Complex division, real part

Percentage Accurate: 61.7% → 84.1%

Time: 8.3s

Alternatives: 14

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 2e+306)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (/ b (hypot d c)) (/ (hypot d c) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 2e+306) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b / hypot(d, c)) / (hypot(d, 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))) <= 2e+306)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b / hypot(d, c)) / Float64(hypot(d, 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], 2e+306], 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[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / d), $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))) < 2.00000000000000003e306

   1. Initial program 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. add-sqr-sqrt 81.0%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. fma-def 81.1%

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

      6. hypot-def 94.7%

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

   3. Applied egg-rr 94.7%

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

   if 2.00000000000000003e306 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 5.6%

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

   2. Taylor expanded in a around 0 4.4%

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

      1. add-sqr-sqrt 4.4%

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

      2. sqrt-div 4.1%

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

      3. hypot-udef 4.1%

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

      4. sqrt-div 4.1%

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

      5. hypot-udef 5.3%

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

      6. times-frac 4.1%

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

      7. add-sqr-sqrt 4.4%

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

      8. times-frac 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. *-commutative 69.6%

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

      2. clear-num 69.7%

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

      3. un-div-inv 69.7%

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

      4. hypot-udef 10.2%

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

      5. +-commutative 10.2%

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

      6. hypot-def 69.7%

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

      7. hypot-udef 10.2%

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

      8. +-commutative 10.2%

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

      9. hypot-def 69.7%

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

   6. Applied egg-rr 69.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \end{array} \]

Alternative 2: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -3.6 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (* (/ d (hypot c d)) (/ b (hypot c d)))))
   (if (<= d -3.6e+46)
     t_1
     (if (<= d -3e-130)
       t_0
       (if (<= d 4.7e-139)
         (* (/ 1.0 c) (+ a (/ d (/ c b))))
         (if (<= d 1.05e+25) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (d / hypot(c, d)) * (b / hypot(c, d));
	double tmp;
	if (d <= -3.6e+46) {
		tmp = t_1;
	} else if (d <= -3e-130) {
		tmp = t_0;
	} else if (d <= 4.7e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 1.05e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (d / Math.hypot(c, d)) * (b / Math.hypot(c, d));
	double tmp;
	if (d <= -3.6e+46) {
		tmp = t_1;
	} else if (d <= -3e-130) {
		tmp = t_0;
	} else if (d <= 4.7e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 1.05e+25) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (d / math.hypot(c, d)) * (b / math.hypot(c, d))
	tmp = 0
	if d <= -3.6e+46:
		tmp = t_1
	elif d <= -3e-130:
		tmp = t_0
	elif d <= 4.7e-139:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif d <= 1.05e+25:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(d / hypot(c, d)) * Float64(b / hypot(c, d)))
	tmp = 0.0
	if (d <= -3.6e+46)
		tmp = t_1;
	elseif (d <= -3e-130)
		tmp = t_0;
	elseif (d <= 4.7e-139)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (d <= 1.05e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (d / hypot(c, d)) * (b / hypot(c, d));
	tmp = 0.0;
	if (d <= -3.6e+46)
		tmp = t_1;
	elseif (d <= -3e-130)
		tmp = t_0;
	elseif (d <= 4.7e-139)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (d <= 1.05e+25)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.6e+46], t$95$1, If[LessEqual[d, -3e-130], t$95$0, If[LessEqual[d, 4.7e-139], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.05e+25], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.5999999999999999e46 or 1.05e25 < d

   1. Initial program 45.3%

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

   2. Taylor expanded in a around 0 42.4%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-div 20.5%

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

      3. hypot-udef 20.5%

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

      4. sqrt-div 20.5%

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

      5. hypot-udef 26.0%

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

      6. times-frac 20.5%

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

      7. add-sqr-sqrt 42.4%

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

      8. times-frac 84.7%

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

   4. Applied egg-rr 84.7%

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

   if -3.5999999999999999e46 < d < -2.99999999999999986e-130 or 4.70000000000000027e-139 < d < 1.05e25

   1. Initial program 90.0%

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

   if -2.99999999999999986e-130 < d < 4.70000000000000027e-139

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.3%

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

      4. hypot-def 74.3%

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

      5. fma-def 74.3%

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

      6. hypot-def 84.8%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in c around inf 42.2%

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

      1. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in c around inf 90.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+46}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-130}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{+25}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 3: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{\frac{b}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (/ (/ b (hypot d c)) (/ (hypot d c) d))))
   (if (<= d -7.5e+48)
     t_1
     (if (<= d -3.05e-130)
       t_0
       (if (<= d 2.8e-139)
         (* (/ 1.0 c) (+ a (/ d (/ c b))))
         (if (<= d 2.65e+24) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / hypot(d, c)) / (hypot(d, c) / d);
	double tmp;
	if (d <= -7.5e+48) {
		tmp = t_1;
	} else if (d <= -3.05e-130) {
		tmp = t_0;
	} else if (d <= 2.8e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 2.65e+24) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / Math.hypot(d, c)) / (Math.hypot(d, c) / d);
	double tmp;
	if (d <= -7.5e+48) {
		tmp = t_1;
	} else if (d <= -3.05e-130) {
		tmp = t_0;
	} else if (d <= 2.8e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 2.65e+24) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / math.hypot(d, c)) / (math.hypot(d, c) / d)
	tmp = 0
	if d <= -7.5e+48:
		tmp = t_1
	elif d <= -3.05e-130:
		tmp = t_0
	elif d <= 2.8e-139:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif d <= 2.65e+24:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / hypot(d, c)) / Float64(hypot(d, c) / d))
	tmp = 0.0
	if (d <= -7.5e+48)
		tmp = t_1;
	elseif (d <= -3.05e-130)
		tmp = t_0;
	elseif (d <= 2.8e-139)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (d <= 2.65e+24)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / hypot(d, c)) / (hypot(d, c) / d);
	tmp = 0.0;
	if (d <= -7.5e+48)
		tmp = t_1;
	elseif (d <= -3.05e-130)
		tmp = t_0;
	elseif (d <= 2.8e-139)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (d <= 2.65e+24)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.5e+48], t$95$1, If[LessEqual[d, -3.05e-130], t$95$0, If[LessEqual[d, 2.8e-139], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.65e+24], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.5000000000000006e48 or 2.6499999999999999e24 < d

   1. Initial program 45.3%

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

   2. Taylor expanded in a around 0 42.4%

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

      1. add-sqr-sqrt 28.8%

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

      2. sqrt-div 20.5%

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

      3. hypot-udef 20.5%

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

      4. sqrt-div 20.5%

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

      5. hypot-udef 26.0%

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

      6. times-frac 20.5%

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

      7. add-sqr-sqrt 42.4%

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

      8. times-frac 84.7%

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

   4. Applied egg-rr 84.7%

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

      1. *-commutative 84.7%

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

      2. clear-num 84.7%

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

      3. un-div-inv 84.8%

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

      4. hypot-udef 47.3%

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

      5. +-commutative 47.3%

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

      6. hypot-def 84.8%

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

      7. hypot-udef 47.3%

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

      8. +-commutative 47.3%

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

      9. hypot-def 84.8%

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

   6. Applied egg-rr 84.8%

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

   if -7.5000000000000006e48 < d < -3.04999999999999998e-130 or 2.7999999999999999e-139 < d < 2.6499999999999999e24

   1. Initial program 90.0%

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

   if -3.04999999999999998e-130 < d < 2.7999999999999999e-139

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.3%

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

      4. hypot-def 74.3%

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

      5. fma-def 74.3%

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

      6. hypot-def 84.8%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in c around inf 42.2%

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

      1. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in c around inf 90.0%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \mathbf{elif}\;d \leq -3.05 \cdot 10^{-130}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 2.65 \cdot 10^{+24}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\mathsf{hypot}\left(d, c\right)}}{\frac{\mathsf{hypot}\left(d, c\right)}{d}}\\ \end{array} \]

Alternative 4: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+68}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= d -1.3e+68)
     (* (+ b (/ c (/ d a))) (/ -1.0 (hypot c d)))
     (if (<= d -2.35e-126)
       t_0
       (if (<= d 4.7e-139)
         (* (/ 1.0 c) (+ a (/ d (/ c b))))
         (if (<= d 4.4e+79) t_0 (+ (/ b d) (* (/ c d) (/ a d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.3e+68) {
		tmp = (b + (c / (d / a))) * (-1.0 / hypot(c, d));
	} else if (d <= -2.35e-126) {
		tmp = t_0;
	} else if (d <= 4.7e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 4.4e+79) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.3e+68) {
		tmp = (b + (c / (d / a))) * (-1.0 / Math.hypot(c, d));
	} else if (d <= -2.35e-126) {
		tmp = t_0;
	} else if (d <= 4.7e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 4.4e+79) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.3e+68:
		tmp = (b + (c / (d / a))) * (-1.0 / math.hypot(c, d))
	elif d <= -2.35e-126:
		tmp = t_0
	elif d <= 4.7e-139:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif d <= 4.4e+79:
		tmp = t_0
	else:
		tmp = (b / d) + ((c / d) * (a / d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.3e+68)
		tmp = Float64(Float64(b + Float64(c / Float64(d / a))) * Float64(-1.0 / hypot(c, d)));
	elseif (d <= -2.35e-126)
		tmp = t_0;
	elseif (d <= 4.7e-139)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (d <= 4.4e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.3e+68)
		tmp = (b + (c / (d / a))) * (-1.0 / hypot(c, d));
	elseif (d <= -2.35e-126)
		tmp = t_0;
	elseif (d <= 4.7e-139)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (d <= 4.4e+79)
		tmp = t_0;
	else
		tmp = (b / d) + ((c / d) * (a / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e+68], N[(N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.35e-126], t$95$0, If[LessEqual[d, 4.7e-139], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e+79], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.2999999999999999e68

   1. Initial program 46.7%

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

      1. *-un-lft-identity 46.7%

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

      2. add-sqr-sqrt 46.7%

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

      3. times-frac 46.7%

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

      4. hypot-def 46.7%

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

      5. fma-def 46.7%

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

      6. hypot-def 64.5%

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

   3. Applied egg-rr 64.5%

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

   4. Taylor expanded in d around -inf 82.7%

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

      1. neg-mul-1 82.7%

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

      2. +-commutative 82.7%

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

      3. unsub-neg 82.7%

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

      4. mul-1-neg 82.7%

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

      5. associate-/l* 86.3%

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

      6. distribute-neg-frac 86.3%

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

   6. Simplified 86.3%

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

   if -1.2999999999999999e68 < d < -2.35000000000000009e-126 or 4.70000000000000027e-139 < d < 4.3999999999999998e79

   1. Initial program 83.6%

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

   if -2.35000000000000009e-126 < d < 4.70000000000000027e-139

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.3%

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

      4. hypot-def 74.3%

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

      5. fma-def 74.3%

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

      6. hypot-def 84.8%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in c around inf 42.2%

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

      1. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in c around inf 90.0%

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

   if 4.3999999999999998e79 < d

   1. Initial program 37.8%

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

   2. Taylor expanded in c around 0 78.5%

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

      1. unpow2 78.5%

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

      2. times-frac 82.4%

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

   4. Simplified 82.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+68}:\\ \;\;\;\;\left(b + \frac{c}{\frac{d}{a}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -2.35 \cdot 10^{-126}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 5: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{if}\;d \leq -4.1 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b d) (* (/ c d) (/ a d)))))
   (if (<= d -4.1e+71)
     t_1
     (if (<= d -9.2e-129)
       t_0
       (if (<= d 3.2e-139)
         (* (/ 1.0 c) (+ a (/ d (/ c b))))
         (if (<= d 1.5e+81) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -4.1e+71) {
		tmp = t_1;
	} else if (d <= -9.2e-129) {
		tmp = t_0;
	} else if (d <= 3.2e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 1.5e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (b / d) + ((c / d) * (a / d))
    if (d <= (-4.1d+71)) then
        tmp = t_1
    else if (d <= (-9.2d-129)) then
        tmp = t_0
    else if (d <= 3.2d-139) then
        tmp = (1.0d0 / c) * (a + (d / (c / b)))
    else if (d <= 1.5d+81) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (b / d) + ((c / d) * (a / d));
	double tmp;
	if (d <= -4.1e+71) {
		tmp = t_1;
	} else if (d <= -9.2e-129) {
		tmp = t_0;
	} else if (d <= 3.2e-139) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (d <= 1.5e+81) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (b / d) + ((c / d) * (a / d))
	tmp = 0
	if d <= -4.1e+71:
		tmp = t_1
	elif d <= -9.2e-129:
		tmp = t_0
	elif d <= 3.2e-139:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif d <= 1.5e+81:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)))
	tmp = 0.0
	if (d <= -4.1e+71)
		tmp = t_1;
	elseif (d <= -9.2e-129)
		tmp = t_0;
	elseif (d <= 3.2e-139)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (d <= 1.5e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (b / d) + ((c / d) * (a / d));
	tmp = 0.0;
	if (d <= -4.1e+71)
		tmp = t_1;
	elseif (d <= -9.2e-129)
		tmp = t_0;
	elseif (d <= 3.2e-139)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (d <= 1.5e+81)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.1e+71], t$95$1, If[LessEqual[d, -9.2e-129], t$95$0, If[LessEqual[d, 3.2e-139], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.5e+81], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.1000000000000002e71 or 1.49999999999999999e81 < d

   1. Initial program 42.4%

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

   2. Taylor expanded in c around 0 76.2%

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

      1. unpow2 76.2%

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

      2. times-frac 83.5%

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

   4. Simplified 83.5%

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

   if -4.1000000000000002e71 < d < -9.1999999999999998e-129 or 3.1999999999999999e-139 < d < 1.49999999999999999e81

   1. Initial program 83.6%

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

   if -9.1999999999999998e-129 < d < 3.1999999999999999e-139

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.3%

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

      4. hypot-def 74.3%

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

      5. fma-def 74.3%

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

      6. hypot-def 84.8%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in c around inf 42.2%

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

      1. associate-/l* 42.3%

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

   6. Simplified 42.3%

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

   7. Taylor expanded in c around inf 90.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 6: 76.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{b} \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e-13)
   (* (/ 1.0 c) (+ a (/ d (/ c b))))
   (if (<= c 1.65e-31)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (+ (/ a c) (/ 1.0 (* (/ c b) (/ c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 1.65e-31) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (1.0 / ((c / 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 (c <= (-5.6d-13)) then
        tmp = (1.0d0 / c) * (a + (d / (c / b)))
    else if (c <= 1.65d-31) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + (1.0d0 / ((c / 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 (c <= -5.6e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 1.65e-31) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + (1.0 / ((c / b) * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e-13:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif c <= 1.65e-31:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + (1.0 / ((c / b) * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e-13)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (c <= 1.65e-31)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(Float64(c / b) * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.6e-13)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (c <= 1.65e-31)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + (1.0 / ((c / b) * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e-13], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.65e-31], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(N[(c / b), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.6000000000000004e-13

   1. Initial program 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. times-frac 52.1%

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

      4. hypot-def 52.1%

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

      5. fma-def 52.1%

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

      6. hypot-def 62.0%

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

   3. Applied egg-rr 62.0%

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

   4. Taylor expanded in c around inf 15.4%

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

      1. associate-/l* 15.6%

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

   6. Simplified 15.6%

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

   7. Taylor expanded in c around inf 80.4%

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

   if -5.6000000000000004e-13 < c < 1.65e-31

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0 82.8%

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

      1. unpow2 82.8%

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

      2. times-frac 86.4%

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

   4. Simplified 86.4%

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

   if 1.65e-31 < c

   1. Initial program 58.9%

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

   2. Taylor expanded in c around inf 68.9%

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

      1. unpow2 68.9%

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

      2. associate-/l* 68.1%

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

   4. Simplified 68.1%

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

      1. clear-num 68.1%

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

      2. inv-pow 68.1%

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

      3. associate-/l* 75.9%

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

   6. Applied egg-rr 75.9%

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

      1. unpow-1 75.9%

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

      2. associate-/l/ 77.2%

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

      3. *-rgt-identity 77.2%

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

      4. times-frac 77.2%

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

      5. associate-/r/ 77.1%

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

      6. associate-*l/ 77.3%

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

      7. *-lft-identity 77.3%

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

   8. Simplified 77.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{b} \cdot \frac{c}{d}}\\ \end{array} \]

Alternative 7: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-13} \lor \neg \left(c \leq 4.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.5e-13) (not (<= c 4.4e-32)))
   (* (/ 1.0 c) (+ a (/ d (/ c b))))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.5e-13) || !(c <= 4.4e-32)) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} 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 <= (-5.5d-13)) .or. (.not. (c <= 4.4d-32))) then
        tmp = (1.0d0 / c) * (a + (d / (c / b)))
    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 <= -5.5e-13) || !(c <= 4.4e-32)) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.5e-13) or not (c <= 4.4e-32):
		tmp = (1.0 / c) * (a + (d / (c / b)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.5e-13) || !(c <= 4.4e-32))
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.5e-13) || ~((c <= 4.4e-32)))
		tmp = (1.0 / c) * (a + (d / (c / b)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.5e-13], N[Not[LessEqual[c, 4.4e-32]], $MachinePrecision]], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.49999999999999979e-13 or 4.4e-32 < c

   1. Initial program 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. add-sqr-sqrt 55.6%

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

      3. times-frac 55.7%

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

      4. hypot-def 55.7%

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

      5. fma-def 55.7%

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

      6. hypot-def 68.5%

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

   3. Applied egg-rr 68.5%

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

   4. Taylor expanded in c around inf 44.1%

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

      1. associate-/l* 47.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in c around inf 78.7%

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

   if -5.49999999999999979e-13 < c < 4.4e-32

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0 74.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-13} \lor \neg \left(c \leq 4.4 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 8: 70.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.1e-13)
   (* (/ 1.0 c) (+ a (/ d (/ c b))))
   (if (<= c 2.65e-33) (/ b d) (+ (/ a c) (* (/ d c) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.1e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 2.65e-33) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-4.1d-13)) then
        tmp = (1.0d0 / c) * (a + (d / (c / b)))
    else if (c <= 2.65d-33) then
        tmp = b / d
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.1e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 2.65e-33) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.1e-13:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif c <= 2.65e-33:
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.1e-13)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (c <= 2.65e-33)
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.1e-13)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (c <= 2.65e-33)
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.1e-13], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.65e-33], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.1000000000000002e-13

   1. Initial program 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. times-frac 52.1%

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

      4. hypot-def 52.1%

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

      5. fma-def 52.1%

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

      6. hypot-def 62.0%

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

   3. Applied egg-rr 62.0%

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

   4. Taylor expanded in c around inf 15.4%

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

      1. associate-/l* 15.6%

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

   6. Simplified 15.6%

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

   7. Taylor expanded in c around inf 80.4%

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

   if -4.1000000000000002e-13 < c < 2.64999999999999984e-33

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0 74.4%

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

   if 2.64999999999999984e-33 < c

   1. Initial program 58.9%

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

   2. Taylor expanded in c around inf 68.9%

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

      1. unpow2 68.9%

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

      2. times-frac 77.2%

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

   4. Simplified 77.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 9: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.7e-13)
   (* (/ 1.0 c) (+ a (/ d (/ c b))))
   (if (<= c 1.65e-31)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (+ (/ a c) (* (/ d c) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 1.65e-31) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.7d-13)) then
        tmp = (1.0d0 / c) * (a + (d / (c / b)))
    else if (c <= 1.65d-31) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.7e-13) {
		tmp = (1.0 / c) * (a + (d / (c / b)));
	} else if (c <= 1.65e-31) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.7e-13:
		tmp = (1.0 / c) * (a + (d / (c / b)))
	elif c <= 1.65e-31:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.7e-13)
		tmp = Float64(Float64(1.0 / c) * Float64(a + Float64(d / Float64(c / b))));
	elseif (c <= 1.65e-31)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.7e-13)
		tmp = (1.0 / c) * (a + (d / (c / b)));
	elseif (c <= 1.65e-31)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.7e-13], N[(N[(1.0 / c), $MachinePrecision] * N[(a + N[(d / N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.65e-31], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.70000000000000011e-13

   1. Initial program 52.1%

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

      1. *-un-lft-identity 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. times-frac 52.1%

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

      4. hypot-def 52.1%

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

      5. fma-def 52.1%

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

      6. hypot-def 62.0%

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

   3. Applied egg-rr 62.0%

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

   4. Taylor expanded in c around inf 15.4%

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

      1. associate-/l* 15.6%

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

   6. Simplified 15.6%

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

   7. Taylor expanded in c around inf 80.4%

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

   if -2.70000000000000011e-13 < c < 1.65e-31

   1. Initial program 72.9%

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

   2. Taylor expanded in c around 0 82.8%

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

      1. unpow2 82.8%

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

      2. times-frac 86.4%

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

   4. Simplified 86.4%

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

   if 1.65e-31 < c

   1. Initial program 58.9%

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

   2. Taylor expanded in c around inf 68.9%

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

      1. unpow2 68.9%

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

      2. times-frac 77.2%

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

   4. Simplified 77.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{c} \cdot \left(a + \frac{d}{\frac{c}{b}}\right)\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 10: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e-13)
   (/ a c)
   (if (<= c 1.45e-30)
     (/ b d)
     (if (<= c 1.9e-5)
       (* b (/ (/ d c) c))
       (if (<= c 1.5e+15) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 1.9e-5) {
		tmp = b * ((d / c) / c);
	} else if (c <= 1.5e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-5.6d-13)) then
        tmp = a / c
    else if (c <= 1.45d-30) then
        tmp = b / d
    else if (c <= 1.9d-5) then
        tmp = b * ((d / c) / c)
    else if (c <= 1.5d+15) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 1.9e-5) {
		tmp = b * ((d / c) / c);
	} else if (c <= 1.5e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e-13:
		tmp = a / c
	elif c <= 1.45e-30:
		tmp = b / d
	elif c <= 1.9e-5:
		tmp = b * ((d / c) / c)
	elif c <= 1.5e+15:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e-13)
		tmp = Float64(a / c);
	elseif (c <= 1.45e-30)
		tmp = Float64(b / d);
	elseif (c <= 1.9e-5)
		tmp = Float64(b * Float64(Float64(d / c) / c));
	elseif (c <= 1.5e+15)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.6e-13)
		tmp = a / c;
	elseif (c <= 1.45e-30)
		tmp = b / d;
	elseif (c <= 1.9e-5)
		tmp = b * ((d / c) / c);
	elseif (c <= 1.5e+15)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e-13], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.45e-30], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.9e-5], N[(b * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.5e+15], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.6000000000000004e-13 or 1.5e15 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in c around inf 66.2%

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

   if -5.6000000000000004e-13 < c < 1.44999999999999995e-30 or 1.9000000000000001e-5 < c < 1.5e15

   1. Initial program 72.5%

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

   2. Taylor expanded in c around 0 74.0%

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

   if 1.44999999999999995e-30 < c < 1.9000000000000001e-5

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 83.5%

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

      1. add-sqr-sqrt 49.9%

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

      2. sqrt-div 49.7%

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

      3. hypot-udef 49.7%

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

      4. sqrt-div 49.5%

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

      5. hypot-udef 49.5%

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

      6. times-frac 49.9%

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

      7. add-sqr-sqrt 83.5%

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

      8. times-frac 83.0%

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in d around 0 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*r/ 83.2%

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

      3. unpow2 83.2%

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

      4. associate-/r* 82.7%

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

   7. Simplified 82.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;b \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 11: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.2e-13)
   (/ a c)
   (if (<= c 1.45e-30)
     (/ b d)
     (if (<= c 3.4e-6)
       (* (/ d c) (/ b c))
       (if (<= c 3.7e+15) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.2e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 3.4e-6) {
		tmp = (d / c) * (b / c);
	} else if (c <= 3.7e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.2d-13)) then
        tmp = a / c
    else if (c <= 1.45d-30) then
        tmp = b / d
    else if (c <= 3.4d-6) then
        tmp = (d / c) * (b / c)
    else if (c <= 3.7d+15) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.2e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 3.4e-6) {
		tmp = (d / c) * (b / c);
	} else if (c <= 3.7e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.2e-13:
		tmp = a / c
	elif c <= 1.45e-30:
		tmp = b / d
	elif c <= 3.4e-6:
		tmp = (d / c) * (b / c)
	elif c <= 3.7e+15:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.2e-13)
		tmp = Float64(a / c);
	elseif (c <= 1.45e-30)
		tmp = Float64(b / d);
	elseif (c <= 3.4e-6)
		tmp = Float64(Float64(d / c) * Float64(b / c));
	elseif (c <= 3.7e+15)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.2e-13)
		tmp = a / c;
	elseif (c <= 1.45e-30)
		tmp = b / d;
	elseif (c <= 3.4e-6)
		tmp = (d / c) * (b / c);
	elseif (c <= 3.7e+15)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.2e-13], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.45e-30], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.4e-6], N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.7e+15], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.2e-13 or 3.7e15 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in c around inf 66.2%

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

   if -3.2e-13 < c < 1.44999999999999995e-30 or 3.40000000000000006e-6 < c < 3.7e15

   1. Initial program 72.5%

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

   2. Taylor expanded in c around 0 74.0%

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

   if 1.44999999999999995e-30 < c < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 83.5%

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

   3. Taylor expanded in c around inf 83.5%

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

      1. unpow2 83.5%

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

   5. Simplified 83.5%

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

      1. times-frac 83.0%

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

   7. Applied egg-rr 83.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 3.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 12: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6e-13)
   (/ a c)
   (if (<= c 1.45e-30)
     (/ b d)
     (if (<= c 3.4e-6)
       (/ (* b d) (* c c))
       (if (<= c 4.4e+15) (/ b d) (/ a c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 3.4e-6) {
		tmp = (b * d) / (c * c);
	} else if (c <= 4.4e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-6d-13)) then
        tmp = a / c
    else if (c <= 1.45d-30) then
        tmp = b / d
    else if (c <= 3.4d-6) then
        tmp = (b * d) / (c * c)
    else if (c <= 4.4d+15) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6e-13) {
		tmp = a / c;
	} else if (c <= 1.45e-30) {
		tmp = b / d;
	} else if (c <= 3.4e-6) {
		tmp = (b * d) / (c * c);
	} else if (c <= 4.4e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6e-13:
		tmp = a / c
	elif c <= 1.45e-30:
		tmp = b / d
	elif c <= 3.4e-6:
		tmp = (b * d) / (c * c)
	elif c <= 4.4e+15:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6e-13)
		tmp = Float64(a / c);
	elseif (c <= 1.45e-30)
		tmp = Float64(b / d);
	elseif (c <= 3.4e-6)
		tmp = Float64(Float64(b * d) / Float64(c * c));
	elseif (c <= 4.4e+15)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6e-13)
		tmp = a / c;
	elseif (c <= 1.45e-30)
		tmp = b / d;
	elseif (c <= 3.4e-6)
		tmp = (b * d) / (c * c);
	elseif (c <= 4.4e+15)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6e-13], N[(a / c), $MachinePrecision], If[LessEqual[c, 1.45e-30], N[(b / d), $MachinePrecision], If[LessEqual[c, 3.4e-6], N[(N[(b * d), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.4e+15], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.99999999999999968e-13 or 4.4e15 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in c around inf 66.2%

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

   if -5.99999999999999968e-13 < c < 1.44999999999999995e-30 or 3.40000000000000006e-6 < c < 4.4e15

   1. Initial program 72.5%

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

   2. Taylor expanded in c around 0 74.0%

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

   if 1.44999999999999995e-30 < c < 3.40000000000000006e-6

   1. Initial program 99.5%

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

   2. Taylor expanded in a around 0 83.5%

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

   3. Taylor expanded in c around inf 83.5%

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

      1. unpow2 83.5%

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

   5. Simplified 83.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 13: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.9e-13) (/ a c) (if (<= c 2e+15) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.9e-13) {
		tmp = a / c;
	} else if (c <= 2e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.9d-13)) then
        tmp = a / c
    else if (c <= 2d+15) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.9e-13) {
		tmp = a / c;
	} else if (c <= 2e+15) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.9e-13:
		tmp = a / c
	elif c <= 2e+15:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.9e-13)
		tmp = Float64(a / c);
	elseif (c <= 2e+15)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.9e-13)
		tmp = a / c;
	elseif (c <= 2e+15)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.9e-13], N[(a / c), $MachinePrecision], If[LessEqual[c, 2e+15], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.8999999999999998e-13 or 2e15 < c

   1. Initial program 53.3%

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

   2. Taylor expanded in c around inf 66.2%

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

   if -2.8999999999999998e-13 < c < 2e15

   1. Initial program 73.7%

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

   2. Taylor expanded in c around 0 70.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 14: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.9%

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

2. Taylor expanded in c around inf 40.5%

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

3. Final simplification 40.5%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023182 
(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))))