Complex division, imag part

Percentage Accurate: 61.7% → 91.7%

Time: 8.7s

Alternatives: 12

Speedup: 1.8×

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: 91.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{if}\;d \leq -1.65 \cdot 10^{+159}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-165}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{d \cdot a}{c} - b\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (/ c (hypot c d))
          (/ b (hypot c d))
          (/ (- a) (/ (pow (hypot c d) 2.0) d)))))
   (if (<= d -1.65e+159)
     (- (* (/ c d) (/ b d)) (/ a d))
     (if (<= d -2.8e-136)
       t_0
       (if (<= d 1.75e-165)
         (* (/ -1.0 c) (- (/ (* d a) c) b))
         (if (<= d 7.2e+149)
           t_0
           (* (/ 1.0 (hypot c d)) (- (/ c (/ d b)) a))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / (pow(hypot(c, d), 2.0) / d)));
	double tmp;
	if (d <= -1.65e+159) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else if (d <= -2.8e-136) {
		tmp = t_0;
	} else if (d <= 1.75e-165) {
		tmp = (-1.0 / c) * (((d * a) / c) - b);
	} else if (d <= 7.2e+149) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * ((c / (d / b)) - a);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)))
	tmp = 0.0
	if (d <= -1.65e+159)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	elseif (d <= -2.8e-136)
		tmp = t_0;
	elseif (d <= 1.75e-165)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(Float64(d * a) / c) - b));
	elseif (d <= 7.2e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(c / Float64(d / b)) - a));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.65e+159], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.8e-136], t$95$0, If[LessEqual[d, 1.75e-165], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.2e+149], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.6499999999999999e159

   1. Initial program 29.9%

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

   2. Taylor expanded in c around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. mul-1-neg 76.2%

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

      3. unsub-neg 76.2%

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

      4. unpow2 76.2%

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

      5. times-frac 80.5%

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

   4. Simplified 80.5%

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

   if -1.6499999999999999e159 < d < -2.8000000000000001e-136 or 1.7500000000000001e-165 < d < 7.1999999999999999e149

   1. Initial program 72.4%

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

      1. div-sub 72.5%

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

      2. *-commutative 72.5%

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

      3. add-sqr-sqrt 72.5%

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

      4. times-frac 74.6%

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

      5. fma-neg 74.6%

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

      6. hypot-def 74.6%

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

      7. hypot-def 90.1%

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

      8. associate-/l* 91.8%

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

      9. add-sqr-sqrt 91.8%

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

      10. pow2 91.8%

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

      11. hypot-def 91.8%

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

   3. Applied egg-rr 91.8%

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

   if -2.8000000000000001e-136 < d < 1.7500000000000001e-165

   1. Initial program 62.3%

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

      1. *-un-lft-identity 62.3%

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

      2. add-sqr-sqrt 62.3%

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

      3. times-frac 62.4%

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

      4. hypot-def 62.4%

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

      5. hypot-def 78.8%

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

   3. Applied egg-rr 78.8%

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

   4. Taylor expanded in c around -inf 50.4%

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

   5. Taylor expanded in c around -inf 94.0%

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

   if 7.1999999999999999e149 < d

   1. Initial program 41.0%

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

      1. *-un-lft-identity 41.0%

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

      2. add-sqr-sqrt 41.0%

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

      3. times-frac 41.0%

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

      4. hypot-def 41.0%

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

      5. hypot-def 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in c around 0 92.3%

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

      1. mul-1-neg 92.3%

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

      2. unsub-neg 92.3%

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

      3. associate-/l* 95.8%

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

   6. Simplified 95.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.65 \cdot 10^{+159}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-165}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{d \cdot a}{c} - b\right)\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \]

Alternative 2: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 2e+303)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (- (/ b c) (* (/ a c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+303) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+303) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 2e+303:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+303)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+303)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+303], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $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))) < 2e303

   1. Initial program 81.1%

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

      1. *-un-lft-identity 81.1%

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

      2. add-sqr-sqrt 81.1%

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

      3. times-frac 81.1%

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

      4. hypot-def 81.1%

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

      5. hypot-def 97.5%

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

   3. Applied egg-rr 97.5%

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

   if 2e303 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 7.5%

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

   2. Taylor expanded in c around inf 48.5%

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

      1. +-commutative 48.5%

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

      2. mul-1-neg 48.5%

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

      3. unsub-neg 48.5%

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

      4. unpow2 48.5%

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

      5. times-frac 57.0%

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

   4. Simplified 57.0%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 3: 78.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{d \cdot \frac{a}{c} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.8e+76)
   (/ (- (* d (/ a c)) b) (hypot c d))
   (if (<= c -9.5e-66)
     (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
     (if (<= c 1.85e+55)
       (- (* (/ c d) (/ b d)) (/ a d))
       (- (/ b c) (* (/ a c) (/ d c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.8e+76) {
		tmp = ((d * (a / c)) - b) / hypot(c, d);
	} else if (c <= -9.5e-66) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 1.85e+55) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.8e+76) {
		tmp = ((d * (a / c)) - b) / Math.hypot(c, d);
	} else if (c <= -9.5e-66) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 1.85e+55) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.8e+76:
		tmp = ((d * (a / c)) - b) / math.hypot(c, d)
	elif c <= -9.5e-66:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif c <= 1.85e+55:
		tmp = ((c / d) * (b / d)) - (a / d)
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.8e+76)
		tmp = Float64(Float64(Float64(d * Float64(a / c)) - b) / hypot(c, d));
	elseif (c <= -9.5e-66)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.85e+55)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.8e+76)
		tmp = ((d * (a / c)) - b) / hypot(c, d);
	elseif (c <= -9.5e-66)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (c <= 1.85e+55)
		tmp = ((c / d) * (b / d)) - (a / d);
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.8e+76], N[(N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -9.5e-66], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.85e+55], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.8e76

   1. Initial program 38.8%

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

      1. *-un-lft-identity 38.8%

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

      2. add-sqr-sqrt 38.8%

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

      3. times-frac 38.8%

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

      4. hypot-def 38.8%

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

      5. hypot-def 61.3%

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

   3. Applied egg-rr 61.3%

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

   4. Taylor expanded in c around -inf 85.9%

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

      1. distribute-lft-in 85.9%

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

      2. associate-/l* 88.3%

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

      3. mul-1-neg 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. distribute-lft-out 88.3%

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

      2. associate-*l/ 88.6%

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

      3. *-lft-identity 88.6%

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

      4. unsub-neg 88.6%

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

      5. associate-/r/ 90.7%

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

      6. *-commutative 90.7%

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

   8. Simplified 90.7%

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

   if -4.8e76 < c < -9.5000000000000004e-66

   1. Initial program 94.8%

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

   if -9.5000000000000004e-66 < c < 1.8500000000000001e55

   1. Initial program 67.2%

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

   2. Taylor expanded in c around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

   if 1.8500000000000001e55 < c

   1. Initial program 51.3%

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

   2. Taylor expanded in c around inf 82.1%

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

      1. +-commutative 82.1%

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

      2. mul-1-neg 82.1%

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

      3. unsub-neg 82.1%

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

      4. unpow2 82.1%

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

      5. times-frac 88.1%

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

   4. Simplified 88.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{d \cdot \frac{a}{c} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+55}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 4: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -3.9 \cdot 10^{+121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.66 \cdot 10^{-65}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ b c) (* (/ a c) (/ d c)))))
   (if (<= c -3.9e+121)
     t_0
     (if (<= c -1.66e-65)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= c 6.8e+53) (- (* (/ c d) (/ b d)) (/ a d)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -3.9e+121) {
		tmp = t_0;
	} else if (c <= -1.66e-65) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 6.8e+53) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b / c) - ((a / c) * (d / c))
    if (c <= (-3.9d+121)) then
        tmp = t_0
    else if (c <= (-1.66d-65)) then
        tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
    else if (c <= 6.8d+53) then
        tmp = ((c / d) * (b / d)) - (a / d)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -3.9e+121) {
		tmp = t_0;
	} else if (c <= -1.66e-65) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 6.8e+53) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / c) - ((a / c) * (d / c))
	tmp = 0
	if c <= -3.9e+121:
		tmp = t_0
	elif c <= -1.66e-65:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif c <= 6.8e+53:
		tmp = ((c / d) * (b / d)) - (a / d)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -3.9e+121)
		tmp = t_0;
	elseif (c <= -1.66e-65)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 6.8e+53)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / c) - ((a / c) * (d / c));
	tmp = 0.0;
	if (c <= -3.9e+121)
		tmp = t_0;
	elseif (c <= -1.66e-65)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (c <= 6.8e+53)
		tmp = ((c / d) * (b / d)) - (a / d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.9e+121], t$95$0, If[LessEqual[c, -1.66e-65], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.8e+53], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.89999999999999984e121 or 6.79999999999999995e53 < c

   1. Initial program 45.0%

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

   2. Taylor expanded in c around inf 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 88.5%

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

   4. Simplified 88.5%

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

   if -3.89999999999999984e121 < c < -1.6599999999999999e-65

   1. Initial program 92.2%

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

   if -1.6599999999999999e-65 < c < 6.79999999999999995e53

   1. Initial program 67.2%

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

   2. Taylor expanded in c around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -1.66 \cdot 10^{-65}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 5: 75.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{\frac{c \cdot c + d \cdot d}{b}}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ b c) (* (/ a c) (/ d c)))))
   (if (<= c -6.8e+75)
     t_0
     (if (<= c -1.7e-64)
       (/ c (/ (+ (* c c) (* d d)) b))
       (if (<= c 3.9e+53) (- (* (/ c d) (/ b d)) (/ a d)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -6.8e+75) {
		tmp = t_0;
	} else if (c <= -1.7e-64) {
		tmp = c / (((c * c) + (d * d)) / b);
	} else if (c <= 3.9e+53) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b / c) - ((a / c) * (d / c))
    if (c <= (-6.8d+75)) then
        tmp = t_0
    else if (c <= (-1.7d-64)) then
        tmp = c / (((c * c) + (d * d)) / b)
    else if (c <= 3.9d+53) then
        tmp = ((c / d) * (b / d)) - (a / d)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -6.8e+75) {
		tmp = t_0;
	} else if (c <= -1.7e-64) {
		tmp = c / (((c * c) + (d * d)) / b);
	} else if (c <= 3.9e+53) {
		tmp = ((c / d) * (b / d)) - (a / d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / c) - ((a / c) * (d / c))
	tmp = 0
	if c <= -6.8e+75:
		tmp = t_0
	elif c <= -1.7e-64:
		tmp = c / (((c * c) + (d * d)) / b)
	elif c <= 3.9e+53:
		tmp = ((c / d) * (b / d)) - (a / d)
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -6.8e+75)
		tmp = t_0;
	elseif (c <= -1.7e-64)
		tmp = Float64(c / Float64(Float64(Float64(c * c) + Float64(d * d)) / b));
	elseif (c <= 3.9e+53)
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / c) - ((a / c) * (d / c));
	tmp = 0.0;
	if (c <= -6.8e+75)
		tmp = t_0;
	elseif (c <= -1.7e-64)
		tmp = c / (((c * c) + (d * d)) / b);
	elseif (c <= 3.9e+53)
		tmp = ((c / d) * (b / d)) - (a / d);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.8e+75], t$95$0, If[LessEqual[c, -1.7e-64], N[(c / N[(N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.9e+53], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.80000000000000022e75 or 3.89999999999999976e53 < c

   1. Initial program 46.6%

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

   2. Taylor expanded in c around inf 80.5%

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

      1. +-commutative 80.5%

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

      2. mul-1-neg 80.5%

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

      3. unsub-neg 80.5%

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

      4. unpow2 80.5%

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

      5. times-frac 88.1%

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

   4. Simplified 88.1%

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

   if -6.80000000000000022e75 < c < -1.70000000000000006e-64

   1. Initial program 94.8%

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

      1. *-un-lft-identity 94.8%

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

      2. add-sqr-sqrt 94.8%

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

      3. times-frac 95.0%

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

      4. hypot-def 95.0%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

   4. Taylor expanded in b around inf 85.5%

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

      1. associate-/l* 85.7%

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

      2. unpow2 85.7%

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

      3. unpow2 85.7%

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

   6. Simplified 85.7%

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

   if -1.70000000000000006e-64 < c < 3.89999999999999976e53

   1. Initial program 67.2%

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

   2. Taylor expanded in c around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{\frac{c \cdot c + d \cdot d}{b}}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+53}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 6: 71.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+41} \lor \neg \left(c \leq 4.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.5e+41) (not (<= c 4.8e+55)))
   (/ b c)
   (- (* (/ c d) (/ b d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.5e+41) || !(c <= 4.8e+55)) {
		tmp = b / c;
	} else {
		tmp = ((c / d) * (b / 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 <= (-5.5d+41)) .or. (.not. (c <= 4.8d+55))) then
        tmp = b / c
    else
        tmp = ((c / d) * (b / 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 <= -5.5e+41) || !(c <= 4.8e+55)) {
		tmp = b / c;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.5e+41) or not (c <= 4.8e+55):
		tmp = b / c
	else:
		tmp = ((c / d) * (b / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.5e+41) || !(c <= 4.8e+55))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.5e+41) || ~((c <= 4.8e+55)))
		tmp = b / c;
	else
		tmp = ((c / d) * (b / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.5e+41], N[Not[LessEqual[c, 4.8e+55]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.5000000000000003e41 or 4.7999999999999998e55 < c

   1. Initial program 48.8%

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

   2. Taylor expanded in c around inf 77.4%

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

   if -5.5000000000000003e41 < c < 4.7999999999999998e55

   1. Initial program 70.5%

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

   2. Taylor expanded in c around 0 78.7%

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

      1. +-commutative 78.7%

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

      2. mul-1-neg 78.7%

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

      3. unsub-neg 78.7%

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

      4. unpow2 78.7%

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

      5. times-frac 80.4%

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

   4. Simplified 80.4%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+41} \lor \neg \left(c \leq 4.8 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 7: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-60} \lor \neg \left(c \leq 2.25 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.1e-60) (not (<= c 2.25e+54)))
   (- (/ b c) (* (/ a c) (/ d c)))
   (- (* (/ c d) (/ b d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.1e-60) || !(c <= 2.25e+54)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = ((c / d) * (b / 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 <= (-1.1d-60)) .or. (.not. (c <= 2.25d+54))) then
        tmp = (b / c) - ((a / c) * (d / c))
    else
        tmp = ((c / d) * (b / 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 <= -1.1e-60) || !(c <= 2.25e+54)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.1e-60) or not (c <= 2.25e+54):
		tmp = (b / c) - ((a / c) * (d / c))
	else:
		tmp = ((c / d) * (b / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.1e-60) || !(c <= 2.25e+54))
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.1e-60) || ~((c <= 2.25e+54)))
		tmp = (b / c) - ((a / c) * (d / c));
	else
		tmp = ((c / d) * (b / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.1e-60], N[Not[LessEqual[c, 2.25e+54]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.0999999999999999e-60 or 2.24999999999999992e54 < c

   1. Initial program 54.4%

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

   2. Taylor expanded in c around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. unsub-neg 76.3%

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

      4. unpow2 76.3%

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

      5. times-frac 82.6%

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

   4. Simplified 82.6%

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

   if -1.0999999999999999e-60 < c < 2.24999999999999992e54

   1. Initial program 67.2%

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

   2. Taylor expanded in c around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. mul-1-neg 83.0%

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

      3. unsub-neg 83.0%

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

      4. unpow2 83.0%

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

      5. times-frac 84.8%

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

   4. Simplified 84.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-60} \lor \neg \left(c \leq 2.25 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 8: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -7.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 10^{-123}:\\ \;\;\;\;\frac{c \cdot b}{d \cdot d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= c -7.8e-61)
     (/ b c)
     (if (<= c 9.2e-150)
       t_0
       (if (<= c 1e-123)
         (/ (* c b) (* d d))
         (if (<= c 5.8e+55) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -7.8e-61) {
		tmp = b / c;
	} else if (c <= 9.2e-150) {
		tmp = t_0;
	} else if (c <= 1e-123) {
		tmp = (c * b) / (d * d);
	} else if (c <= 5.8e+55) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (c <= (-7.8d-61)) then
        tmp = b / c
    else if (c <= 9.2d-150) then
        tmp = t_0
    else if (c <= 1d-123) then
        tmp = (c * b) / (d * d)
    else if (c <= 5.8d+55) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -7.8e-61) {
		tmp = b / c;
	} else if (c <= 9.2e-150) {
		tmp = t_0;
	} else if (c <= 1e-123) {
		tmp = (c * b) / (d * d);
	} else if (c <= 5.8e+55) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if c <= -7.8e-61:
		tmp = b / c
	elif c <= 9.2e-150:
		tmp = t_0
	elif c <= 1e-123:
		tmp = (c * b) / (d * d)
	elif c <= 5.8e+55:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (c <= -7.8e-61)
		tmp = Float64(b / c);
	elseif (c <= 9.2e-150)
		tmp = t_0;
	elseif (c <= 1e-123)
		tmp = Float64(Float64(c * b) / Float64(d * d));
	elseif (c <= 5.8e+55)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (c <= -7.8e-61)
		tmp = b / c;
	elseif (c <= 9.2e-150)
		tmp = t_0;
	elseif (c <= 1e-123)
		tmp = (c * b) / (d * d);
	elseif (c <= 5.8e+55)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[c, -7.8e-61], N[(b / c), $MachinePrecision], If[LessEqual[c, 9.2e-150], t$95$0, If[LessEqual[c, 1e-123], N[(N[(c * b), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.8e+55], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.80000000000000065e-61 or 5.7999999999999997e55 < c

   1. Initial program 54.4%

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

   2. Taylor expanded in c around inf 72.7%

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

   if -7.80000000000000065e-61 < c < 9.20000000000000011e-150 or 1.0000000000000001e-123 < c < 5.7999999999999997e55

   1. Initial program 65.2%

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

   2. Taylor expanded in c around 0 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

   4. Simplified 74.9%

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

   if 9.20000000000000011e-150 < c < 1.0000000000000001e-123

   1. Initial program 89.5%

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

   2. Taylor expanded in b around inf 70.8%

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

      1. unpow2 70.8%

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

      2. unpow2 70.8%

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

   4. Simplified 70.8%

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

      1. *-un-lft-identity 70.8%

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

      2. times-frac 42.6%

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

      3. pow1 42.6%

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

      4. metadata-eval 42.6%

         \[\leadsto \frac{c}{1} \cdot \frac{b}{{\left(d \cdot d + c \cdot c\right)}^{\color{blue}{\left(\frac{2}{2}\right)}}} \]

      5. sqrt-pow2 42.6%

         \[\leadsto \frac{c}{1} \cdot \frac{b}{\color{blue}{{\left(\sqrt{d \cdot d + c \cdot c}\right)}^{2}}} \]

      6. +-commutative 42.6%

         \[\leadsto \frac{c}{1} \cdot \frac{b}{{\left(\sqrt{\color{blue}{c \cdot c + d \cdot d}}\right)}^{2}} \]

      7. hypot-udef 42.6%

         \[\leadsto \frac{c}{1} \cdot \frac{b}{{\color{blue}{\left(\mathsf{hypot}\left(c, d\right)\right)}}^{2}} \]

   6. Applied egg-rr 42.6%

      \[\leadsto \color{blue}{\frac{c}{1} \cdot \frac{b}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]

   7. Taylor expanded in c around 0 70.7%

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

      1. unpow2 70.7%

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

   9. Simplified 70.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 10^{-123}:\\ \;\;\;\;\frac{c \cdot b}{d \cdot d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -9.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= c -9.4e-61)
     (/ b c)
     (if (<= c 1.95e-150)
       t_0
       (if (<= c 1.02e-123)
         (/ (* (/ c d) b) d)
         (if (<= c 2.2e+54) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -9.4e-61) {
		tmp = b / c;
	} else if (c <= 1.95e-150) {
		tmp = t_0;
	} else if (c <= 1.02e-123) {
		tmp = ((c / d) * b) / d;
	} else if (c <= 2.2e+54) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (c <= (-9.4d-61)) then
        tmp = b / c
    else if (c <= 1.95d-150) then
        tmp = t_0
    else if (c <= 1.02d-123) then
        tmp = ((c / d) * b) / d
    else if (c <= 2.2d+54) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -9.4e-61) {
		tmp = b / c;
	} else if (c <= 1.95e-150) {
		tmp = t_0;
	} else if (c <= 1.02e-123) {
		tmp = ((c / d) * b) / d;
	} else if (c <= 2.2e+54) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if c <= -9.4e-61:
		tmp = b / c
	elif c <= 1.95e-150:
		tmp = t_0
	elif c <= 1.02e-123:
		tmp = ((c / d) * b) / d
	elif c <= 2.2e+54:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (c <= -9.4e-61)
		tmp = Float64(b / c);
	elseif (c <= 1.95e-150)
		tmp = t_0;
	elseif (c <= 1.02e-123)
		tmp = Float64(Float64(Float64(c / d) * b) / d);
	elseif (c <= 2.2e+54)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (c <= -9.4e-61)
		tmp = b / c;
	elseif (c <= 1.95e-150)
		tmp = t_0;
	elseif (c <= 1.02e-123)
		tmp = ((c / d) * b) / d;
	elseif (c <= 2.2e+54)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[c, -9.4e-61], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.95e-150], t$95$0, If[LessEqual[c, 1.02e-123], N[(N[(N[(c / d), $MachinePrecision] * b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.2e+54], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.3999999999999993e-61 or 2.1999999999999999e54 < c

   1. Initial program 54.4%

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

   2. Taylor expanded in c around inf 72.7%

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

   if -9.3999999999999993e-61 < c < 1.9500000000000001e-150 or 1.02e-123 < c < 2.1999999999999999e54

   1. Initial program 65.2%

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

   2. Taylor expanded in c around 0 74.9%

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

      1. associate-*r/ 74.9%

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

      2. neg-mul-1 74.9%

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

   4. Simplified 74.9%

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

   if 1.9500000000000001e-150 < c < 1.02e-123

   1. Initial program 89.5%

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

   2. Taylor expanded in b around inf 70.8%

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

      1. unpow2 70.8%

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

      2. unpow2 70.8%

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

   4. Simplified 70.8%

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

   5. Taylor expanded in c around 0 70.7%

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

      1. unpow2 70.7%

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

      2. times-frac 52.1%

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

   7. Simplified 52.1%

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

      1. associate-*r/ 70.7%

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

   9. Applied egg-rr 70.7%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.4 \cdot 10^{-61}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-150}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-123}:\\ \;\;\;\;\frac{\frac{c}{d} \cdot b}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 10: 63.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.95 \cdot 10^{-61} \lor \neg \left(c \leq 2.3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.95e-61) (not (<= c 2.3e+53))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.95e-61) || !(c <= 2.3e+53)) {
		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 <= (-2.95d-61)) .or. (.not. (c <= 2.3d+53))) 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 <= -2.95e-61) || !(c <= 2.3e+53)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.95e-61) or not (c <= 2.3e+53):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.95e-61) || !(c <= 2.3e+53))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.95e-61) || ~((c <= 2.3e+53)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.95e-61], N[Not[LessEqual[c, 2.3e+53]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.94999999999999986e-61 or 2.3000000000000002e53 < c

   1. Initial program 54.4%

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

   2. Taylor expanded in c around inf 72.7%

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

   if -2.94999999999999986e-61 < c < 2.3000000000000002e53

   1. Initial program 67.2%

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

   2. Taylor expanded in c around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   4. Simplified 70.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.95 \cdot 10^{-61} \lor \neg \left(c \leq 2.3 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 11: 9.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. *-un-lft-identity 60.4%

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

   2. add-sqr-sqrt 60.4%

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

   3. times-frac 60.4%

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

   4. hypot-def 60.4%

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

   5. hypot-def 73.6%

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

3. Applied egg-rr 73.6%

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

4. Taylor expanded in c around -inf 31.0%

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

5. Taylor expanded in c around 0 8.9%

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

6. Final simplification 8.9%

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

Alternative 12: 43.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

2. Taylor expanded in c around inf 45.4%

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

3. Final simplification 45.4%

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

Developer target: 99.3% 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 2023240 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (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))))