1.9e-34)) {
		tmp = a / c;
	} else {
		tmp = (b / d) + (((a * c) / d) / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.3d+18)) .or. (.not. (c <= 1.9d-34))) then
        tmp = a / c
    else
        tmp = (b / d) + (((a * c) / d) / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.3e+18) || !(c <= 1.9e-34)) {
		tmp = a / c;
	} else {
		tmp = (b / d) + (((a * c) / d) / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.3e+18) or not (c <= 1.9e-34):
		tmp = a / c
	else:
		tmp = (b / d) + (((a * c) / d) / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.3e+18) || !(c <= 1.9e-34))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.3e+18) || ~((c <= 1.9e-34)))
		tmp = a / c;
	else
		tmp = (b / d) + (((a * c) / d) / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.3e+18], N[Not[LessEqual[c, 1.9e-34]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.3e18 or 1.9000000000000001e-34 < c

   1. Initial program 61.9%

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

   3. Taylor expanded in c around inf 67.9%

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

   if -2.3e18 < c < 1.9000000000000001e-34

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0 76.8%

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

      1. associate-/l* 74.9%

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

      2. associate-/r/ 74.1%

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

   5. Simplified 74.1%

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

      1. pow2 74.1%

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

      2. associate-*l/ 76.8%

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

      3. *-commutative 76.8%

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

      4. associate-/r* 81.9%

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

      5. *-commutative 81.9%

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

   7. Applied egg-rr 81.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{+18} \lor \neg \left(c \leq 1.9 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.5 \lor \neg \left(d \leq 4.8 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.5) (not (<= d 4.8e-46)))
   (/ (- (- b) (* a (/ c d))) (- d))
   (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.5) || !(d <= 4.8e-46)) {
		tmp = (-b - (a * (c / d))) / -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 <= (-7.5d0)) .or. (.not. (d <= 4.8d-46))) then
        tmp = (-b - (a * (c / d))) / -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 <= -7.5) || !(d <= 4.8e-46)) {
		tmp = (-b - (a * (c / d))) / -d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.5) or not (d <= 4.8e-46):
		tmp = (-b - (a * (c / d))) / -d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.5) || !(d <= 4.8e-46))
		tmp = Float64(Float64(Float64(-b) - Float64(a * Float64(c / d))) / Float64(-d));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.5) || ~((d <= 4.8e-46)))
		tmp = (-b - (a * (c / d))) / -d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.5], N[Not[LessEqual[d, 4.8e-46]], $MachinePrecision]], N[(N[((-b) - N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.5 or 4.80000000000000027e-46 < d

   1. Initial program 53.8%

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

   3. Taylor expanded in c around 0 69.0%

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

      1. associate-/l* 68.0%

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

      2. associate-/r/ 71.6%

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

   5. Simplified 71.6%

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

      1. pow2 71.6%

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

      2. associate-*l/ 69.0%

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

      3. *-commutative 69.0%

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

      4. associate-/r* 72.8%

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

      5. *-commutative 72.8%

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

   7. Applied egg-rr 72.8%

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

      1. div-inv 72.8%

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

      2. associate-/l* 76.7%

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

      3. add-sqr-sqrt 36.4%

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

      4. sqrt-unprod 56.7%

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

      5. sqr-neg 56.7%

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

      6. sqrt-unprod 28.5%

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

      7. add-sqr-sqrt 57.9%

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

      8. distribute-frac-neg 57.9%

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

      9. associate-/l* 56.3%

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

      10. cancel-sign-sub-inv 56.3%

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

      11. div-inv 56.3%

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

      12. frac-2neg 56.3%

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

      13. frac-2neg 56.3%

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

      14. associate-/l* 57.9%

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

      15. distribute-frac-neg 57.9%

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

      16. sub-div 57.9%

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

   9. Applied egg-rr 79.4%

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

       1. associate-*l/ 72.8%

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

       2. associate-*r/ 77.1%

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

   11. Simplified 77.1%

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

   if -7.5 < d < 4.80000000000000027e-46

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf 74.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \lor \neg \left(d \leq 4.8 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\left(-b\right) - a \cdot \frac{c}{d}}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \lor \neg \left(d \leq 110\right):\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.5) (not (<= d 110.0)))
   (/ (- (- b) (* c (/ a d))) (- d))
   (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.5) || !(d <= 110.0)) {
		tmp = (-b - (c * (a / d))) / -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 <= (-6.5d0)) .or. (.not. (d <= 110.0d0))) then
        tmp = (-b - (c * (a / d))) / -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 <= -6.5) || !(d <= 110.0)) {
		tmp = (-b - (c * (a / d))) / -d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.5) or not (d <= 110.0):
		tmp = (-b - (c * (a / d))) / -d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.5) || !(d <= 110.0))
		tmp = Float64(Float64(Float64(-b) - Float64(c * Float64(a / d))) / Float64(-d));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -6.5) || ~((d <= 110.0)))
		tmp = (-b - (c * (a / d))) / -d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.5], N[Not[LessEqual[d, 110.0]], $MachinePrecision]], N[(N[((-b) - N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-d)), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5 or 110 < d

   1. Initial program 52.8%

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

   3. Taylor expanded in c around 0 69.3%

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

      1. associate-/l* 68.3%

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

      2. associate-/r/ 72.7%

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

   5. Simplified 72.7%

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

      1. pow2 72.7%

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

      2. associate-*l/ 69.3%

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

      3. *-commutative 69.3%

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

      4. associate-/r* 73.3%

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

      5. *-commutative 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. div-inv 73.3%

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

      2. associate-/l* 77.3%

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

      3. add-sqr-sqrt 36.3%

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

      4. sqrt-unprod 57.3%

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

      5. sqr-neg 57.3%

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

      6. sqrt-unprod 29.6%

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

      7. add-sqr-sqrt 60.1%

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

      8. distribute-frac-neg 60.1%

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

      9. associate-/l* 58.4%

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

      10. cancel-sign-sub-inv 58.4%

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

      11. div-inv 58.4%

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

      12. frac-2neg 58.4%

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

      13. frac-2neg 58.4%

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

      14. associate-/l* 60.1%

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

      15. distribute-frac-neg 60.1%

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

      16. sub-div 60.1%

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

   9. Applied egg-rr 80.8%

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

   if -6.5 < d < 110

   1. Initial program 75.2%

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

   3. Taylor expanded in c around inf 73.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \lor \neg \left(d \leq 110\right):\\ \;\;\;\;\frac{\left(-b\right) - c \cdot \frac{a}{d}}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+84} \lor \neg \left(d \leq 470\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9e+84) (not (<= d 470.0))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e+84) || !(d <= 470.0)) {
		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 <= (-9d+84)) .or. (.not. (d <= 470.0d0))) 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 <= -9e+84) || !(d <= 470.0)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9e+84) or not (d <= 470.0):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9e+84) || !(d <= 470.0))
		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 <= -9e+84) || ~((d <= 470.0)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9e+84], N[Not[LessEqual[d, 470.0]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.9999999999999994e84 or 470 < d

   1. Initial program 49.2%

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

   3. Taylor expanded in c around 0 68.2%

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

   if -8.9999999999999994e84 < d < 470

   1. Initial program 75.4%

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

   3. Taylor expanded in c around inf 68.1%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+84} \lor \neg \left(d \leq 470\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in c around inf 46.0%

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

4. Final simplification 46.0%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024041 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d)))))

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