Complex division, imag part

Percentage Accurate: 61.7% → 85.2%

Time: 10.9s

Alternatives: 12

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a \cdot d - b \cdot c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) 2e+293)
   (* (/ (- (* a d) (* b c)) (hypot c d)) (/ -1.0 (hypot c d)))
   (/ (- (/ c (/ d b)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+293) {
		tmp = (((a * d) - (b * c)) / hypot(c, d)) * (-1.0 / hypot(c, d));
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+293) {
		tmp = (((a * d) - (b * c)) / Math.hypot(c, d)) * (-1.0 / Math.hypot(c, d));
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+293:
		tmp = (((a * d) - (b * c)) / math.hypot(c, d)) * (-1.0 / math.hypot(c, d))
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+293)
		tmp = Float64(Float64(Float64(Float64(a * d) - Float64(b * c)) / hypot(c, d)) * Float64(-1.0 / hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+293)
		tmp = (((a * d) - (b * c)) / hypot(c, d)) * (-1.0 / hypot(c, d));
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+293], N[(N[(N[(N[(a * d), $MachinePrecision] - N[(b * c), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.9999999999999998e293

   1. Initial program 76.0%

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

      1. *-un-lft-identity 76.0%

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

      2. add-sqr-sqrt 76.0%

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

      3. times-frac 76.0%

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

      4. hypot-define 76.0%

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

      5. hypot-define 95.6%

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

   4. Applied egg-rr 95.6%

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

   if 1.9999999999999998e293 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 9.0%

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

   3. Taylor expanded in c around 0 45.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 45.1%

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

      2. mul-1-neg 45.1%

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

      3. unsub-neg 45.1%

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

      4. unpow2 45.1%

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

      5. associate-/r* 50.6%

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

      6. div-sub 52.4%

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

      7. *-commutative 52.4%

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

      8. associate-/l* 57.8%

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

   5. Simplified 57.8%

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

      1. clear-num 57.8%

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

      2. un-div-inv 57.9%

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

   7. Applied egg-rr 57.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{a \cdot d - b \cdot c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{-1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+97}:\\ \;\;\;\;b \cdot \frac{\frac{c - \frac{d}{\frac{b}{a}}}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.4e+106)
   (/ (- (/ c (/ d b)) a) d)
   (if (<= d 5.5e+97)
     (* b (/ (/ (- c (/ d (/ b a))) (hypot c d)) (hypot c d)))
     (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.4e+106) {
		tmp = ((c / (d / b)) - a) / d;
	} else if (d <= 5.5e+97) {
		tmp = b * (((c - (d / (b / a))) / hypot(c, d)) / hypot(c, d));
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.4e+106) {
		tmp = ((c / (d / b)) - a) / d;
	} else if (d <= 5.5e+97) {
		tmp = b * (((c - (d / (b / a))) / Math.hypot(c, d)) / Math.hypot(c, d));
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.4e+106:
		tmp = ((c / (d / b)) - a) / d
	elif d <= 5.5e+97:
		tmp = b * (((c - (d / (b / a))) / math.hypot(c, d)) / math.hypot(c, d))
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.4e+106)
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	elseif (d <= 5.5e+97)
		tmp = Float64(b * Float64(Float64(Float64(c - Float64(d / Float64(b / a))) / hypot(c, d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -2.4e+106)
		tmp = ((c / (d / b)) - a) / d;
	elseif (d <= 5.5e+97)
		tmp = b * (((c - (d / (b / a))) / hypot(c, d)) / hypot(c, d));
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.4e+106], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 5.5e+97], N[(b * N[(N[(N[(c - N[(d / N[(b / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.4000000000000001e106

   1. Initial program 34.4%

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

   3. Taylor expanded in c around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/r* 80.1%

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

      6. div-sub 80.2%

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

      7. *-commutative 80.2%

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

      8. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

      1. clear-num 83.4%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

   if -2.4000000000000001e106 < d < 5.50000000000000021e97

   1. Initial program 72.9%

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

      1. *-un-lft-identity 72.9%

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

      2. add-sqr-sqrt 72.9%

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

      3. times-frac 72.8%

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

      4. hypot-define 72.8%

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

      5. hypot-define 85.6%

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

   4. Applied egg-rr 85.6%

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

   5. Taylor expanded in b around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. unsub-neg 82.2%

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

      3. *-commutative 82.2%

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

      4. associate-/l* 76.1%

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

   7. Simplified 76.1%

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

      1. associate-*l/ 76.3%

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

      2. *-un-lft-identity 76.3%

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

      3. clear-num 75.7%

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

      4. associate-/l* 88.0%

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

      5. clear-num 87.9%

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

      6. un-div-inv 89.3%

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

   9. Applied egg-rr 89.3%

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

       1. associate-/r/ 89.8%

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

       2. associate-*r/ 77.9%

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

       3. associate-/l* 78.0%

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

       4. *-commutative 78.0%

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

       5. associate-*r/ 78.1%

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

       6. *-rgt-identity 78.1%

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

       7. associate-*r/ 90.1%

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

       8. associate-/l* 88.8%

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

   11. Simplified 88.8%

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

   if 5.50000000000000021e97 < d

   1. Initial program 37.5%

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

   3. Taylor expanded in c around 0 73.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 73.9%

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

      2. mul-1-neg 73.9%

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

      3. unsub-neg 73.9%

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

      4. unpow2 73.9%

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

      5. associate-/r* 78.0%

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

      6. div-sub 78.0%

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

      7. *-commutative 78.0%

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

      8. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

Alternative 3: 82.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{a}{c} - b\right) \cdot \frac{-1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -2.2e+36)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -8.5e-116)
       t_0
       (if (<= c 8.2e-117)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 9.5e+55) t_0 (* (- (* d (/ a c)) b) (/ -1.0 c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.2e+36) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -8.5e-116) {
		tmp = t_0;
	} else if (c <= 8.2e-117) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 9.5e+55) {
		tmp = t_0;
	} else {
		tmp = ((d * (a / c)) - b) * (-1.0 / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (c <= (-2.2d+36)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= (-8.5d-116)) then
        tmp = t_0
    else if (c <= 8.2d-117) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 9.5d+55) then
        tmp = t_0
    else
        tmp = ((d * (a / c)) - b) * ((-1.0d0) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.2e+36) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -8.5e-116) {
		tmp = t_0;
	} else if (c <= 8.2e-117) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 9.5e+55) {
		tmp = t_0;
	} else {
		tmp = ((d * (a / c)) - b) * (-1.0 / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.2e+36:
		tmp = (b - (a * (d / c))) / c
	elif c <= -8.5e-116:
		tmp = t_0
	elif c <= 8.2e-117:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 9.5e+55:
		tmp = t_0
	else:
		tmp = ((d * (a / c)) - b) * (-1.0 / c)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.2e+36)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -8.5e-116)
		tmp = t_0;
	elseif (c <= 8.2e-117)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 9.5e+55)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(d * Float64(a / c)) - b) * Float64(-1.0 / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -2.2e+36)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -8.5e-116)
		tmp = t_0;
	elseif (c <= 8.2e-117)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 9.5e+55)
		tmp = t_0;
	else
		tmp = ((d * (a / c)) - b) * (-1.0 / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.2e+36], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -8.5e-116], t$95$0, If[LessEqual[c, 8.2e-117], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.5e+55], t$95$0, N[(N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.2e36

   1. Initial program 49.5%

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

      1. *-un-lft-identity 49.5%

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

      2. add-sqr-sqrt 49.5%

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

      3. times-frac 49.5%

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

      4. hypot-define 49.5%

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

      5. hypot-define 72.1%

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

   4. Applied egg-rr 72.1%

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

   5. Taylor expanded in b around inf 62.1%

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

      1. mul-1-neg 62.1%

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

      2. unsub-neg 62.1%

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

      3. *-commutative 62.1%

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

      4. associate-/l* 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in c around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. unsub-neg 84.3%

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

      3. associate-/l* 86.4%

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

   10. Simplified 86.4%

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

   if -2.2e36 < c < -8.4999999999999995e-116 or 8.20000000000000063e-117 < c < 9.49999999999999989e55

   1. Initial program 81.3%

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

   if -8.4999999999999995e-116 < c < 8.20000000000000063e-117

   1. Initial program 67.3%

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

   3. Taylor expanded in c around 0 79.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 79.4%

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

      2. mul-1-neg 79.4%

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

      3. unsub-neg 79.4%

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

      4. unpow2 79.4%

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

      5. associate-/r* 88.6%

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

      6. div-sub 90.0%

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

      7. *-commutative 90.0%

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

      8. associate-/l* 87.6%

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

   5. Simplified 87.6%

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

      1. clear-num 87.5%

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

      2. un-div-inv 87.5%

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

   7. Applied egg-rr 87.5%

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

      1. associate-/r/ 90.1%

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

   9. Simplified 90.1%

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

   if 9.49999999999999989e55 < c

   1. Initial program 30.6%

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

   3. Taylor expanded in c around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

   5. Simplified 74.9%

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

      1. frac-2neg 74.9%

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

      2. div-inv 74.7%

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

      3. sub-neg 74.7%

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

      4. distribute-neg-in 74.7%

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

      5. distribute-neg-frac2 74.7%

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

      6. distribute-frac-neg 74.7%

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

      7. frac-2neg 74.7%

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

      8. associate-/l* 84.4%

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

   7. Applied egg-rr 84.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{a}{c} - b\right) \cdot \frac{-1}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{a}{c} - b\right) \cdot \frac{-1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.5e-31)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 2.8e+87)
     (/ (- (* b (/ c d)) a) d)
     (* (- (* d (/ a c)) b) (/ -1.0 c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.5e-31) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 2.8e+87) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = ((d * (a / c)) - b) * (-1.0 / 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 <= (-3.5d-31)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 2.8d+87) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = ((d * (a / c)) - b) * ((-1.0d0) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.5e-31) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 2.8e+87) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = ((d * (a / c)) - b) * (-1.0 / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.5e-31:
		tmp = (b - (a * (d / c))) / c
	elif c <= 2.8e+87:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = ((d * (a / c)) - b) * (-1.0 / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.5e-31)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 2.8e+87)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(Float64(d * Float64(a / c)) - b) * Float64(-1.0 / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.5e-31)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 2.8e+87)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = ((d * (a / c)) - b) * (-1.0 / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.5e-31], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.8e+87], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.49999999999999985e-31

   1. Initial program 59.4%

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

      1. *-un-lft-identity 59.4%

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

      2. add-sqr-sqrt 59.4%

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

      3. times-frac 59.4%

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

      4. hypot-define 59.4%

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

      5. hypot-define 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in b around inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. unsub-neg 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-/l* 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in c around inf 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

      3. associate-/l* 81.4%

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

   10. Simplified 81.4%

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

   if -3.49999999999999985e-31 < c < 2.80000000000000015e87

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0 68.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 76.2%

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

      6. div-sub 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. clear-num 76.4%

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

      2. un-div-inv 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/r/ 77.8%

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

   9. Simplified 77.8%

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

   if 2.80000000000000015e87 < c

   1. Initial program 32.9%

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

   3. Taylor expanded in c around inf 79.0%

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

      1. mul-1-neg 79.0%

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

      2. unsub-neg 79.0%

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

      3. *-commutative 79.0%

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

   5. Simplified 79.0%

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

      1. frac-2neg 79.0%

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

      2. div-inv 78.7%

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

      3. sub-neg 78.7%

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

      4. distribute-neg-in 78.7%

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

      5. distribute-neg-frac2 78.7%

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

      6. distribute-frac-neg 78.7%

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

      7. frac-2neg 78.7%

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

      8. associate-/l* 89.3%

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

   7. Applied egg-rr 89.3%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\left(d \cdot \frac{a}{c} - b\right) \cdot \frac{-1}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \left(d \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.9e-27)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 3.4e+87)
     (/ (- (* b (/ c d)) a) d)
     (/ (+ b (* a (* d (/ -1.0 c)))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.9e-27) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 3.4e+87) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b + (a * (d * (-1.0 / c)))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-5.9d-27)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 3.4d+87) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b + (a * (d * ((-1.0d0) / c)))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.9e-27) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 3.4e+87) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b + (a * (d * (-1.0 / c)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.9e-27:
		tmp = (b - (a * (d / c))) / c
	elif c <= 3.4e+87:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b + (a * (d * (-1.0 / c)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.9e-27)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 3.4e+87)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(d * Float64(-1.0 / c)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.9e-27)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 3.4e+87)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b + (a * (d * (-1.0 / c)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.9e-27], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.4e+87], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b + N[(a * N[(d * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.8999999999999998e-27

   1. Initial program 59.4%

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

      1. *-un-lft-identity 59.4%

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

      2. add-sqr-sqrt 59.4%

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

      3. times-frac 59.4%

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

      4. hypot-define 59.4%

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

      5. hypot-define 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in b around inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. unsub-neg 67.8%

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

      3. *-commutative 67.8%

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

      4. associate-/l* 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in c around inf 79.9%

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

      1. mul-1-neg 79.9%

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

      2. unsub-neg 79.9%

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

      3. associate-/l* 81.4%

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

   10. Simplified 81.4%

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

   if -5.8999999999999998e-27 < c < 3.4000000000000002e87

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0 68.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 76.2%

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

      6. div-sub 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. clear-num 76.4%

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

      2. un-div-inv 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/r/ 77.8%

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

   9. Simplified 77.8%

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

   if 3.4000000000000002e87 < c

   1. Initial program 32.9%

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

   3. Taylor expanded in c around inf 79.0%

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

      1. mul-1-neg 79.0%

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

      2. unsub-neg 79.0%

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

      3. *-commutative 79.0%

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

   5. Simplified 79.0%

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

      1. div-inv 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-*l* 88.2%

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

   7. Applied egg-rr 88.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \left(d \cdot \frac{-1}{c}\right)}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-33} \lor \neg \left(c \leq 1.18 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.15e-33) (not (<= c 1.18e+86)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.15e-33) || !(c <= 1.18e+86)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.15d-33)) .or. (.not. (c <= 1.18d+86))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.15e-33) || !(c <= 1.18e+86)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.15e-33) or not (c <= 1.18e+86):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.15e-33) || !(c <= 1.18e+86))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.15e-33) || ~((c <= 1.18e+86)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.15e-33], N[Not[LessEqual[c, 1.18e+86]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.14999999999999993e-33 or 1.18e86 < c

   1. Initial program 48.6%

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

      1. *-un-lft-identity 48.6%

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

      2. add-sqr-sqrt 48.6%

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

      3. times-frac 48.6%

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

      4. hypot-define 48.6%

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

      5. hypot-define 69.7%

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in b around inf 61.9%

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

      1. mul-1-neg 61.9%

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

      2. unsub-neg 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-/l* 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in c around inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. unsub-neg 79.5%

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

      3. associate-/l* 84.1%

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

   10. Simplified 84.1%

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

   if -1.14999999999999993e-33 < c < 1.18e86

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0 68.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 76.2%

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

      6. div-sub 77.1%

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

      7. *-commutative 77.1%

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

      8. associate-/l* 76.4%

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

   5. Simplified 76.4%

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

      1. clear-num 76.4%

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

      2. un-div-inv 76.5%

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

   7. Applied egg-rr 76.5%

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

      1. associate-/r/ 77.8%

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

   9. Simplified 77.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-33} \lor \neg \left(c \leq 1.18 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-20} \lor \neg \left(d \leq 2.3 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.7e-20) (not (<= d 2.3e-50)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.7e-20) || !(d <= 2.3e-50)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.7d-20)) .or. (.not. (d <= 2.3d-50))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.7e-20) || !(d <= 2.3e-50)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.7e-20) or not (d <= 2.3e-50):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.7e-20) || !(d <= 2.3e-50))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.7e-20) || ~((d <= 2.3e-50)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.7e-20], N[Not[LessEqual[d, 2.3e-50]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.7e-20 or 2.3000000000000002e-50 < d

   1. Initial program 49.7%

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

   3. Taylor expanded in c around 0 69.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 69.3%

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

      2. mul-1-neg 69.3%

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

      3. unsub-neg 69.3%

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

      4. unpow2 69.3%

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

      5. associate-/r* 73.0%

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

      6. div-sub 73.0%

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

      7. *-commutative 73.0%

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

      8. associate-/l* 75.4%

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

   5. Simplified 75.4%

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

   if -2.7e-20 < d < 2.3000000000000002e-50

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.7%

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

      3. times-frac 72.6%

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

      4. hypot-define 72.7%

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

      5. hypot-define 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in b around inf 84.9%

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

      1. mul-1-neg 84.9%

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

      2. unsub-neg 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-/l* 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in c around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

      3. associate-/l* 85.4%

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

   10. Simplified 85.4%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{-20} \lor \neg \left(d \leq 2.3 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{-20} \lor \neg \left(d \leq 1.45 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3e-20) (not (<= d 1.45e-53)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e-20) || !(d <= 1.45e-53)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-3d-20)) .or. (.not. (d <= 1.45d-53))) then
        tmp = a / -d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e-20) || !(d <= 1.45e-53)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3e-20) or not (d <= 1.45e-53):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3e-20) || !(d <= 1.45e-53))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3e-20) || ~((d <= 1.45e-53)))
		tmp = a / -d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3e-20], N[Not[LessEqual[d, 1.45e-53]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.00000000000000029e-20 or 1.4499999999999999e-53 < d

   1. Initial program 49.7%

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

   3. Taylor expanded in c around 0 65.0%

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

      1. associate-*r/ 65.0%

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

      2. neg-mul-1 65.0%

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

   5. Simplified 65.0%

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

   if -3.00000000000000029e-20 < d < 1.4499999999999999e-53

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.7%

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

      3. times-frac 72.6%

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

      4. hypot-define 72.7%

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

      5. hypot-define 88.0%

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

   4. Applied egg-rr 88.0%

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

   5. Taylor expanded in b around inf 84.9%

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

      1. mul-1-neg 84.9%

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

      2. unsub-neg 84.9%

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

      3. *-commutative 84.9%

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

      4. associate-/l* 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in c around inf 85.2%

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

      1. mul-1-neg 85.2%

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

      2. unsub-neg 85.2%

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

      3. associate-/l* 85.4%

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

   10. Simplified 85.4%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{-20} \lor \neg \left(d \leq 1.45 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-32} \lor \neg \left(c \leq 7 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.9e-32) (not (<= c 7e+45))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-32) || !(c <= 7e+45)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.9d-32)) .or. (.not. (c <= 7d+45))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-32) || !(c <= 7e+45)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.9e-32) or not (c <= 7e+45):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.9e-32) || !(c <= 7e+45))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.9e-32) || ~((c <= 7e+45)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.9e-32], N[Not[LessEqual[c, 7e+45]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.90000000000000004e-32 or 7.00000000000000046e45 < c

   1. Initial program 48.0%

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

   3. Taylor expanded in c around inf 68.1%

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

   if -1.90000000000000004e-32 < c < 7.00000000000000046e45

   1. Initial program 72.5%

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

   3. Taylor expanded in c around 0 62.6%

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

      1. associate-*r/ 62.6%

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

      2. neg-mul-1 62.6%

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

   5. Simplified 62.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-32} \lor \neg \left(c \leq 7 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 15.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+124} \lor \neg \left(d \leq 6 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.7e+124) (not (<= d 6e+107))) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e+124) || !(d <= 6e+107)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.7d+124)) .or. (.not. (d <= 6d+107))) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e+124) || !(d <= 6e+107)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.7e+124) or not (d <= 6e+107):
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.7e+124) || !(d <= 6e+107))
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.7e+124) || ~((d <= 6e+107)))
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.7e+124], N[Not[LessEqual[d, 6e+107]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.70000000000000021e124 or 6.00000000000000046e107 < d

   1. Initial program 32.1%

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

      1. *-un-lft-identity 32.1%

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

      2. add-sqr-sqrt 32.1%

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

      3. times-frac 32.1%

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

      4. hypot-define 32.1%

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

      5. hypot-define 55.6%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in c around 0 52.5%

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

      1. neg-mul-1 52.5%

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

   7. Simplified 52.5%

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

   8. Taylor expanded in d around -inf 24.4%

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

   if -5.70000000000000021e124 < d < 6.00000000000000046e107

   1. Initial program 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. add-sqr-sqrt 73.6%

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

      3. times-frac 73.6%

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

      4. hypot-define 73.6%

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

      5. hypot-define 86.0%

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

   4. Applied egg-rr 86.0%

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

   5. Taylor expanded in c around 0 20.8%

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

      1. neg-mul-1 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in c around -inf 11.4%

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

4. Final simplification 15.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+124} \lor \neg \left(d \leq 6 \cdot 10^{+107}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 3.2 \cdot 10^{+210}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 3.2e+210) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 3.2e+210) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 3.2d+210) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 3.2e+210) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 3.2e+210:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 3.2e+210)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 3.2e+210)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 3.2e+210], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3.2000000000000002e210

   1. Initial program 63.3%

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

   3. Taylor expanded in c around inf 44.8%

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

   if 3.2000000000000002e210 < d

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-define 36.4%

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

      5. hypot-define 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in c around 0 86.6%

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

      1. neg-mul-1 86.6%

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

   7. Simplified 86.6%

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

   8. Taylor expanded in d around -inf 38.1%

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

Alternative 12: 9.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.1%

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

   1. *-un-lft-identity 61.1%

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

   2. add-sqr-sqrt 61.1%

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

   3. times-frac 61.1%

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

   4. hypot-define 61.1%

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

   5. hypot-define 76.8%

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

4. Applied egg-rr 76.8%

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

5. Taylor expanded in c around 0 30.4%

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

   1. neg-mul-1 30.4%

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

7. Simplified 30.4%

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

8. Taylor expanded in c around -inf 9.9%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
9. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (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(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (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[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024149 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))