Complex division, real part

Percentage Accurate: 61.7% → 85.4%

Time: 15.0s

Alternatives: 13

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. associate-*r/ 73.6%

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

      3. fma-def 73.6%

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

      4. add-sqr-sqrt 73.6%

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

      5. times-frac 73.6%

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

      6. fma-def 73.6%

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

      7. hypot-def 73.6%

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

      8. fma-def 73.6%

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

      9. fma-def 73.6%

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

      10. hypot-def 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. associate-*l/ 93.5%

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

      2. *-un-lft-identity 93.5%

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

   6. Applied egg-rr 93.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf 1.5%

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

      1. associate-/l* 3.9%

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

      2. +-commutative 3.9%

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

      3. unpow2 3.9%

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

      4. fma-udef 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-div 2.2%

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

      3. fma-udef 2.2%

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

      4. +-commutative 2.2%

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

      5. pow2 2.2%

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

      6. hypot-udef 2.2%

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

      7. sqrt-div 2.2%

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

      8. fma-udef 2.2%

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

      9. +-commutative 2.2%

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

      10. pow2 2.2%

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

      11. hypot-udef 30.4%

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

      12. times-frac 2.2%

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

      13. add-sqr-sqrt 3.9%

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

      14. *-un-lft-identity 3.9%

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

      15. times-frac 55.1%

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

      16. /-rgt-identity 55.1%

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

   7. Applied egg-rr 55.1%

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

4. Final simplification 86.5%

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

Alternative 2: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (/ (/ t_0 (hypot c d)) (hypot c d))
     (/ a (* (hypot c d) (/ (hypot c d) c))))))

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))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = a / (hypot(c, d) * (hypot(c, d) / c));
	}
	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))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(c, d)) / Math.hypot(c, d);
	} else {
		tmp = a / (Math.hypot(c, d) * (Math.hypot(c, d) / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= math.inf:
		tmp = (t_0 / math.hypot(c, d)) / math.hypot(c, d)
	else:
		tmp = a / (math.hypot(c, d) * (math.hypot(c, d) / c))
	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))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(a / Float64(hypot(c, d) * Float64(hypot(c, d) / c)));
	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))) <= Inf)
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	else
		tmp = a / (hypot(c, d) * (hypot(c, d) / c));
	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], Infinity], N[(N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. associate-*r/ 73.6%

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

      3. fma-def 73.6%

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

      4. add-sqr-sqrt 73.6%

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

      5. times-frac 73.6%

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

      6. fma-def 73.6%

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

      7. hypot-def 73.6%

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

      8. fma-def 73.6%

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

      9. fma-def 73.6%

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

      10. hypot-def 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. associate-*l/ 93.5%

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

      2. *-un-lft-identity 93.5%

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

   6. Applied egg-rr 93.5%

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

      1. fma-def 93.5%

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

      2. +-commutative 93.5%

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

   8. Applied egg-rr 93.5%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in a around inf 1.5%

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

      1. associate-/l* 3.9%

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

      2. +-commutative 3.9%

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

      3. unpow2 3.9%

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

      4. fma-udef 3.9%

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

   5. Simplified 3.9%

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

      1. add-sqr-sqrt 2.2%

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

      2. sqrt-div 2.2%

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

      3. fma-udef 2.2%

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

      4. +-commutative 2.2%

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

      5. pow2 2.2%

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

      6. hypot-udef 2.2%

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

      7. sqrt-div 2.2%

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

      8. fma-udef 2.2%

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

      9. +-commutative 2.2%

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

      10. pow2 2.2%

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

      11. hypot-udef 30.4%

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

      12. times-frac 2.2%

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

      13. add-sqr-sqrt 3.9%

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

      14. *-un-lft-identity 3.9%

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

      15. times-frac 55.1%

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

      16. /-rgt-identity 55.1%

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

   7. Applied egg-rr 55.1%

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

4. Final simplification 86.5%

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

Alternative 3: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{if}\;c \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.22 \cdot 10^{-146}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (/ a (/ d c))))))
   (if (<= c -1.25e+119)
     (/ (- (/ (- b) (/ c d)) a) (hypot c d))
     (if (<= c -1.22e-146)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (if (<= c 9.5e-118)
         t_0
         (if (<= c 2.5e+46)
           (/ (fma a c (* b d)) (fma c c (* d d)))
           (if (<= c 1.4e+102) t_0 (/ (+ a (/ b (/ c d))) (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -1.25e+119) {
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	} else if (c <= -1.22e-146) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else if (c <= 9.5e-118) {
		tmp = t_0;
	} else if (c <= 2.5e+46) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else if (c <= 1.4e+102) {
		tmp = t_0;
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))))
	tmp = 0.0
	if (c <= -1.25e+119)
		tmp = Float64(Float64(Float64(Float64(-b) / Float64(c / d)) - a) / hypot(c, d));
	elseif (c <= -1.22e-146)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 9.5e-118)
		tmp = t_0;
	elseif (c <= 2.5e+46)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	elseif (c <= 1.4e+102)
		tmp = t_0;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.25e+119], N[(N[(N[((-b) / N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.22e-146], 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, 9.5e-118], t$95$0, If[LessEqual[c, 2.5e+46], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.4e+102], t$95$0, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.25e119

   1. Initial program 43.3%

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

      1. *-un-lft-identity 43.3%

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

      2. associate-*r/ 43.3%

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

      3. fma-def 43.3%

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

      4. add-sqr-sqrt 43.3%

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

      5. times-frac 43.4%

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

      6. fma-def 43.4%

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

      7. hypot-def 43.4%

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

      8. fma-def 43.4%

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

      9. fma-def 43.4%

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

      10. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in c around -inf 79.8%

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

      1. neg-mul-1 79.8%

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

      2. +-commutative 79.8%

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

      3. unsub-neg 79.8%

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

      4. mul-1-neg 79.8%

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

      5. associate-/l* 86.0%

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

      6. distribute-neg-frac 86.0%

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

   9. Simplified 86.0%

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

   if -1.25e119 < c < -1.2200000000000001e-146

   1. Initial program 84.0%

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

   if -1.2200000000000001e-146 < c < 9.49999999999999931e-118 or 2.5000000000000001e46 < c < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. associate-*r/ 53.8%

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

      3. fma-def 53.8%

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

      4. add-sqr-sqrt 53.7%

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

      5. times-frac 53.8%

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

      6. fma-def 53.8%

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

      7. hypot-def 53.9%

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

      8. fma-def 53.9%

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

      9. fma-def 53.9%

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

      10. hypot-def 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in c around 0 39.1%

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

      1. associate-/l* 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in c around 0 82.1%

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

   if 9.49999999999999931e-118 < c < 2.5000000000000001e46

   1. Initial program 87.0%

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

      1. fma-def 87.1%

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

      2. fma-def 87.1%

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

   3. Simplified 87.1%

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

   if 1.40000000000000009e102 < c

   1. Initial program 33.4%

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

      1. *-un-lft-identity 33.4%

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

      2. associate-*r/ 33.4%

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

      3. fma-def 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. times-frac 33.4%

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

      6. fma-def 33.4%

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

      7. hypot-def 33.4%

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

      8. fma-def 33.4%

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

      9. fma-def 33.4%

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

      10. hypot-def 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in c around inf 85.6%

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

      1. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.22 \cdot 10^{-146}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{if}\;c \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.04 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5.1 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (* (/ 1.0 d) (+ b (/ a (/ d c))))))
   (if (<= c -4.5e+118)
     (+ (/ a c) (* d (* (/ 1.0 c) (/ b c))))
     (if (<= c -2.8e-142)
       t_0
       (if (<= c 1.04e-122)
         t_1
         (if (<= c 5.1e+48)
           t_0
           (if (<= c 1.4e+102) t_1 (/ (+ a (/ b (/ c d))) (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 t_1 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -4.5e+118) {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	} else if (c <= -2.8e-142) {
		tmp = t_0;
	} else if (c <= 1.04e-122) {
		tmp = t_1;
	} else if (c <= 5.1e+48) {
		tmp = t_0;
	} else if (c <= 1.4e+102) {
		tmp = t_1;
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -4.5e+118) {
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	} else if (c <= -2.8e-142) {
		tmp = t_0;
	} else if (c <= 1.04e-122) {
		tmp = t_1;
	} else if (c <= 5.1e+48) {
		tmp = t_0;
	} else if (c <= 1.4e+102) {
		tmp = t_1;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (1.0 / d) * (b + (a / (d / c)))
	tmp = 0
	if c <= -4.5e+118:
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)))
	elif c <= -2.8e-142:
		tmp = t_0
	elif c <= 1.04e-122:
		tmp = t_1
	elif c <= 5.1e+48:
		tmp = t_0
	elif c <= 1.4e+102:
		tmp = t_1
	else:
		tmp = (a + (b / (c / d))) / 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)))
	t_1 = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))))
	tmp = 0.0
	if (c <= -4.5e+118)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(1.0 / c) * Float64(b / c))));
	elseif (c <= -2.8e-142)
		tmp = t_0;
	elseif (c <= 1.04e-122)
		tmp = t_1;
	elseif (c <= 5.1e+48)
		tmp = t_0;
	elseif (c <= 1.4e+102)
		tmp = t_1;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (1.0 / d) * (b + (a / (d / c)));
	tmp = 0.0;
	if (c <= -4.5e+118)
		tmp = (a / c) + (d * ((1.0 / c) * (b / c)));
	elseif (c <= -2.8e-142)
		tmp = t_0;
	elseif (c <= 1.04e-122)
		tmp = t_1;
	elseif (c <= 5.1e+48)
		tmp = t_0;
	elseif (c <= 1.4e+102)
		tmp = t_1;
	else
		tmp = (a + (b / (c / d))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(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[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.5e+118], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.8e-142], t$95$0, If[LessEqual[c, 1.04e-122], t$95$1, If[LessEqual[c, 5.1e+48], t$95$0, If[LessEqual[c, 1.4e+102], t$95$1, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.50000000000000002e118

   1. Initial program 43.3%

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

   3. Taylor expanded in c around inf 74.6%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   5. Simplified 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. pow2 78.3%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if -4.50000000000000002e118 < c < -2.80000000000000004e-142 or 1.03999999999999998e-122 < c < 5.0999999999999998e48

   1. Initial program 85.7%

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

   if -2.80000000000000004e-142 < c < 1.03999999999999998e-122 or 5.0999999999999998e48 < c < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. associate-*r/ 53.8%

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

      3. fma-def 53.8%

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

      4. add-sqr-sqrt 53.7%

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

      5. times-frac 53.8%

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

      6. fma-def 53.8%

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

      7. hypot-def 53.9%

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

      8. fma-def 53.9%

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

      9. fma-def 53.9%

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

      10. hypot-def 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in c around 0 39.1%

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

      1. associate-/l* 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in c around 0 82.1%

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

   if 1.40000000000000009e102 < c

   1. Initial program 33.4%

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

      1. *-un-lft-identity 33.4%

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

      2. associate-*r/ 33.4%

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

      3. fma-def 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. times-frac 33.4%

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

      6. fma-def 33.4%

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

      7. hypot-def 33.4%

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

      8. fma-def 33.4%

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

      9. fma-def 33.4%

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

      10. hypot-def 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in c around inf 85.6%

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

      1. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-142}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.04 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 5.1 \cdot 10^{+48}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{if}\;c \leq -3.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (* (/ 1.0 d) (+ b (/ a (/ d c))))))
   (if (<= c -3.3e+119)
     (/ (- (/ (- b) (/ c d)) a) (hypot c d))
     (if (<= c -1.35e-144)
       t_0
       (if (<= c 1.1e-123)
         t_1
         (if (<= c 5e+48)
           t_0
           (if (<= c 1.4e+102) t_1 (/ (+ a (/ b (/ c d))) (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 t_1 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -3.3e+119) {
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	} else if (c <= -1.35e-144) {
		tmp = t_0;
	} else if (c <= 1.1e-123) {
		tmp = t_1;
	} else if (c <= 5e+48) {
		tmp = t_0;
	} else if (c <= 1.4e+102) {
		tmp = t_1;
	} else {
		tmp = (a + (b / (c / d))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -3.3e+119) {
		tmp = ((-b / (c / d)) - a) / Math.hypot(c, d);
	} else if (c <= -1.35e-144) {
		tmp = t_0;
	} else if (c <= 1.1e-123) {
		tmp = t_1;
	} else if (c <= 5e+48) {
		tmp = t_0;
	} else if (c <= 1.4e+102) {
		tmp = t_1;
	} else {
		tmp = (a + (b / (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (1.0 / d) * (b + (a / (d / c)))
	tmp = 0
	if c <= -3.3e+119:
		tmp = ((-b / (c / d)) - a) / math.hypot(c, d)
	elif c <= -1.35e-144:
		tmp = t_0
	elif c <= 1.1e-123:
		tmp = t_1
	elif c <= 5e+48:
		tmp = t_0
	elif c <= 1.4e+102:
		tmp = t_1
	else:
		tmp = (a + (b / (c / d))) / 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)))
	t_1 = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))))
	tmp = 0.0
	if (c <= -3.3e+119)
		tmp = Float64(Float64(Float64(Float64(-b) / Float64(c / d)) - a) / hypot(c, d));
	elseif (c <= -1.35e-144)
		tmp = t_0;
	elseif (c <= 1.1e-123)
		tmp = t_1;
	elseif (c <= 5e+48)
		tmp = t_0;
	elseif (c <= 1.4e+102)
		tmp = t_1;
	else
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (1.0 / d) * (b + (a / (d / c)));
	tmp = 0.0;
	if (c <= -3.3e+119)
		tmp = ((-b / (c / d)) - a) / hypot(c, d);
	elseif (c <= -1.35e-144)
		tmp = t_0;
	elseif (c <= 1.1e-123)
		tmp = t_1;
	elseif (c <= 5e+48)
		tmp = t_0;
	elseif (c <= 1.4e+102)
		tmp = t_1;
	else
		tmp = (a + (b / (c / d))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(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[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.3e+119], N[(N[(N[((-b) / N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.35e-144], t$95$0, If[LessEqual[c, 1.1e-123], t$95$1, If[LessEqual[c, 5e+48], t$95$0, If[LessEqual[c, 1.4e+102], t$95$1, N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.3000000000000002e119

   1. Initial program 43.3%

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

      1. *-un-lft-identity 43.3%

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

      2. associate-*r/ 43.3%

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

      3. fma-def 43.3%

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

      4. add-sqr-sqrt 43.3%

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

      5. times-frac 43.4%

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

      6. fma-def 43.4%

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

      7. hypot-def 43.4%

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

      8. fma-def 43.4%

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

      9. fma-def 43.4%

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

      10. hypot-def 61.5%

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

   4. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   6. Applied egg-rr 61.6%

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

   7. Taylor expanded in c around -inf 79.8%

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

      1. neg-mul-1 79.8%

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

      2. +-commutative 79.8%

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

      3. unsub-neg 79.8%

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

      4. mul-1-neg 79.8%

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

      5. associate-/l* 86.0%

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

      6. distribute-neg-frac 86.0%

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

   9. Simplified 86.0%

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

   if -3.3000000000000002e119 < c < -1.34999999999999988e-144 or 1.10000000000000003e-123 < c < 4.99999999999999973e48

   1. Initial program 85.7%

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

   if -1.34999999999999988e-144 < c < 1.10000000000000003e-123 or 4.99999999999999973e48 < c < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. associate-*r/ 53.8%

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

      3. fma-def 53.8%

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

      4. add-sqr-sqrt 53.7%

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

      5. times-frac 53.8%

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

      6. fma-def 53.8%

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

      7. hypot-def 53.9%

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

      8. fma-def 53.9%

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

      9. fma-def 53.9%

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

      10. hypot-def 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in c around 0 39.1%

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

      1. associate-/l* 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in c around 0 82.1%

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

   if 1.40000000000000009e102 < c

   1. Initial program 33.4%

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

      1. *-un-lft-identity 33.4%

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

      2. associate-*r/ 33.4%

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

      3. fma-def 33.4%

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

      4. add-sqr-sqrt 33.4%

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

      5. times-frac 33.4%

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

      6. fma-def 33.4%

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

      7. hypot-def 33.4%

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

      8. fma-def 33.4%

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

      9. fma-def 33.4%

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

      10. hypot-def 57.4%

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

   4. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in c around inf 85.6%

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

      1. associate-/l* 88.2%

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

   9. Simplified 88.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{\frac{-b}{\frac{c}{d}} - a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a + \frac{b}{\frac{c}{d}}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ t_2 := \frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.75 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ a c) (* d (* (/ 1.0 c) (/ b c)))))
        (t_2 (* (/ 1.0 d) (+ b (/ a (/ d c))))))
   (if (<= c -6.2e+119)
     t_1
     (if (<= c -1.75e-145)
       t_0
       (if (<= c 2.9e-123)
         t_2
         (if (<= c 4e+48) t_0 (if (<= c 5e+102) t_2 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 * ((1.0 / c) * (b / c)));
	double t_2 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -6.2e+119) {
		tmp = t_1;
	} else if (c <= -1.75e-145) {
		tmp = t_0;
	} else if (c <= 2.9e-123) {
		tmp = t_2;
	} else if (c <= 4e+48) {
		tmp = t_0;
	} else if (c <= 5e+102) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (a / c) + (d * ((1.0d0 / c) * (b / c)))
    t_2 = (1.0d0 / d) * (b + (a / (d / c)))
    if (c <= (-6.2d+119)) then
        tmp = t_1
    else if (c <= (-1.75d-145)) then
        tmp = t_0
    else if (c <= 2.9d-123) then
        tmp = t_2
    else if (c <= 4d+48) then
        tmp = t_0
    else if (c <= 5d+102) then
        tmp = t_2
    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 * ((1.0 / c) * (b / c)));
	double t_2 = (1.0 / d) * (b + (a / (d / c)));
	double tmp;
	if (c <= -6.2e+119) {
		tmp = t_1;
	} else if (c <= -1.75e-145) {
		tmp = t_0;
	} else if (c <= 2.9e-123) {
		tmp = t_2;
	} else if (c <= 4e+48) {
		tmp = t_0;
	} else if (c <= 5e+102) {
		tmp = t_2;
	} 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 * ((1.0 / c) * (b / c)))
	t_2 = (1.0 / d) * (b + (a / (d / c)))
	tmp = 0
	if c <= -6.2e+119:
		tmp = t_1
	elif c <= -1.75e-145:
		tmp = t_0
	elif c <= 2.9e-123:
		tmp = t_2
	elif c <= 4e+48:
		tmp = t_0
	elif c <= 5e+102:
		tmp = t_2
	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(Float64(1.0 / c) * Float64(b / c))))
	t_2 = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))))
	tmp = 0.0
	if (c <= -6.2e+119)
		tmp = t_1;
	elseif (c <= -1.75e-145)
		tmp = t_0;
	elseif (c <= 2.9e-123)
		tmp = t_2;
	elseif (c <= 4e+48)
		tmp = t_0;
	elseif (c <= 5e+102)
		tmp = t_2;
	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 * ((1.0 / c) * (b / c)));
	t_2 = (1.0 / d) * (b + (a / (d / c)));
	tmp = 0.0;
	if (c <= -6.2e+119)
		tmp = t_1;
	elseif (c <= -1.75e-145)
		tmp = t_0;
	elseif (c <= 2.9e-123)
		tmp = t_2;
	elseif (c <= 4e+48)
		tmp = t_0;
	elseif (c <= 5e+102)
		tmp = t_2;
	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[(N[(1.0 / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.2e+119], t$95$1, If[LessEqual[c, -1.75e-145], t$95$0, If[LessEqual[c, 2.9e-123], t$95$2, If[LessEqual[c, 4e+48], t$95$0, If[LessEqual[c, 5e+102], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.1999999999999999e119 or 5e102 < c

   1. Initial program 38.3%

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

   3. Taylor expanded in c around inf 77.6%

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

      1. associate-/l* 79.5%

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

      2. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. pow2 79.5%

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

      3. times-frac 86.9%

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

   7. Applied egg-rr 86.9%

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

   if -6.1999999999999999e119 < c < -1.74999999999999998e-145 or 2.90000000000000004e-123 < c < 4.00000000000000018e48

   1. Initial program 85.7%

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

   if -1.74999999999999998e-145 < c < 2.90000000000000004e-123 or 4.00000000000000018e48 < c < 5e102

   1. Initial program 53.8%

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

      1. *-un-lft-identity 53.8%

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

      2. associate-*r/ 53.8%

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

      3. fma-def 53.8%

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

      4. add-sqr-sqrt 53.7%

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

      5. times-frac 53.8%

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

      6. fma-def 53.8%

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

      7. hypot-def 53.9%

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

      8. fma-def 53.9%

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

      9. fma-def 53.9%

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

      10. hypot-def 77.3%

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

   4. Applied egg-rr 77.3%

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

   5. Taylor expanded in c around 0 39.1%

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

      1. associate-/l* 40.1%

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

   7. Simplified 40.1%

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

   8. Taylor expanded in c around 0 82.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -1.75 \cdot 10^{-145}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \left(\frac{1}{c} \cdot \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{+52} \lor \neg \left(c \leq 1200000000 \lor \neg \left(c \leq 3.4 \cdot 10^{+40}\right) \land c \leq 3.6 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.75e+52)
         (not
          (or (<= c 1200000000.0) (and (not (<= c 3.4e+40)) (<= c 3.6e+102)))))
   (/ a c)
   (* (/ 1.0 d) (+ b (* c (/ a d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.75e+52) || !((c <= 1200000000.0) || (!(c <= 3.4e+40) && (c <= 3.6e+102)))) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.75d+52)) .or. (.not. (c <= 1200000000.0d0) .or. (.not. (c <= 3.4d+40)) .and. (c <= 3.6d+102))) then
        tmp = a / c
    else
        tmp = (1.0d0 / d) * (b + (c * (a / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.75e+52) || !((c <= 1200000000.0) || (!(c <= 3.4e+40) && (c <= 3.6e+102)))) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.75e+52) or not ((c <= 1200000000.0) or (not (c <= 3.4e+40) and (c <= 3.6e+102))):
		tmp = a / c
	else:
		tmp = (1.0 / d) * (b + (c * (a / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.75e+52) || !((c <= 1200000000.0) || (!(c <= 3.4e+40) && (c <= 3.6e+102))))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(c * Float64(a / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.75e+52) || ~(((c <= 1200000000.0) || (~((c <= 3.4e+40)) && (c <= 3.6e+102)))))
		tmp = a / c;
	else
		tmp = (1.0 / d) * (b + (c * (a / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.75e+52], N[Not[Or[LessEqual[c, 1200000000.0], And[N[Not[LessEqual[c, 3.4e+40]], $MachinePrecision], LessEqual[c, 3.6e+102]]]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.75e52 or 1.2e9 < c < 3.39999999999999989e40 or 3.6000000000000002e102 < c

   1. Initial program 50.0%

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

   3. Taylor expanded in c around inf 72.7%

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

   if -1.75e52 < c < 1.2e9 or 3.39999999999999989e40 < c < 3.6000000000000002e102

   1. Initial program 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. associate-*r/ 66.4%

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

      3. fma-def 66.5%

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

      4. add-sqr-sqrt 66.4%

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

      5. times-frac 66.4%

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

      6. fma-def 66.4%

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

      7. hypot-def 66.5%

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

      8. fma-def 66.5%

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

      9. fma-def 66.5%

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

      10. hypot-def 82.7%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in c around 0 38.7%

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

      1. associate-/l* 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c around 0 74.8%

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

      1. associate-/r/ 73.6%

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

   10. Applied egg-rr 73.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{+52} \lor \neg \left(c \leq 1200000000 \lor \neg \left(c \leq 3.4 \cdot 10^{+40}\right) \land c \leq 3.6 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+49} \lor \neg \left(c \leq 1100000000 \lor \neg \left(c \leq 1.15 \cdot 10^{+46}\right) \land c \leq 2.1 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6.2e+49)
         (not
          (or (<= c 1100000000.0)
              (and (not (<= c 1.15e+46)) (<= c 2.1e+103)))))
   (/ a c)
   (* (/ 1.0 d) (+ b (/ a (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.2e+49) || !((c <= 1100000000.0) || (!(c <= 1.15e+46) && (c <= 2.1e+103)))) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-6.2d+49)) .or. (.not. (c <= 1100000000.0d0) .or. (.not. (c <= 1.15d+46)) .and. (c <= 2.1d+103))) then
        tmp = a / c
    else
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.2e+49) || !((c <= 1100000000.0) || (!(c <= 1.15e+46) && (c <= 2.1e+103)))) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6.2e+49) or not ((c <= 1100000000.0) or (not (c <= 1.15e+46) and (c <= 2.1e+103))):
		tmp = a / c
	else:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6.2e+49) || !((c <= 1100000000.0) || (!(c <= 1.15e+46) && (c <= 2.1e+103))))
		tmp = Float64(a / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -6.2e+49) || ~(((c <= 1100000000.0) || (~((c <= 1.15e+46)) && (c <= 2.1e+103)))))
		tmp = a / c;
	else
		tmp = (1.0 / d) * (b + (a / (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6.2e+49], N[Not[Or[LessEqual[c, 1100000000.0], And[N[Not[LessEqual[c, 1.15e+46]], $MachinePrecision], LessEqual[c, 2.1e+103]]]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.19999999999999985e49 or 1.1e9 < c < 1.15e46 or 2.1000000000000002e103 < c

   1. Initial program 50.0%

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

   3. Taylor expanded in c around inf 72.7%

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

   if -6.19999999999999985e49 < c < 1.1e9 or 1.15e46 < c < 2.1000000000000002e103

   1. Initial program 66.4%

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

      1. *-un-lft-identity 66.4%

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

      2. associate-*r/ 66.4%

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

      3. fma-def 66.5%

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

      4. add-sqr-sqrt 66.4%

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

      5. times-frac 66.4%

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

      6. fma-def 66.4%

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

      7. hypot-def 66.5%

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

      8. fma-def 66.5%

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

      9. fma-def 66.5%

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

      10. hypot-def 82.7%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in c around 0 38.7%

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

      1. associate-/l* 39.3%

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

   7. Simplified 39.3%

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

   8. Taylor expanded in c around 0 74.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+49} \lor \neg \left(c \leq 1100000000 \lor \neg \left(c \leq 1.15 \cdot 10^{+46}\right) \land c \leq 2.1 \cdot 10^{+103}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* c (/ a d))))))
   (if (<= d -1.15e+62)
     t_0
     (if (<= d -3.9e-23)
       (/ (* b d) (+ (* c c) (* d d)))
       (if (<= d -5.7e-40)
         t_0
         (if (<= d 1.25e+27)
           (+ (/ a c) (/ (/ (* b d) c) c))
           (* (/ 1.0 d) (+ b (/ a (/ d c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (c * (a / d)));
	double tmp;
	if (d <= -1.15e+62) {
		tmp = t_0;
	} else if (d <= -3.9e-23) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= -5.7e-40) {
		tmp = t_0;
	} else if (d <= 1.25e+27) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / d) * (b + (c * (a / d)))
    if (d <= (-1.15d+62)) then
        tmp = t_0
    else if (d <= (-3.9d-23)) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= (-5.7d-40)) then
        tmp = t_0
    else if (d <= 1.25d+27) then
        tmp = (a / c) + (((b * d) / c) / c)
    else
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (c * (a / d)));
	double tmp;
	if (d <= -1.15e+62) {
		tmp = t_0;
	} else if (d <= -3.9e-23) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= -5.7e-40) {
		tmp = t_0;
	} else if (d <= 1.25e+27) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (c * (a / d)))
	tmp = 0
	if d <= -1.15e+62:
		tmp = t_0
	elif d <= -3.9e-23:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= -5.7e-40:
		tmp = t_0
	elif d <= 1.25e+27:
		tmp = (a / c) + (((b * d) / c) / c)
	else:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(c * Float64(a / d))))
	tmp = 0.0
	if (d <= -1.15e+62)
		tmp = t_0;
	elseif (d <= -3.9e-23)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= -5.7e-40)
		tmp = t_0;
	elseif (d <= 1.25e+27)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (c * (a / d)));
	tmp = 0.0;
	if (d <= -1.15e+62)
		tmp = t_0;
	elseif (d <= -3.9e-23)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= -5.7e-40)
		tmp = t_0;
	elseif (d <= 1.25e+27)
		tmp = (a / c) + (((b * d) / c) / c);
	else
		tmp = (1.0 / d) * (b + (a / (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.15e+62], t$95$0, If[LessEqual[d, -3.9e-23], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -5.7e-40], t$95$0, If[LessEqual[d, 1.25e+27], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.14999999999999992e62 or -3.9e-23 < d < -5.69999999999999984e-40

   1. Initial program 48.9%

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

      1. *-un-lft-identity 48.9%

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

      2. associate-*r/ 48.9%

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

      3. fma-def 48.9%

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

      4. add-sqr-sqrt 48.9%

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

      5. times-frac 49.0%

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

      6. fma-def 49.0%

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

      7. hypot-def 49.0%

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

      8. fma-def 49.0%

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

      9. fma-def 49.0%

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

      10. hypot-def 70.5%

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

   4. Applied egg-rr 70.5%

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

   5. Taylor expanded in c around 0 14.1%

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

      1. associate-/l* 14.3%

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

   7. Simplified 14.3%

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

   8. Taylor expanded in c around 0 79.6%

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

      1. associate-/r/ 81.4%

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

   10. Applied egg-rr 81.4%

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

   if -1.14999999999999992e62 < d < -3.9e-23

   1. Initial program 85.1%

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

   3. Taylor expanded in a around 0 80.5%

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

   if -5.69999999999999984e-40 < d < 1.24999999999999995e27

   1. Initial program 68.0%

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

   3. Taylor expanded in c around inf 74.5%

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

      1. associate-/l* 75.9%

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

      2. associate-/r/ 73.5%

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

   5. Simplified 73.5%

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

      1. associate-*l/ 74.5%

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

      2. pow2 74.5%

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

      3. associate-/r* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 1.24999999999999995e27 < d

   1. Initial program 47.0%

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

      1. *-un-lft-identity 47.0%

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

      2. associate-*r/ 47.0%

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

      3. fma-def 47.1%

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

      4. add-sqr-sqrt 47.0%

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

      5. times-frac 47.0%

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

      6. fma-def 47.0%

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

      7. hypot-def 47.0%

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

      8. fma-def 47.0%

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

      9. fma-def 47.0%

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

      10. hypot-def 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 73.5%

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

      1. associate-/l* 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in c around 0 78.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq -3.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq 1.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{b \cdot d}{t_0}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{a \cdot c}{t_0}\\ \mathbf{elif}\;d \leq 10^{+28}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))))
   (if (<= d -1.6e+62)
     (* (/ 1.0 d) (+ b (* c (/ a d))))
     (if (<= d -6.8e-19)
       (/ (* b d) t_0)
       (if (<= d -1.35e-40)
         (/ (* a c) t_0)
         (if (<= d 1e+28)
           (+ (/ a c) (/ (/ (* b d) c) c))
           (* (/ 1.0 d) (+ b (/ a (/ d c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double tmp;
	if (d <= -1.6e+62) {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	} else if (d <= -6.8e-19) {
		tmp = (b * d) / t_0;
	} else if (d <= -1.35e-40) {
		tmp = (a * c) / t_0;
	} else if (d <= 1e+28) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c * c) + (d * d)
    if (d <= (-1.6d+62)) then
        tmp = (1.0d0 / d) * (b + (c * (a / d)))
    else if (d <= (-6.8d-19)) then
        tmp = (b * d) / t_0
    else if (d <= (-1.35d-40)) then
        tmp = (a * c) / t_0
    else if (d <= 1d+28) then
        tmp = (a / c) + (((b * d) / c) / c)
    else
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double tmp;
	if (d <= -1.6e+62) {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	} else if (d <= -6.8e-19) {
		tmp = (b * d) / t_0;
	} else if (d <= -1.35e-40) {
		tmp = (a * c) / t_0;
	} else if (d <= 1e+28) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * c) + (d * d)
	tmp = 0
	if d <= -1.6e+62:
		tmp = (1.0 / d) * (b + (c * (a / d)))
	elif d <= -6.8e-19:
		tmp = (b * d) / t_0
	elif d <= -1.35e-40:
		tmp = (a * c) / t_0
	elif d <= 1e+28:
		tmp = (a / c) + (((b * d) / c) / c)
	else:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (d <= -1.6e+62)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(c * Float64(a / d))));
	elseif (d <= -6.8e-19)
		tmp = Float64(Float64(b * d) / t_0);
	elseif (d <= -1.35e-40)
		tmp = Float64(Float64(a * c) / t_0);
	elseif (d <= 1e+28)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * c) + (d * d);
	tmp = 0.0;
	if (d <= -1.6e+62)
		tmp = (1.0 / d) * (b + (c * (a / d)));
	elseif (d <= -6.8e-19)
		tmp = (b * d) / t_0;
	elseif (d <= -1.35e-40)
		tmp = (a * c) / t_0;
	elseif (d <= 1e+28)
		tmp = (a / c) + (((b * d) / c) / c);
	else
		tmp = (1.0 / d) * (b + (a / (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.6e+62], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6.8e-19], N[(N[(b * d), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, -1.35e-40], N[(N[(a * c), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[d, 1e+28], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.59999999999999992e62

   1. Initial program 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. associate-*r/ 43.7%

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

      3. fma-def 43.7%

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

      4. add-sqr-sqrt 43.7%

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

      5. times-frac 43.8%

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

      6. fma-def 43.8%

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

      7. hypot-def 43.8%

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

      8. fma-def 43.8%

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

      9. fma-def 43.8%

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

      10. hypot-def 67.4%

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

   4. Applied egg-rr 67.4%

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

   5. Taylor expanded in c around 0 15.5%

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

      1. associate-/l* 15.7%

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

   7. Simplified 15.7%

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

   8. Taylor expanded in c around 0 81.4%

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

      1. associate-/r/ 83.4%

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

   10. Applied egg-rr 83.4%

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

   if -1.59999999999999992e62 < d < -6.8000000000000004e-19

   1. Initial program 83.5%

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

   3. Taylor expanded in a around 0 83.7%

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

   if -6.8000000000000004e-19 < d < -1.35e-40

   1. Initial program 99.3%

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

   3. Taylor expanded in a around inf 61.2%

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

      1. *-commutative 61.2%

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

   5. Simplified 61.2%

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

   if -1.35e-40 < d < 9.99999999999999958e27

   1. Initial program 68.0%

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

   3. Taylor expanded in c around inf 74.5%

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

      1. associate-/l* 75.9%

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

      2. associate-/r/ 73.5%

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

   5. Simplified 73.5%

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

      1. associate-*l/ 74.5%

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

      2. pow2 74.5%

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

      3. associate-/r* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 9.99999999999999958e27 < d

   1. Initial program 47.0%

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

      1. *-un-lft-identity 47.0%

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

      2. associate-*r/ 47.0%

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

      3. fma-def 47.1%

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

      4. add-sqr-sqrt 47.0%

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

      5. times-frac 47.0%

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

      6. fma-def 47.0%

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

      7. hypot-def 47.0%

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

      8. fma-def 47.0%

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

      9. fma-def 47.0%

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

      10. hypot-def 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 73.5%

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

      1. associate-/l* 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in c around 0 78.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq -6.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{a \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 10^{+28}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{+46} \lor \neg \left(c \leq 4 \cdot 10^{-23} \lor \neg \left(c \leq 3.4 \cdot 10^{+45}\right) \land c \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.75e+46)
         (not (or (<= c 4e-23) (and (not (<= c 3.4e+45)) (<= c 1.4e+102)))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.75e+46) || !((c <= 4e-23) || (!(c <= 3.4e+45) && (c <= 1.4e+102)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.75d+46)) .or. (.not. (c <= 4d-23) .or. (.not. (c <= 3.4d+45)) .and. (c <= 1.4d+102))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.75e+46) || !((c <= 4e-23) || (!(c <= 3.4e+45) && (c <= 1.4e+102)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.75e+46) or not ((c <= 4e-23) or (not (c <= 3.4e+45) and (c <= 1.4e+102))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.75e+46) || !((c <= 4e-23) || (!(c <= 3.4e+45) && (c <= 1.4e+102))))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.75e+46) || ~(((c <= 4e-23) || (~((c <= 3.4e+45)) && (c <= 1.4e+102)))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.75e+46], N[Not[Or[LessEqual[c, 4e-23], And[N[Not[LessEqual[c, 3.4e+45]], $MachinePrecision], LessEqual[c, 1.4e+102]]]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.74999999999999992e46 or 3.99999999999999984e-23 < c < 3.4e45 or 1.40000000000000009e102 < c

   1. Initial program 54.4%

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

   3. Taylor expanded in c around inf 69.3%

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

   if -1.74999999999999992e46 < c < 3.99999999999999984e-23 or 3.4e45 < c < 1.40000000000000009e102

   1. Initial program 64.4%

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

   3. Taylor expanded in c around 0 61.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{+46} \lor \neg \left(c \leq 4 \cdot 10^{-23} \lor \neg \left(c \leq 3.4 \cdot 10^{+45}\right) \land c \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.9e-40)
   (* (/ 1.0 d) (+ b (* c (/ a d))))
   (if (<= d 8.2e+26)
     (+ (/ a c) (/ (/ (* b d) c) c))
     (* (/ 1.0 d) (+ b (/ a (/ d c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.9e-40) {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	} else if (d <= 8.2e+26) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3.9d-40)) then
        tmp = (1.0d0 / d) * (b + (c * (a / d)))
    else if (d <= 8.2d+26) then
        tmp = (a / c) + (((b * d) / c) / c)
    else
        tmp = (1.0d0 / d) * (b + (a / (d / c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.9e-40) {
		tmp = (1.0 / d) * (b + (c * (a / d)));
	} else if (d <= 8.2e+26) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (1.0 / d) * (b + (a / (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.9e-40:
		tmp = (1.0 / d) * (b + (c * (a / d)))
	elif d <= 8.2e+26:
		tmp = (a / c) + (((b * d) / c) / c)
	else:
		tmp = (1.0 / d) * (b + (a / (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.9e-40)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(c * Float64(a / d))));
	elseif (d <= 8.2e+26)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a / Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.9e-40)
		tmp = (1.0 / d) * (b + (c * (a / d)));
	elseif (d <= 8.2e+26)
		tmp = (a / c) + (((b * d) / c) / c);
	else
		tmp = (1.0 / d) * (b + (a / (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.9e-40], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(c * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 8.2e+26], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.89999999999999981e-40

   1. Initial program 58.8%

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

      1. *-un-lft-identity 58.8%

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

      2. associate-*r/ 58.8%

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

      3. fma-def 58.8%

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

      4. add-sqr-sqrt 58.8%

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

      5. times-frac 58.9%

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

      6. fma-def 58.9%

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

      7. hypot-def 58.9%

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

      8. fma-def 58.9%

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

      9. fma-def 58.9%

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

      10. hypot-def 77.1%

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

   4. Applied egg-rr 77.1%

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

   5. Taylor expanded in c around 0 12.1%

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

      1. associate-/l* 12.3%

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

   7. Simplified 12.3%

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

   8. Taylor expanded in c around 0 73.4%

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

      1. associate-/r/ 74.7%

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

   10. Applied egg-rr 74.7%

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

   if -3.89999999999999981e-40 < d < 8.19999999999999967e26

   1. Initial program 68.0%

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

   3. Taylor expanded in c around inf 74.5%

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

      1. associate-/l* 75.9%

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

      2. associate-/r/ 73.5%

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

   5. Simplified 73.5%

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

      1. associate-*l/ 74.5%

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

      2. pow2 74.5%

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

      3. associate-/r* 83.1%

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

   7. Applied egg-rr 83.1%

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

   if 8.19999999999999967e26 < d

   1. Initial program 47.0%

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

      1. *-un-lft-identity 47.0%

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

      2. associate-*r/ 47.0%

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

      3. fma-def 47.1%

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

      4. add-sqr-sqrt 47.0%

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

      5. times-frac 47.0%

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

      6. fma-def 47.0%

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

      7. hypot-def 47.0%

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

      8. fma-def 47.0%

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

      9. fma-def 47.0%

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

      10. hypot-def 62.5%

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

   4. Applied egg-rr 62.5%

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

   5. Taylor expanded in c around 0 73.5%

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

      1. associate-/l* 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in c around 0 78.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.9 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + c \cdot \frac{a}{d}\right)\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a}{\frac{d}{c}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.7% 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 60.1%

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

3. Taylor expanded in c around inf 41.6%

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

4. Final simplification 41.6%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024024 
(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))))