Complex division, real part

Percentage Accurate: 62.2% → 81.0%

Time: 11.1s

Alternatives: 12

Speedup: 1.3×

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: 62.2% 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: 81.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.18 \cdot 10^{+104}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-131}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.18e+104)
     (* (+ a (/ b (/ c d))) (/ -1.0 (hypot c d)))
     (if (<= c -5.2e-27)
       t_0
       (if (<= c 1.95e-131)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 2.7e+88)
           t_0
           (if (<= c 3.3e+107)
             (* (/ d (hypot c d)) (/ b (hypot c d)))
             (+ (/ a c) (/ b (* c (/ c d)))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.18e+104) {
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	} else if (c <= -5.2e-27) {
		tmp = t_0;
	} else if (c <= 1.95e-131) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.7e+88) {
		tmp = t_0;
	} else if (c <= 3.3e+107) {
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.18e+104) {
		tmp = (a + (b / (c / d))) * (-1.0 / Math.hypot(c, d));
	} else if (c <= -5.2e-27) {
		tmp = t_0;
	} else if (c <= 1.95e-131) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.7e+88) {
		tmp = t_0;
	} else if (c <= 3.3e+107) {
		tmp = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.18e+104:
		tmp = (a + (b / (c / d))) * (-1.0 / math.hypot(c, d))
	elif c <= -5.2e-27:
		tmp = t_0
	elif c <= 1.95e-131:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 2.7e+88:
		tmp = t_0
	elif c <= 3.3e+107:
		tmp = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.18e+104)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -5.2e-27)
		tmp = t_0;
	elseif (c <= 1.95e-131)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 2.7e+88)
		tmp = t_0;
	elseif (c <= 3.3e+107)
		tmp = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)));
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.18e+104)
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	elseif (c <= -5.2e-27)
		tmp = t_0;
	elseif (c <= 1.95e-131)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 2.7e+88)
		tmp = t_0;
	elseif (c <= 3.3e+107)
		tmp = (d / hypot(c, d)) * (b / hypot(c, d));
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.18e+104], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.2e-27], t$95$0, If[LessEqual[c, 1.95e-131], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.7e+88], t$95$0, If[LessEqual[c, 3.3e+107], N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.18e104

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. +-commutative 28.6%

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

      3. fma-udef 28.6%

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

      4. add-sqr-sqrt 28.6%

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

      5. times-frac 28.6%

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

      6. fma-udef 28.6%

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

      7. +-commutative 28.6%

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

      8. hypot-def 28.6%

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

      9. fma-def 28.8%

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

      10. fma-udef 28.8%

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

      11. +-commutative 28.8%

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

      12. hypot-def 54.2%

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

   4. Applied egg-rr 54.2%

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

   5. Taylor expanded in c around -inf 77.6%

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

      1. neg-mul-1 77.6%

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

      2. +-commutative 77.6%

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

      3. unsub-neg 77.6%

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

      4. mul-1-neg 77.6%

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

      5. associate-/l* 83.4%

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

      6. distribute-neg-frac 83.4%

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

   7. Simplified 83.4%

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

   if -1.18e104 < c < -5.20000000000000034e-27 or 1.9500000000000001e-131 < c < 2.70000000000000016e88

   1. Initial program 90.5%

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

   if -5.20000000000000034e-27 < c < 1.9500000000000001e-131

   1. Initial program 71.5%

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

   3. Taylor expanded in c around 0 83.2%

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

      1. associate-/l* 84.1%

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

      2. associate-/r/ 80.2%

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

   5. Simplified 80.2%

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

      1. associate-*l/ 83.2%

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

      2. unpow2 83.2%

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

      3. associate-/r* 91.4%

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

      4. *-commutative 91.4%

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

   7. Applied egg-rr 91.4%

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

   if 2.70000000000000016e88 < c < 3.30000000000000032e107

   1. Initial program 32.6%

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

   3. Taylor expanded in a around 0 32.6%

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

      1. *-commutative 32.6%

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

      2. add-sqr-sqrt 32.6%

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

      3. hypot-udef 32.6%

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

      4. hypot-udef 32.6%

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

      5. times-frac 98.8%

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

   5. Applied egg-rr 98.8%

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

   if 3.30000000000000032e107 < c

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf 77.5%

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

      1. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. pow2 79.4%

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

      2. *-un-lft-identity 79.4%

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

      3. times-frac 87.8%

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

   7. Applied egg-rr 87.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.18 \cdot 10^{+104}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-131}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $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 78.4%

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

      1. *-un-lft-identity 78.4%

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

      2. +-commutative 78.4%

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

      3. fma-udef 78.4%

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

      4. add-sqr-sqrt 78.4%

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

      5. times-frac 78.3%

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

      6. fma-udef 78.3%

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

      7. +-commutative 78.3%

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

      8. hypot-def 78.3%

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

      9. fma-def 78.3%

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

      10. fma-udef 78.3%

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

      11. +-commutative 78.3%

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

      12. hypot-def 95.8%

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

   4. Applied egg-rr 95.8%

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

   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 c around inf 46.1%

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

      1. associate-/l* 51.8%

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

   5. Simplified 51.8%

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

      1. pow2 51.8%

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

      2. *-un-lft-identity 51.8%

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

      3. times-frac 57.2%

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

   7. Applied egg-rr 57.2%

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

4. Final simplification 87.3%

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

Alternative 3: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.14 \cdot 10^{+104}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-131}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.14e+104)
     (* (+ a (/ b (/ c d))) (/ -1.0 (hypot c d)))
     (if (<= c -5.5e-27)
       t_0
       (if (<= c 1.05e-131)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 8.5e+153) t_0 (+ (/ a c) (/ b (* c (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.14e+104) {
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	} else if (c <= -5.5e-27) {
		tmp = t_0;
	} else if (c <= 1.05e-131) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 8.5e+153) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.14e+104) {
		tmp = (a + (b / (c / d))) * (-1.0 / Math.hypot(c, d));
	} else if (c <= -5.5e-27) {
		tmp = t_0;
	} else if (c <= 1.05e-131) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 8.5e+153) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.14e+104:
		tmp = (a + (b / (c / d))) * (-1.0 / math.hypot(c, d))
	elif c <= -5.5e-27:
		tmp = t_0
	elif c <= 1.05e-131:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 8.5e+153:
		tmp = t_0
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.14e+104)
		tmp = Float64(Float64(a + Float64(b / Float64(c / d))) * Float64(-1.0 / hypot(c, d)));
	elseif (c <= -5.5e-27)
		tmp = t_0;
	elseif (c <= 1.05e-131)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 8.5e+153)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.14e+104)
		tmp = (a + (b / (c / d))) * (-1.0 / hypot(c, d));
	elseif (c <= -5.5e-27)
		tmp = t_0;
	elseif (c <= 1.05e-131)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 8.5e+153)
		tmp = t_0;
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.14e+104], N[(N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.5e-27], t$95$0, If[LessEqual[c, 1.05e-131], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.5e+153], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.14000000000000007e104

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. +-commutative 28.6%

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

      3. fma-udef 28.6%

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

      4. add-sqr-sqrt 28.6%

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

      5. times-frac 28.6%

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

      6. fma-udef 28.6%

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

      7. +-commutative 28.6%

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

      8. hypot-def 28.6%

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

      9. fma-def 28.8%

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

      10. fma-udef 28.8%

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

      11. +-commutative 28.8%

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

      12. hypot-def 54.2%

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

   4. Applied egg-rr 54.2%

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

   5. Taylor expanded in c around -inf 77.6%

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

      1. neg-mul-1 77.6%

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

      2. +-commutative 77.6%

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

      3. unsub-neg 77.6%

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

      4. mul-1-neg 77.6%

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

      5. associate-/l* 83.4%

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

      6. distribute-neg-frac 83.4%

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

   7. Simplified 83.4%

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

   if -1.14000000000000007e104 < c < -5.5000000000000002e-27 or 1.04999999999999999e-131 < c < 8.49999999999999935e153

   1. Initial program 81.8%

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

   if -5.5000000000000002e-27 < c < 1.04999999999999999e-131

   1. Initial program 71.5%

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

   3. Taylor expanded in c around 0 83.2%

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

      1. associate-/l* 84.1%

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

      2. associate-/r/ 80.2%

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

   5. Simplified 80.2%

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

      1. associate-*l/ 83.2%

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

      2. unpow2 83.2%

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

      3. associate-/r* 91.4%

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

      4. *-commutative 91.4%

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

   7. Applied egg-rr 91.4%

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

   if 8.49999999999999935e153 < c

   1. Initial program 28.7%

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

   3. Taylor expanded in c around inf 80.6%

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

      1. associate-/l* 83.0%

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

   5. Simplified 83.0%

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

      1. pow2 83.0%

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

      2. *-un-lft-identity 83.0%

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

      3. times-frac 93.6%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.14 \cdot 10^{+104}:\\ \;\;\;\;\left(a + \frac{b}{\frac{c}{d}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-131}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.14 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -1.14e+104)
     (* (/ 1.0 c) (+ a (/ b (/ c d))))
     (if (<= c -5.2e-27)
       t_0
       (if (<= c 4.2e-132)
         (+ (/ b d) (/ (/ (* a c) d) d))
         (if (<= c 2.4e+153) t_0 (+ (/ a c) (/ b (* c (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.14e+104) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else if (c <= -5.2e-27) {
		tmp = t_0;
	} else if (c <= 4.2e-132) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.4e+153) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (b / (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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-1.14d+104)) then
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    else if (c <= (-5.2d-27)) then
        tmp = t_0
    else if (c <= 4.2d-132) then
        tmp = (b / d) + (((a * c) / d) / d)
    else if (c <= 2.4d+153) then
        tmp = t_0
    else
        tmp = (a / c) + (b / (c * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.14e+104) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else if (c <= -5.2e-27) {
		tmp = t_0;
	} else if (c <= 4.2e-132) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.4e+153) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1.14e+104:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	elif c <= -5.2e-27:
		tmp = t_0
	elif c <= 4.2e-132:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 2.4e+153:
		tmp = t_0
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.14e+104)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	elseif (c <= -5.2e-27)
		tmp = t_0;
	elseif (c <= 4.2e-132)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 2.4e+153)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1.14e+104)
		tmp = (1.0 / c) * (a + (b / (c / d)));
	elseif (c <= -5.2e-27)
		tmp = t_0;
	elseif (c <= 4.2e-132)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 2.4e+153)
		tmp = t_0;
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.14e+104], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.2e-27], t$95$0, If[LessEqual[c, 4.2e-132], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.4e+153], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.14000000000000007e104

   1. Initial program 28.6%

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

      1. *-un-lft-identity 28.6%

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

      2. +-commutative 28.6%

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

      3. fma-udef 28.6%

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

      4. add-sqr-sqrt 28.6%

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

      5. times-frac 28.6%

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

      6. fma-udef 28.6%

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

      7. +-commutative 28.6%

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

      8. hypot-def 28.6%

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

      9. fma-def 28.8%

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

      10. fma-udef 28.8%

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

      11. +-commutative 28.8%

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

      12. hypot-def 54.2%

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

   4. Applied egg-rr 54.2%

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

   5. Taylor expanded in c around inf 26.0%

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

      1. associate-/l* 26.1%

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

   7. Simplified 26.1%

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

   8. Taylor expanded in c around inf 83.2%

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

   if -1.14000000000000007e104 < c < -5.20000000000000034e-27 or 4.2000000000000002e-132 < c < 2.39999999999999992e153

   1. Initial program 81.8%

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

   if -5.20000000000000034e-27 < c < 4.2000000000000002e-132

   1. Initial program 71.5%

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

   3. Taylor expanded in c around 0 83.2%

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

      1. associate-/l* 84.1%

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

      2. associate-/r/ 80.2%

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

   5. Simplified 80.2%

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

      1. associate-*l/ 83.2%

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

      2. unpow2 83.2%

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

      3. associate-/r* 91.4%

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

      4. *-commutative 91.4%

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

   7. Applied egg-rr 91.4%

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

   if 2.39999999999999992e153 < c

   1. Initial program 28.7%

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

   3. Taylor expanded in c around inf 80.6%

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

      1. associate-/l* 83.0%

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

   5. Simplified 83.0%

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

      1. pow2 83.0%

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

      2. *-un-lft-identity 83.0%

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

      3. times-frac 93.6%

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

   7. Applied egg-rr 93.6%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.14 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq -5.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{-132}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{if}\;c \leq -1.76 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-32}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88} \lor \neg \left(c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 c) (+ a (/ b (/ c d))))))
   (if (<= c -1.76e-26)
     t_0
     (if (<= c 6e-32)
       (/ b d)
       (if (or (<= c 2.7e+88) (not (<= c 1.25e+107))) t_0 (/ 1.0 (/ d b)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -1.76e-26) {
		tmp = t_0;
	} else if (c <= 6e-32) {
		tmp = b / d;
	} else if ((c <= 2.7e+88) || !(c <= 1.25e+107)) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (d / b);
	}
	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 / c) * (a + (b / (c / d)))
    if (c <= (-1.76d-26)) then
        tmp = t_0
    else if (c <= 6d-32) then
        tmp = b / d
    else if ((c <= 2.7d+88) .or. (.not. (c <= 1.25d+107))) then
        tmp = t_0
    else
        tmp = 1.0d0 / (d / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -1.76e-26) {
		tmp = t_0;
	} else if (c <= 6e-32) {
		tmp = b / d;
	} else if ((c <= 2.7e+88) || !(c <= 1.25e+107)) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (d / b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / c) * (a + (b / (c / d)))
	tmp = 0
	if c <= -1.76e-26:
		tmp = t_0
	elif c <= 6e-32:
		tmp = b / d
	elif (c <= 2.7e+88) or not (c <= 1.25e+107):
		tmp = t_0
	else:
		tmp = 1.0 / (d / b)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))))
	tmp = 0.0
	if (c <= -1.76e-26)
		tmp = t_0;
	elseif (c <= 6e-32)
		tmp = Float64(b / d);
	elseif ((c <= 2.7e+88) || !(c <= 1.25e+107))
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(d / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / c) * (a + (b / (c / d)));
	tmp = 0.0;
	if (c <= -1.76e-26)
		tmp = t_0;
	elseif (c <= 6e-32)
		tmp = b / d;
	elseif ((c <= 2.7e+88) || ~((c <= 1.25e+107)))
		tmp = t_0;
	else
		tmp = 1.0 / (d / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.76e-26], t$95$0, If[LessEqual[c, 6e-32], N[(b / d), $MachinePrecision], If[Or[LessEqual[c, 2.7e+88], N[Not[LessEqual[c, 1.25e+107]], $MachinePrecision]], t$95$0, N[(1.0 / N[(d / b), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.7599999999999999e-26 or 6.0000000000000001e-32 < c < 2.70000000000000016e88 or 1.25e107 < c

   1. Initial program 50.7%

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

      1. *-un-lft-identity 50.7%

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

      2. +-commutative 50.7%

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

      3. fma-udef 50.7%

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

      4. add-sqr-sqrt 50.7%

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

      5. times-frac 50.7%

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

      6. fma-udef 50.7%

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

      7. +-commutative 50.7%

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

      8. hypot-def 50.7%

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

      9. fma-def 50.7%

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

      10. fma-udef 50.7%

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

      11. +-commutative 50.7%

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

      12. hypot-def 67.2%

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in c around inf 52.3%

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

      1. associate-/l* 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in c around inf 79.9%

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

   if -1.7599999999999999e-26 < c < 6.0000000000000001e-32

   1. Initial program 74.6%

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

   3. Taylor expanded in c around 0 74.8%

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

   if 2.70000000000000016e88 < c < 1.25e107

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. +-commutative 32.6%

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

      3. fma-udef 32.6%

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

      4. add-sqr-sqrt 32.6%

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

      5. times-frac 32.3%

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

      6. fma-udef 32.3%

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

      7. +-commutative 32.3%

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

      8. hypot-def 32.3%

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

      9. fma-def 32.3%

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

      10. fma-udef 32.3%

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

      11. +-commutative 32.3%

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

      12. hypot-def 71.5%

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in c around 0 58.0%

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

   6. Taylor expanded in c around 0 84.8%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot b \]
   7. Step-by-step derivation

      1. associate-/r/ 85.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]

   8. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.76 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq 6 \cdot 10^{-32}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88} \lor \neg \left(c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{if}\;c \leq -2.12 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 c) (+ a (/ b (/ c d))))))
   (if (<= c -2.12e-26)
     t_0
     (if (<= c 3.4e-28)
       (/ b d)
       (if (<= c 2.7e+88)
         (+ (/ a c) (* d (/ (/ b c) c)))
         (if (<= c 1.25e+107) (/ 1.0 (/ d b)) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -2.12e-26) {
		tmp = t_0;
	} else if (c <= 3.4e-28) {
		tmp = b / d;
	} else if (c <= 2.7e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.25e+107) {
		tmp = 1.0 / (d / b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) * (a + (b / (c / d)))
    if (c <= (-2.12d-26)) then
        tmp = t_0
    else if (c <= 3.4d-28) then
        tmp = b / d
    else if (c <= 2.7d+88) then
        tmp = (a / c) + (d * ((b / c) / c))
    else if (c <= 1.25d+107) then
        tmp = 1.0d0 / (d / b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -2.12e-26) {
		tmp = t_0;
	} else if (c <= 3.4e-28) {
		tmp = b / d;
	} else if (c <= 2.7e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.25e+107) {
		tmp = 1.0 / (d / b);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / c) * (a + (b / (c / d)))
	tmp = 0
	if c <= -2.12e-26:
		tmp = t_0
	elif c <= 3.4e-28:
		tmp = b / d
	elif c <= 2.7e+88:
		tmp = (a / c) + (d * ((b / c) / c))
	elif c <= 1.25e+107:
		tmp = 1.0 / (d / b)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))))
	tmp = 0.0
	if (c <= -2.12e-26)
		tmp = t_0;
	elseif (c <= 3.4e-28)
		tmp = Float64(b / d);
	elseif (c <= 2.7e+88)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(b / c) / c)));
	elseif (c <= 1.25e+107)
		tmp = Float64(1.0 / Float64(d / b));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / c) * (a + (b / (c / d)));
	tmp = 0.0;
	if (c <= -2.12e-26)
		tmp = t_0;
	elseif (c <= 3.4e-28)
		tmp = b / d;
	elseif (c <= 2.7e+88)
		tmp = (a / c) + (d * ((b / c) / c));
	elseif (c <= 1.25e+107)
		tmp = 1.0 / (d / b);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.12e-26], t$95$0, If[LessEqual[c, 3.4e-28], N[(b / d), $MachinePrecision], If[LessEqual[c, 2.7e+88], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(b / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.25e+107], N[(1.0 / N[(d / b), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.12000000000000002e-26 or 1.25e107 < c

   1. Initial program 44.0%

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

      1. *-un-lft-identity 44.0%

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

      2. +-commutative 44.0%

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

      3. fma-udef 44.0%

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

      4. add-sqr-sqrt 44.0%

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

      5. times-frac 43.9%

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

      6. fma-udef 43.9%

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

      7. +-commutative 43.9%

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

      8. hypot-def 43.9%

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

      9. fma-def 43.9%

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

      10. fma-udef 43.9%

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

      11. +-commutative 43.9%

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

      12. hypot-def 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in c around inf 48.5%

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

      1. associate-/l* 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in c around inf 81.3%

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

   if -2.12000000000000002e-26 < c < 3.4000000000000001e-28

   1. Initial program 74.6%

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

   3. Taylor expanded in c around 0 74.8%

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

   if 3.4000000000000001e-28 < c < 2.70000000000000016e88

   1. Initial program 96.0%

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

   3. Taylor expanded in c around inf 77.1%

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

      1. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

      1. pow2 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. times-frac 71.5%

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

   7. Applied egg-rr 71.5%

      \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{\frac{c}{1} \cdot \frac{c}{d}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 71.5%

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

      2. associate-/r* 71.4%

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

      3. associate-/r/ 77.3%

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

   9. Applied egg-rr 77.3%

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

   if 2.70000000000000016e88 < c < 1.25e107

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. +-commutative 32.6%

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

      3. fma-udef 32.6%

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

      4. add-sqr-sqrt 32.6%

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

      5. times-frac 32.3%

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

      6. fma-udef 32.3%

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

      7. +-commutative 32.3%

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

      8. hypot-def 32.3%

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

      9. fma-def 32.3%

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

      10. fma-udef 32.3%

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

      11. +-commutative 32.3%

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

      12. hypot-def 71.5%

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in c around 0 58.0%

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

   6. Taylor expanded in c around 0 84.8%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot b \]
   7. Step-by-step derivation

      1. associate-/r/ 85.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]

   8. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.12 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{if}\;c \leq -9.8 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 c) (+ a (/ b (/ c d))))))
   (if (<= c -9.8e-26)
     t_0
     (if (<= c 9e-25)
       (+ (/ b d) (/ (/ (* a c) d) d))
       (if (<= c 2.5e+88)
         (+ (/ a c) (* d (/ (/ b c) c)))
         (if (<= c 1.35e+107) (+ (/ b d) (* c (/ (/ a d) d))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -9.8e-26) {
		tmp = t_0;
	} else if (c <= 9e-25) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.5e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.35e+107) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / c) * (a + (b / (c / d)))
    if (c <= (-9.8d-26)) then
        tmp = t_0
    else if (c <= 9d-25) then
        tmp = (b / d) + (((a * c) / d) / d)
    else if (c <= 2.5d+88) then
        tmp = (a / c) + (d * ((b / c) / c))
    else if (c <= 1.35d+107) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / c) * (a + (b / (c / d)));
	double tmp;
	if (c <= -9.8e-26) {
		tmp = t_0;
	} else if (c <= 9e-25) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.5e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.35e+107) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / c) * (a + (b / (c / d)))
	tmp = 0
	if c <= -9.8e-26:
		tmp = t_0
	elif c <= 9e-25:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 2.5e+88:
		tmp = (a / c) + (d * ((b / c) / c))
	elif c <= 1.35e+107:
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))))
	tmp = 0.0
	if (c <= -9.8e-26)
		tmp = t_0;
	elseif (c <= 9e-25)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 2.5e+88)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(b / c) / c)));
	elseif (c <= 1.35e+107)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / c) * (a + (b / (c / d)));
	tmp = 0.0;
	if (c <= -9.8e-26)
		tmp = t_0;
	elseif (c <= 9e-25)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 2.5e+88)
		tmp = (a / c) + (d * ((b / c) / c));
	elseif (c <= 1.35e+107)
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.8e-26], t$95$0, If[LessEqual[c, 9e-25], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.5e+88], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(b / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.35e+107], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.7999999999999998e-26 or 1.3500000000000001e107 < c

   1. Initial program 44.0%

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

      1. *-un-lft-identity 44.0%

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

      2. +-commutative 44.0%

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

      3. fma-udef 44.0%

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

      4. add-sqr-sqrt 44.0%

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

      5. times-frac 43.9%

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

      6. fma-udef 43.9%

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

      7. +-commutative 43.9%

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

      8. hypot-def 43.9%

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

      9. fma-def 43.9%

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

      10. fma-udef 43.9%

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

      11. +-commutative 43.9%

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

      12. hypot-def 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in c around inf 48.5%

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

      1. associate-/l* 51.1%

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

   7. Simplified 51.1%

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

   8. Taylor expanded in c around inf 81.3%

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

   if -9.7999999999999998e-26 < c < 9.0000000000000002e-25

   1. Initial program 74.6%

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

   3. Taylor expanded in c around 0 82.1%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

      1. associate-*l/ 82.1%

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

      2. unpow2 82.1%

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

      3. associate-/r* 88.9%

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

      4. *-commutative 88.9%

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

   7. Applied egg-rr 88.9%

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

   if 9.0000000000000002e-25 < c < 2.49999999999999999e88

   1. Initial program 96.0%

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

   3. Taylor expanded in c around inf 77.1%

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

      1. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

      1. pow2 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. times-frac 71.5%

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

   7. Applied egg-rr 71.5%

      \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{\frac{c}{1} \cdot \frac{c}{d}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 71.5%

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

      2. associate-/r* 71.4%

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

      3. associate-/r/ 77.3%

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

   9. Applied egg-rr 77.3%

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

   if 2.49999999999999999e88 < c < 1.3500000000000001e107

   1. Initial program 32.6%

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

   3. Taylor expanded in c around 0 85.0%

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

      1. associate-/l* 85.0%

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

      2. associate-/r/ 85.0%

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

   5. Simplified 85.0%

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

      1. *-un-lft-identity 85.0%

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

      2. unpow2 85.0%

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

      3. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

   9. Simplified 86.2%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7e-27)
   (* (/ 1.0 c) (+ a (/ b (/ c d))))
   (if (<= c 1.1e-30)
     (+ (/ b d) (/ (/ (* a c) d) d))
     (if (<= c 2.7e+88)
       (+ (/ a c) (* d (/ (/ b c) c)))
       (if (<= c 1.25e+107)
         (+ (/ b d) (* c (/ (/ a d) d)))
         (+ (/ a c) (/ b (* c (/ c d)))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7e-27) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else if (c <= 1.1e-30) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.7e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.25e+107) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (a / c) + (b / (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 (c <= (-7d-27)) then
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    else if (c <= 1.1d-30) then
        tmp = (b / d) + (((a * c) / d) / d)
    else if (c <= 2.7d+88) then
        tmp = (a / c) + (d * ((b / c) / c))
    else if (c <= 1.25d+107) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = (a / c) + (b / (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 (c <= -7e-27) {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	} else if (c <= 1.1e-30) {
		tmp = (b / d) + (((a * c) / d) / d);
	} else if (c <= 2.7e+88) {
		tmp = (a / c) + (d * ((b / c) / c));
	} else if (c <= 1.25e+107) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (a / c) + (b / (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7e-27:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	elif c <= 1.1e-30:
		tmp = (b / d) + (((a * c) / d) / d)
	elif c <= 2.7e+88:
		tmp = (a / c) + (d * ((b / c) / c))
	elif c <= 1.25e+107:
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = (a / c) + (b / (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7e-27)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	elseif (c <= 1.1e-30)
		tmp = Float64(Float64(b / d) + Float64(Float64(Float64(a * c) / d) / d));
	elseif (c <= 2.7e+88)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(Float64(b / c) / c)));
	elseif (c <= 1.25e+107)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7e-27)
		tmp = (1.0 / c) * (a + (b / (c / d)));
	elseif (c <= 1.1e-30)
		tmp = (b / d) + (((a * c) / d) / d);
	elseif (c <= 2.7e+88)
		tmp = (a / c) + (d * ((b / c) / c));
	elseif (c <= 1.25e+107)
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = (a / c) + (b / (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7e-27], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.1e-30], N[(N[(b / d), $MachinePrecision] + N[(N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.7e+88], N[(N[(a / c), $MachinePrecision] + N[(d * N[(N[(b / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.25e+107], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -7.0000000000000003e-27

   1. Initial program 51.4%

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

      1. *-un-lft-identity 51.4%

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

      2. +-commutative 51.4%

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

      3. fma-udef 51.4%

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

      4. add-sqr-sqrt 51.3%

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

      5. times-frac 51.3%

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

      6. fma-udef 51.3%

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

      7. +-commutative 51.3%

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

      8. hypot-def 51.3%

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

      9. fma-def 51.3%

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

      10. fma-udef 51.3%

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

      11. +-commutative 51.3%

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

      12. hypot-def 68.5%

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

   4. Applied egg-rr 68.5%

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

   5. Taylor expanded in c around inf 16.9%

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

      1. associate-/l* 16.9%

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

   7. Simplified 16.9%

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

   8. Taylor expanded in c around inf 75.4%

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

   if -7.0000000000000003e-27 < c < 1.09999999999999992e-30

   1. Initial program 74.6%

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

   3. Taylor expanded in c around 0 82.1%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 79.5%

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

   5. Simplified 79.5%

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

      1. associate-*l/ 82.1%

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

      2. unpow2 82.1%

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

      3. associate-/r* 88.9%

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

      4. *-commutative 88.9%

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

   7. Applied egg-rr 88.9%

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

   if 1.09999999999999992e-30 < c < 2.70000000000000016e88

   1. Initial program 96.0%

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

   3. Taylor expanded in c around inf 77.1%

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

      1. associate-/l* 71.5%

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

   5. Simplified 71.5%

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

      1. pow2 71.5%

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

      2. *-un-lft-identity 71.5%

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

      3. times-frac 71.5%

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

   7. Applied egg-rr 71.5%

      \[\leadsto \frac{a}{c} + \frac{b}{\color{blue}{\frac{c}{1} \cdot \frac{c}{d}}} \]
   8. Step-by-step derivation

      1. /-rgt-identity 71.5%

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

      2. associate-/r* 71.4%

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

      3. associate-/r/ 77.3%

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

   9. Applied egg-rr 77.3%

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

   if 2.70000000000000016e88 < c < 1.25e107

   1. Initial program 32.6%

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

   3. Taylor expanded in c around 0 85.0%

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

      1. associate-/l* 85.0%

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

      2. associate-/r/ 85.0%

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

   5. Simplified 85.0%

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

      1. *-un-lft-identity 85.0%

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

      2. unpow2 85.0%

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

      3. times-frac 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. associate-*l/ 86.2%

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

      2. *-lft-identity 86.2%

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

   9. Simplified 86.2%

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

   if 1.25e107 < c

   1. Initial program 36.0%

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

   3. Taylor expanded in c around inf 77.5%

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

      1. associate-/l* 79.4%

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

   5. Simplified 79.4%

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

      1. pow2 79.4%

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

      2. *-un-lft-identity 79.4%

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

      3. times-frac 87.8%

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

   7. Applied egg-rr 87.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{-27}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d} + \frac{\frac{a \cdot c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{\frac{b}{c}}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-46} \lor \neg \left(d \leq 3.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9e-46) (not (<= d 3.3e-47)))
   (+ (/ b d) (* c (/ (/ a d) d)))
   (* (/ 1.0 c) (+ a (/ b (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e-46) || !(d <= 3.3e-47)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (1.0 / c) * (a + (b / (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 ((d <= (-9d-46)) .or. (.not. (d <= 3.3d-47))) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = (1.0d0 / c) * (a + (b / (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e-46) || !(d <= 3.3e-47)) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (1.0 / c) * (a + (b / (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9e-46) or not (d <= 3.3e-47):
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = (1.0 / c) * (a + (b / (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9e-46) || !(d <= 3.3e-47))
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(b / Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -9e-46) || ~((d <= 3.3e-47)))
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = (1.0 / c) * (a + (b / (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9e-46], N[Not[LessEqual[d, 3.3e-47]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(b / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.00000000000000001e-46 or 3.30000000000000004e-47 < d

   1. Initial program 56.6%

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

   3. Taylor expanded in c around 0 72.4%

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

      1. associate-/l* 71.1%

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

      2. associate-/r/ 72.6%

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

   5. Simplified 72.6%

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

      1. *-un-lft-identity 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. associate-*l/ 73.3%

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

      2. *-lft-identity 73.3%

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

   9. Simplified 73.3%

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

   if -9.00000000000000001e-46 < d < 3.30000000000000004e-47

   1. Initial program 67.4%

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

      1. *-un-lft-identity 67.4%

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

      2. +-commutative 67.4%

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

      3. fma-udef 67.4%

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

      4. add-sqr-sqrt 67.4%

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

      5. times-frac 67.3%

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

      6. fma-udef 67.3%

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

      7. +-commutative 67.3%

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

      8. hypot-def 67.3%

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

      9. fma-def 67.3%

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

      10. fma-udef 67.3%

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

      11. +-commutative 67.3%

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

      12. hypot-def 80.2%

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

   4. Applied egg-rr 80.2%

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

   5. Taylor expanded in c around inf 55.5%

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

      1. associate-/l* 54.7%

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

   7. Simplified 54.7%

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

   8. Taylor expanded in c around inf 83.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-46} \lor \neg \left(d \leq 3.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{b}{\frac{c}{d}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88} \lor \neg \left(c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.35e-20)
   (/ a c)
   (if (<= c 2.75e-30)
     (/ b d)
     (if (or (<= c 2.7e+88) (not (<= c 1.25e+107))) (/ a c) (/ 1.0 (/ d b))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.35e-20) {
		tmp = a / c;
	} else if (c <= 2.75e-30) {
		tmp = b / d;
	} else if ((c <= 2.7e+88) || !(c <= 1.25e+107)) {
		tmp = a / c;
	} else {
		tmp = 1.0 / (d / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.35d-20)) then
        tmp = a / c
    else if (c <= 2.75d-30) then
        tmp = b / d
    else if ((c <= 2.7d+88) .or. (.not. (c <= 1.25d+107))) then
        tmp = a / c
    else
        tmp = 1.0d0 / (d / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.35e-20) {
		tmp = a / c;
	} else if (c <= 2.75e-30) {
		tmp = b / d;
	} else if ((c <= 2.7e+88) || !(c <= 1.25e+107)) {
		tmp = a / c;
	} else {
		tmp = 1.0 / (d / b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.35e-20:
		tmp = a / c
	elif c <= 2.75e-30:
		tmp = b / d
	elif (c <= 2.7e+88) or not (c <= 1.25e+107):
		tmp = a / c
	else:
		tmp = 1.0 / (d / b)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.35e-20)
		tmp = Float64(a / c);
	elseif (c <= 2.75e-30)
		tmp = Float64(b / d);
	elseif ((c <= 2.7e+88) || !(c <= 1.25e+107))
		tmp = Float64(a / c);
	else
		tmp = Float64(1.0 / Float64(d / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.35e-20)
		tmp = a / c;
	elseif (c <= 2.75e-30)
		tmp = b / d;
	elseif ((c <= 2.7e+88) || ~((c <= 1.25e+107)))
		tmp = a / c;
	else
		tmp = 1.0 / (d / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.35e-20], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.75e-30], N[(b / d), $MachinePrecision], If[Or[LessEqual[c, 2.7e+88], N[Not[LessEqual[c, 1.25e+107]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(1.0 / N[(d / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.35000000000000007e-20 or 2.74999999999999988e-30 < c < 2.70000000000000016e88 or 1.25e107 < 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 68.7%

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

   if -2.35000000000000007e-20 < c < 2.74999999999999988e-30

   1. Initial program 75.0%

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

   3. Taylor expanded in c around 0 73.6%

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

   if 2.70000000000000016e88 < c < 1.25e107

   1. Initial program 32.6%

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

      1. *-un-lft-identity 32.6%

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

      2. +-commutative 32.6%

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

      3. fma-udef 32.6%

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

      4. add-sqr-sqrt 32.6%

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

      5. times-frac 32.3%

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

      6. fma-udef 32.3%

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

      7. +-commutative 32.3%

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

      8. hypot-def 32.3%

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

      9. fma-def 32.3%

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

      10. fma-udef 32.3%

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

      11. +-commutative 32.3%

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

      12. hypot-def 71.5%

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

   4. Applied egg-rr 71.5%

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

   5. Taylor expanded in c around 0 58.0%

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

   6. Taylor expanded in c around 0 84.8%

      \[\leadsto \color{blue}{\frac{1}{d}} \cdot b \]
   7. Step-by-step derivation

      1. associate-/r/ 85.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]

   8. Applied egg-rr 85.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{d}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+88} \lor \neg \left(c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{d}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.36 \cdot 10^{-19} \lor \neg \left(c \leq 2 \cdot 10^{-26} \lor \neg \left(c \leq 2.7 \cdot 10^{+88}\right) \land c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.36e-19)
         (not (or (<= c 2e-26) (and (not (<= c 2.7e+88)) (<= c 1.25e+107)))))
   (/ a c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.36e-19) || !((c <= 2e-26) || (!(c <= 2.7e+88) && (c <= 1.25e+107)))) {
		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 <= (-1.36d-19)) .or. (.not. (c <= 2d-26) .or. (.not. (c <= 2.7d+88)) .and. (c <= 1.25d+107))) 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 <= -1.36e-19) || !((c <= 2e-26) || (!(c <= 2.7e+88) && (c <= 1.25e+107)))) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.36e-19) or not ((c <= 2e-26) or (not (c <= 2.7e+88) and (c <= 1.25e+107))):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.36e-19) || !((c <= 2e-26) || (!(c <= 2.7e+88) && (c <= 1.25e+107))))
		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 <= -1.36e-19) || ~(((c <= 2e-26) || (~((c <= 2.7e+88)) && (c <= 1.25e+107)))))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.36e-19], N[Not[Or[LessEqual[c, 2e-26], And[N[Not[LessEqual[c, 2.7e+88]], $MachinePrecision], LessEqual[c, 1.25e+107]]]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.3599999999999999e-19 or 2.0000000000000001e-26 < c < 2.70000000000000016e88 or 1.25e107 < 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 68.7%

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

   if -1.3599999999999999e-19 < c < 2.0000000000000001e-26 or 2.70000000000000016e88 < c < 1.25e107

   1. Initial program 72.7%

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

   3. Taylor expanded in c around 0 74.2%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.36 \cdot 10^{-19} \lor \neg \left(c \leq 2 \cdot 10^{-26} \lor \neg \left(c \leq 2.7 \cdot 10^{+88}\right) \land c \leq 1.25 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.2%

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

3. Taylor expanded in c around inf 40.8%

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

4. Final simplification 40.8%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024011 
(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))))