Complex division, imag part

Percentage Accurate: 61.2% → 95.6%

Time: 9.8s

Alternatives: 11

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: 61.2% 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: 95.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{-\mathsf{hypot}\left(d, c\right)}\right) \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	return fma((c / hypot(c, d)), (b / hypot(c, d)), (a * ((d / hypot(d, c)) / -hypot(d, c))));
}

Julia

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

Wolfram

code[a_, b_, c_, d_] := 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[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

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

   1. div-sub 58.4%

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

   2. *-commutative 58.4%

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

   3. add-sqr-sqrt 58.4%

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

   4. times-frac 61.5%

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

   5. fma-neg 61.5%

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

   6. hypot-define 61.5%

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

   7. hypot-define 78.2%

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

   8. associate-/l* 81.9%

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

   9. add-sqr-sqrt 81.9%

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

   10. pow2 81.9%

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

   11. hypot-define 81.9%

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

4. Applied egg-rr 81.9%

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

   1. *-un-lft-identity 81.9%

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

   2. unpow2 81.9%

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

   3. times-frac 96.7%

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

   4. hypot-undefine 81.9%

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

   5. +-commutative 81.9%

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

   6. hypot-define 96.7%

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

   7. hypot-undefine 81.9%

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

   8. +-commutative 81.9%

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

   9. hypot-define 96.7%

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

6. Applied egg-rr 96.7%

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

   1. associate-*l/ 96.7%

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

   2. *-lft-identity 96.7%

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

8. Simplified 96.7%

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

9. Final simplification 96.7%

   \[\leadsto \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{-\mathsf{hypot}\left(d, c\right)}\right) \]
10. Add Preprocessing

Alternative 2: 93.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ t_3 := \mathsf{fma}\left(t\_0, t\_1, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -7 \cdot 10^{+131}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 10^{-183}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d)))
        (t_1 (/ b (hypot c d)))
        (t_2 (fma t_0 t_1 (* a (/ d (- (pow (hypot c d) 2.0))))))
        (t_3 (fma t_0 t_1 (/ a (- d)))))
   (if (<= d -7e+131)
     t_3
     (if (<= d -9e-39)
       t_2
       (if (<= d 1e-183)
         (* (/ (- c (* a (/ d b))) (hypot d c)) (/ b (hypot d c)))
         (if (<= d 5.6e+147) t_2 t_3))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double t_2 = fma(t_0, t_1, (a * (d / -pow(hypot(c, d), 2.0))));
	double t_3 = fma(t_0, t_1, (a / -d));
	double tmp;
	if (d <= -7e+131) {
		tmp = t_3;
	} else if (d <= -9e-39) {
		tmp = t_2;
	} else if (d <= 1e-183) {
		tmp = ((c - (a * (d / b))) / hypot(d, c)) * (b / hypot(d, c));
	} else if (d <= 5.6e+147) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	t_2 = fma(t_0, t_1, Float64(a * Float64(d / Float64(-(hypot(c, d) ^ 2.0)))))
	t_3 = fma(t_0, t_1, Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -7e+131)
		tmp = t_3;
	elseif (d <= -9e-39)
		tmp = t_2;
	elseif (d <= 1e-183)
		tmp = Float64(Float64(Float64(c - Float64(a * Float64(d / b))) / hypot(d, c)) * Float64(b / hypot(d, c)));
	elseif (d <= 5.6e+147)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[(a * N[(d / (-N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1 + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7e+131], t$95$3, If[LessEqual[d, -9e-39], t$95$2, If[LessEqual[d, 1e-183], N[(N[(N[(c - N[(a * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.6e+147], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.9999999999999998e131 or 5.6000000000000002e147 < d

   1. Initial program 31.5%

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

      1. div-sub 31.5%

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

      2. *-commutative 31.5%

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

      3. add-sqr-sqrt 31.5%

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

      4. times-frac 31.8%

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

      5. fma-neg 31.8%

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

      6. hypot-define 31.8%

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

      7. hypot-define 49.6%

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

      8. associate-/l* 53.9%

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

      9. add-sqr-sqrt 53.9%

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

      10. pow2 53.9%

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

      11. hypot-define 53.9%

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

   4. Applied egg-rr 53.9%

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

   5. Taylor expanded in d around inf 93.9%

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

   if -6.9999999999999998e131 < d < -9.0000000000000002e-39 or 1.00000000000000001e-183 < d < 5.6000000000000002e147

   1. Initial program 72.4%

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

      1. div-sub 71.4%

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

      2. *-commutative 71.4%

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

      3. add-sqr-sqrt 71.4%

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

      4. times-frac 75.3%

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

      5. fma-neg 75.3%

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

      6. hypot-define 75.3%

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

      7. hypot-define 86.7%

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

      8. associate-/l* 92.9%

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

      9. add-sqr-sqrt 92.9%

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

      10. pow2 92.9%

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

      11. hypot-define 92.9%

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

   4. Applied egg-rr 92.9%

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

   if -9.0000000000000002e-39 < d < 1.00000000000000001e-183

   1. Initial program 64.9%

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

   3. Taylor expanded in b around inf 63.8%

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

      1. mul-1-neg 63.8%

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

      2. unsub-neg 63.8%

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

      3. associate-/l* 63.9%

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

   5. Simplified 63.9%

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

      1. *-commutative 63.9%

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

      2. +-commutative 63.9%

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

      3. add-sqr-sqrt 63.9%

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

      4. hypot-undefine 63.9%

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

      5. hypot-undefine 63.9%

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

      6. times-frac 95.6%

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

   7. Applied egg-rr 95.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{elif}\;d \leq 10^{-183}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;d \leq 5.6 \cdot 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{+57} \lor \neg \left(d \leq 8.5 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.12e+57) (not (<= d 8.5e+73)))
   (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ a (- d)))
   (* (/ (- c (* a (/ d b))) (hypot d c)) (/ b (hypot d c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.12e+57) || !(d <= 8.5e+73)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a / -d));
	} else {
		tmp = ((c - (a * (d / b))) / hypot(d, c)) * (b / hypot(d, c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.12e+57) || !(d <= 8.5e+73))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a / Float64(-d)));
	else
		tmp = Float64(Float64(Float64(c - Float64(a * Float64(d / b))) / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.12e+57], N[Not[LessEqual[d, 8.5e+73]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a / (-d)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c - N[(a * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.12000000000000003e57 or 8.4999999999999998e73 < d

   1. Initial program 40.4%

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

      1. div-sub 40.4%

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

      2. *-commutative 40.4%

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

      3. add-sqr-sqrt 40.4%

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

      4. times-frac 42.9%

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

      5. fma-neg 42.9%

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

      6. hypot-define 42.9%

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

      7. hypot-define 57.6%

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

      8. associate-/l* 66.7%

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

      9. add-sqr-sqrt 66.7%

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

      10. pow2 66.7%

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

      11. hypot-define 66.7%

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

   4. Applied egg-rr 66.7%

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

   5. Taylor expanded in d around inf 88.8%

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

   if -1.12000000000000003e57 < d < 8.4999999999999998e73

   1. Initial program 71.3%

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

   3. Taylor expanded in b around inf 67.2%

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

      1. mul-1-neg 67.2%

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

      2. unsub-neg 67.2%

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

      3. associate-/l* 67.1%

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

   5. Simplified 67.1%

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

      1. *-commutative 67.1%

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

      2. +-commutative 67.1%

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

      3. add-sqr-sqrt 67.1%

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

      4. hypot-undefine 67.1%

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

      5. hypot-undefine 67.1%

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

      6. times-frac 91.1%

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

   7. Applied egg-rr 91.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.12 \cdot 10^{+57} \lor \neg \left(d \leq 8.5 \cdot 10^{+73}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{if}\;c \leq -1.02 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-115}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 820000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+224}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ (- c (* a (/ d b))) (hypot d c)) (/ b (hypot d c)))))
   (if (<= c -1.02e-51)
     t_0
     (if (<= c 1.3e-210)
       (/ (- (* b (/ c d)) a) d)
       (if (<= c 3.4e-115)
         t_0
         (if (<= c 820000.0)
           (/ (fma b c (* d (- a))) (fma d d (* c c)))
           (if (<= c 2.4e+224) t_0 (/ (- b (* a (/ d c))) c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c - (a * (d / b))) / hypot(d, c)) * (b / hypot(d, c));
	double tmp;
	if (c <= -1.02e-51) {
		tmp = t_0;
	} else if (c <= 1.3e-210) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 3.4e-115) {
		tmp = t_0;
	} else if (c <= 820000.0) {
		tmp = fma(b, c, (d * -a)) / fma(d, d, (c * c));
	} else if (c <= 2.4e+224) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c - Float64(a * Float64(d / b))) / hypot(d, c)) * Float64(b / hypot(d, c)))
	tmp = 0.0
	if (c <= -1.02e-51)
		tmp = t_0;
	elseif (c <= 1.3e-210)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 3.4e-115)
		tmp = t_0;
	elseif (c <= 820000.0)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(d, d, Float64(c * c)));
	elseif (c <= 2.4e+224)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c - N[(a * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.02e-51], t$95$0, If[LessEqual[c, 1.3e-210], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3.4e-115], t$95$0, If[LessEqual[c, 820000.0], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.4e+224], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.01999999999999998e-51 or 1.2999999999999999e-210 < c < 3.3999999999999998e-115 or 8.2e5 < c < 2.40000000000000001e224

   1. Initial program 53.0%

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

   3. Taylor expanded in b around inf 50.7%

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

      1. mul-1-neg 50.7%

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

      2. unsub-neg 50.7%

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

      3. associate-/l* 50.7%

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

   5. Simplified 50.7%

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

      1. *-commutative 50.7%

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

      2. +-commutative 50.7%

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

      3. add-sqr-sqrt 50.7%

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

      4. hypot-undefine 50.7%

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

      5. hypot-undefine 50.7%

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

      6. times-frac 89.9%

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

   7. Applied egg-rr 89.9%

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

   if -1.01999999999999998e-51 < c < 1.2999999999999999e-210

   1. Initial program 69.9%

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

      1. div-sub 64.5%

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

      2. *-commutative 64.5%

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

      3. add-sqr-sqrt 64.5%

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

      4. times-frac 63.5%

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

      5. fma-neg 63.5%

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

      6. hypot-define 63.5%

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

      7. hypot-define 66.6%

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

      8. associate-/l* 76.6%

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

      9. add-sqr-sqrt 76.6%

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

      10. pow2 76.6%

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

      11. hypot-define 76.6%

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

   4. Applied egg-rr 76.6%

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

   5. Taylor expanded in c around 0 84.1%

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

      1. *-commutative 84.1%

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

      2. unpow2 84.1%

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

      3. associate-/l/ 91.8%

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

      4. div-sub 91.9%

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

   7. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. *-un-lft-identity 91.9%

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

      3. times-frac 91.9%

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

   9. Applied egg-rr 91.9%

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

   if 3.3999999999999998e-115 < c < 8.2e5

   1. Initial program 99.9%

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

      1. fma-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-define 100.0%

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

   3. Simplified 100.0%

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

   if 2.40000000000000001e224 < c

   1. Initial program 40.1%

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

   3. Taylor expanded in c around inf 86.7%

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

      1. remove-double-neg 86.7%

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

      2. mul-1-neg 86.7%

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

      3. neg-mul-1 86.7%

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

      4. distribute-lft-in 86.7%

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

      5. mul-1-neg 86.7%

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

      6. distribute-neg-in 86.7%

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

      7. mul-1-neg 86.7%

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

      8. remove-double-neg 86.7%

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

      9. unsub-neg 86.7%

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

      10. associate-/l* 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{-51}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-210}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 820000:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+224}:\\ \;\;\;\;\frac{c - a \cdot \frac{d}{b}}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.32 \cdot 10^{+100}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 460:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+79}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -1.32e+100)
     (/ (- (* c (/ b d)) a) d)
     (if (<= d -7.2e-44)
       t_0
       (if (<= d 460.0)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 2.2e+79) t_0 (/ (- (* b (/ c d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.32e+100) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -7.2e-44) {
		tmp = t_0;
	} else if (d <= 460.0) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 2.2e+79) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-1.32d+100)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= (-7.2d-44)) then
        tmp = t_0
    else if (d <= 460.0d0) then
        tmp = (b - (a * (d / c))) / c
    else if (d <= 2.2d+79) then
        tmp = t_0
    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 t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.32e+100) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -7.2e-44) {
		tmp = t_0;
	} else if (d <= 460.0) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 2.2e+79) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.32e+100:
		tmp = ((c * (b / d)) - a) / d
	elif d <= -7.2e-44:
		tmp = t_0
	elif d <= 460.0:
		tmp = (b - (a * (d / c))) / c
	elif d <= 2.2e+79:
		tmp = t_0
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.32e+100)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -7.2e-44)
		tmp = t_0;
	elseif (d <= 460.0)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 2.2e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.32e+100)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= -7.2e-44)
		tmp = t_0;
	elseif (d <= 460.0)
		tmp = (b - (a * (d / c))) / c;
	elseif (d <= 2.2e+79)
		tmp = t_0;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.32e+100], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -7.2e-44], t$95$0, If[LessEqual[d, 460.0], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.2e+79], t$95$0, N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.32e100

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. unpow2 71.8%

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

      5. associate-/r* 79.2%

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

      6. div-sub 79.2%

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

      7. *-commutative 79.2%

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

      8. associate-/l* 87.8%

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

   5. Simplified 87.8%

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

   if -1.32e100 < d < -7.1999999999999998e-44 or 460 < d < 2.1999999999999999e79

   1. Initial program 86.2%

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

   if -7.1999999999999998e-44 < d < 460

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf 91.5%

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

      1. remove-double-neg 91.5%

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

      2. mul-1-neg 91.5%

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

      3. neg-mul-1 91.5%

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

      4. distribute-lft-in 91.5%

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

      5. mul-1-neg 91.5%

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

      6. distribute-neg-in 91.5%

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

      7. mul-1-neg 91.5%

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

      8. remove-double-neg 91.5%

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

      9. unsub-neg 91.5%

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

      10. associate-/l* 92.3%

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

   5. Simplified 92.3%

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

   if 2.1999999999999999e79 < d

   1. Initial program 38.1%

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

      1. div-sub 38.1%

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

      2. *-commutative 38.1%

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

      3. add-sqr-sqrt 38.1%

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

      4. times-frac 40.4%

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

      5. fma-neg 40.4%

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

      6. hypot-define 40.4%

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

      7. hypot-define 51.2%

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

      8. associate-/l* 63.2%

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

      9. add-sqr-sqrt 63.2%

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

      10. pow2 63.2%

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

      11. hypot-define 63.2%

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

   4. Applied egg-rr 63.2%

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

   5. Taylor expanded in c around 0 70.1%

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

      1. *-commutative 70.1%

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

      2. unpow2 70.1%

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

      3. associate-/l/ 71.7%

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

      4. div-sub 71.7%

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

   7. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. *-un-lft-identity 71.7%

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

      3. times-frac 78.7%

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

   9. Applied egg-rr 78.7%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.32 \cdot 10^{+100}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 460:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{-51} \lor \neg \left(c \leq 1.1 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.6e-51) (not (<= c 1.1e-46)))
   (/ (- 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 <= -1.6e-51) || !(c <= 1.1e-46)) {
		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 <= (-1.6d-51)) .or. (.not. (c <= 1.1d-46))) 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 <= -1.6e-51) || !(c <= 1.1e-46)) {
		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 <= -1.6e-51) or not (c <= 1.1e-46):
		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 <= -1.6e-51) || !(c <= 1.1e-46))
		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 <= -1.6e-51) || ~((c <= 1.1e-46)))
		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, -1.6e-51], N[Not[LessEqual[c, 1.1e-46]], $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 < -1.6e-51 or 1.1e-46 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 80.5%

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

      1. remove-double-neg 80.5%

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

      2. mul-1-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. distribute-lft-in 80.5%

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

      5. mul-1-neg 80.5%

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

      6. distribute-neg-in 80.5%

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

      7. mul-1-neg 80.5%

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

      8. remove-double-neg 80.5%

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

      9. unsub-neg 80.5%

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

      10. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if -1.6e-51 < c < 1.1e-46

   1. Initial program 68.5%

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

      1. div-sub 64.1%

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

      2. *-commutative 64.1%

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

      3. add-sqr-sqrt 64.1%

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

      4. times-frac 63.3%

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

      5. fma-neg 63.3%

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

      6. hypot-define 63.4%

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

      7. hypot-define 66.6%

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

      8. associate-/l* 75.3%

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

      9. add-sqr-sqrt 75.3%

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

      10. pow2 75.3%

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

      11. hypot-define 75.3%

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

   4. Applied egg-rr 75.3%

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

   5. Taylor expanded in c around 0 77.3%

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

      1. *-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. associate-/l/ 83.5%

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

      4. div-sub 83.6%

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

   7. Simplified 83.6%

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

      1. *-commutative 83.6%

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

      2. *-un-lft-identity 83.6%

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

      3. times-frac 83.7%

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{-51} \lor \neg \left(c \leq 1.1 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{-51} \lor \neg \left(c \leq 7.8 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.2e-51) (not (<= c 7.8e-47)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ (* c b) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.2e-51) || !(c <= 7.8e-47)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * 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 <= (-4.2d-51)) .or. (.not. (c <= 7.8d-47))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = (((c * 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 <= -4.2e-51) || !(c <= 7.8e-47)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.2e-51) or not (c <= 7.8e-47):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.2e-51) || !(c <= 7.8e-47))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.2e-51) || ~((c <= 7.8e-47)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.2e-51], N[Not[LessEqual[c, 7.8e-47]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.20000000000000003e-51 or 7.79999999999999956e-47 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 80.5%

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

      1. remove-double-neg 80.5%

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

      2. mul-1-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. distribute-lft-in 80.5%

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

      5. mul-1-neg 80.5%

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

      6. distribute-neg-in 80.5%

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

      7. mul-1-neg 80.5%

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

      8. remove-double-neg 80.5%

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

      9. unsub-neg 80.5%

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

      10. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if -4.20000000000000003e-51 < c < 7.79999999999999956e-47

   1. Initial program 68.5%

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

      1. div-sub 64.1%

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

      2. *-commutative 64.1%

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

      3. add-sqr-sqrt 64.1%

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

      4. times-frac 63.3%

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

      5. fma-neg 63.3%

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

      6. hypot-define 63.4%

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

      7. hypot-define 66.6%

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

      8. associate-/l* 75.3%

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

      9. add-sqr-sqrt 75.3%

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

      10. pow2 75.3%

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

      11. hypot-define 75.3%

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

   4. Applied egg-rr 75.3%

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

   5. Taylor expanded in c around 0 77.3%

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

      1. *-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. associate-/l/ 83.5%

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

      4. div-sub 83.6%

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

   7. Simplified 83.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{-51} \lor \neg \left(c \leq 7.8 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-51} \lor \neg \left(c \leq 3.6 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.2e-51) (not (<= c 3.6e-48)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.2e-51) || !(c <= 3.6e-48)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c * (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 <= (-7.2d-51)) .or. (.not. (c <= 3.6d-48))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((c * (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 <= -7.2e-51) || !(c <= 3.6e-48)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.2e-51) or not (c <= 3.6e-48):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.2e-51) || !(c <= 3.6e-48))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	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 ((c <= -7.2e-51) || ~((c <= 3.6e-48)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.2e-51], N[Not[LessEqual[c, 3.6e-48]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.2000000000000001e-51 or 3.6000000000000002e-48 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 80.5%

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

      1. remove-double-neg 80.5%

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

      2. mul-1-neg 80.5%

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

      3. neg-mul-1 80.5%

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

      4. distribute-lft-in 80.5%

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

      5. mul-1-neg 80.5%

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

      6. distribute-neg-in 80.5%

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

      7. mul-1-neg 80.5%

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

      8. remove-double-neg 80.5%

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

      9. unsub-neg 80.5%

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

      10. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

   if -7.2000000000000001e-51 < c < 3.6000000000000002e-48

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 83.5%

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

      6. div-sub 83.6%

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

      7. *-commutative 83.6%

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

      8. associate-/l* 81.8%

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

   5. Simplified 81.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-51} \lor \neg \left(c \leq 3.6 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-51} \lor \neg \left(c \leq 1.5 \cdot 10^{-212}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3e-51) (not (<= c 1.5e-212)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3e-51) || !(c <= 1.5e-212)) {
		tmp = (b - (a * (d / c))) / 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 <= (-3d-51)) .or. (.not. (c <= 1.5d-212))) then
        tmp = (b - (a * (d / c))) / 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 <= -3e-51) || !(c <= 1.5e-212)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3e-51) or not (c <= 1.5e-212):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3e-51) || !(c <= 1.5e-212))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / 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 <= -3e-51) || ~((c <= 1.5e-212)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3e-51], N[Not[LessEqual[c, 1.5e-212]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.00000000000000002e-51 or 1.5000000000000001e-212 < c

   1. Initial program 55.7%

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

   3. Taylor expanded in c around inf 75.9%

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

      1. remove-double-neg 75.9%

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

      2. mul-1-neg 75.9%

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

      3. neg-mul-1 75.9%

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

      4. distribute-lft-in 75.9%

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

      5. mul-1-neg 75.9%

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

      6. distribute-neg-in 75.9%

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

      7. mul-1-neg 75.9%

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

      8. remove-double-neg 75.9%

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

      9. unsub-neg 75.9%

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

      10. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

   if -3.00000000000000002e-51 < c < 1.5000000000000001e-212

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0 81.9%

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

      1. associate-*r/ 81.9%

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

      2. neg-mul-1 81.9%

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

   5. Simplified 81.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-51} \lor \neg \left(c \leq 1.5 \cdot 10^{-212}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{-51} \lor \neg \left(c \leq 2.8 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e-51) (not (<= c 2.8e-50))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e-51) || !(c <= 2.8e-50)) {
		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 <= (-2.4d-51)) .or. (.not. (c <= 2.8d-50))) 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 <= -2.4e-51) || !(c <= 2.8e-50)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e-51) or not (c <= 2.8e-50):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e-51) || !(c <= 2.8e-50))
		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 <= -2.4e-51) || ~((c <= 2.8e-50)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e-51], N[Not[LessEqual[c, 2.8e-50]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.4e-51 or 2.7999999999999998e-50 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 67.8%

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

   if -2.4e-51 < c < 2.7999999999999998e-50

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. neg-mul-1 73.9%

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

   5. Simplified 73.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{-51} \lor \neg \left(c \leq 2.8 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.5% 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 60.1%

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

3. Taylor expanded in c around inf 45.9%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Add Preprocessing

Developer target: 99.3% accurate, 0.1× speedup

Math

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

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 2024096 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))