Complex division, imag part

Percentage Accurate: 62.1% → 95.9%

Time: 8.3s

Alternatives: 13

Speedup: 1.0×

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: 62.1% 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: 95.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -a \cdot \left(\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{d}{\mathsf{hypot}\left(c, d\right)}\right)\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (fma
  (/ c (hypot c d))
  (/ b (hypot c d))
  (- (* a (* (/ 1.0 (hypot c d)) (/ d (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	return fma((c / hypot(c, d)), (b / hypot(c, d)), -(a * ((1.0 / hypot(c, d)) * (d / hypot(c, d)))));
}

Julia

function code(a, b, c, d)
	return fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(-Float64(a * Float64(Float64(1.0 / hypot(c, d)) * Float64(d / hypot(c, d))))))
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[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

   1. div-sub 55.8%

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

   2. *-commutative 55.8%

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

   3. add-sqr-sqrt 55.8%

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

   4. times-frac 56.9%

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

   5. fma-neg 56.9%

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

   6. hypot-define 56.9%

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

   7. hypot-define 76.3%

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

   8. associate-/l* 79.3%

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

   9. add-sqr-sqrt 79.3%

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

   10. pow2 79.3%

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

   11. hypot-define 79.3%

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

4. Applied egg-rr 79.3%

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

   1. *-un-lft-identity 79.3%

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

   2. unpow2 79.3%

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

   3. times-frac 94.9%

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

6. Applied egg-rr 94.9%

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

Alternative 2: 90.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, -\frac{a}{d}\right)\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 6.3 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_1, -a \cdot \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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)))))
   (if (<= d -1.4e+117)
     t_2
     (if (<= d 6.3e+125)
       (fma t_0 t_1 (- (* a (/ d (pow (hypot c d) 2.0)))))
       t_2))))

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));
	double tmp;
	if (d <= -1.4e+117) {
		tmp = t_2;
	} else if (d <= 6.3e+125) {
		tmp = fma(t_0, t_1, -(a * (d / pow(hypot(c, d), 2.0))));
	} else {
		tmp = t_2;
	}
	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(-Float64(a / d)))
	tmp = 0.0
	if (d <= -1.4e+117)
		tmp = t_2;
	elseif (d <= 6.3e+125)
		tmp = fma(t_0, t_1, Float64(-Float64(a * Float64(d / (hypot(c, d) ^ 2.0)))));
	else
		tmp = t_2;
	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 / d), $MachinePrecision])), $MachinePrecision]}, If[LessEqual[d, -1.4e+117], t$95$2, If[LessEqual[d, 6.3e+125], 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], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.39999999999999999e117 or 6.3000000000000002e125 < d

   1. Initial program 30.8%

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

      1. div-sub 30.8%

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

      2. *-commutative 30.8%

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

      3. add-sqr-sqrt 30.8%

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

      4. times-frac 31.0%

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

      5. fma-neg 31.0%

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

      6. hypot-define 31.0%

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

      7. hypot-define 45.9%

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

      8. associate-/l* 52.9%

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

      9. add-sqr-sqrt 52.9%

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

      10. pow2 52.9%

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

      11. hypot-define 52.9%

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

   4. Applied egg-rr 52.9%

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

   5. Taylor expanded in d around inf 89.6%

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

   if -1.39999999999999999e117 < d < 6.3000000000000002e125

   1. Initial program 71.2%

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

      1. div-sub 67.6%

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

      2. *-commutative 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. times-frac 69.1%

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

      5. fma-neg 69.1%

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

      6. hypot-define 69.1%

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

      7. hypot-define 90.6%

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

      8. associate-/l* 91.7%

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

      9. add-sqr-sqrt 91.7%

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

      10. pow2 91.7%

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

      11. hypot-define 91.7%

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

   4. Applied egg-rr 91.7%

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

Alternative 3: 81.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}\\ \mathbf{if}\;c \leq -12000000000000:\\ \;\;\;\;-1 \cdot \left(\frac{d}{c} \cdot \frac{a}{c}\right) + \frac{b}{c}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 7.8 \cdot 10^{+158}:\\ \;\;\;\;b \cdot \frac{c}{t\_0} - a \cdot \frac{d}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{d}{\frac{c}{a}}}{c} + \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (pow (hypot c d) 2.0)))
   (if (<= c -12000000000000.0)
     (+ (* -1.0 (* (/ d c) (/ a c))) (/ b c))
     (if (<= c 2.9e-141)
       (/ (- (/ b (/ d c)) a) d)
       (if (<= c 7.8e+158)
         (- (* b (/ c t_0)) (* a (/ d t_0)))
         (+ (* -1.0 (/ (/ d (/ c a)) c)) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = pow(hypot(c, d), 2.0);
	double tmp;
	if (c <= -12000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 2.9e-141) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 7.8e+158) {
		tmp = (b * (c / t_0)) - (a * (d / t_0));
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = Math.pow(Math.hypot(c, d), 2.0);
	double tmp;
	if (c <= -12000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 2.9e-141) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 7.8e+158) {
		tmp = (b * (c / t_0)) - (a * (d / t_0));
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = math.pow(math.hypot(c, d), 2.0)
	tmp = 0
	if c <= -12000000000000.0:
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c)
	elif c <= 2.9e-141:
		tmp = ((b / (d / c)) - a) / d
	elif c <= 7.8e+158:
		tmp = (b * (c / t_0)) - (a * (d / t_0))
	else:
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = hypot(c, d) ^ 2.0
	tmp = 0.0
	if (c <= -12000000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / c) * Float64(a / c))) + Float64(b / c));
	elseif (c <= 2.9e-141)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 7.8e+158)
		tmp = Float64(Float64(b * Float64(c / t_0)) - Float64(a * Float64(d / t_0)));
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / Float64(c / a)) / c)) + Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = hypot(c, d) ^ 2.0;
	tmp = 0.0;
	if (c <= -12000000000000.0)
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	elseif (c <= 2.9e-141)
		tmp = ((b / (d / c)) - a) / d;
	elseif (c <= 7.8e+158)
		tmp = (b * (c / t_0)) - (a * (d / t_0));
	else
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[c, -12000000000000.0], N[(N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.9e-141], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.8e+158], N[(N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.2e13

   1. Initial program 40.0%

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

   3. Taylor expanded in d around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. times-frac 76.2%

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

   5. Applied egg-rr 76.2%

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

   if -1.2e13 < c < 2.9e-141

   1. Initial program 71.0%

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

   3. Taylor expanded in d around inf 92.0%

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

      1. +-commutative 92.0%

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

      2. mul-1-neg 92.0%

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

      3. unsub-neg 92.0%

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

      4. associate-/l* 92.9%

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

   5. Applied egg-rr 92.9%

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

      1. clear-num 93.0%

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

      2. un-div-inv 93.0%

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

   7. Applied egg-rr 93.0%

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

   if 2.9e-141 < c < 7.8e158

   1. Initial program 76.8%

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

      1. div-sub 76.8%

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

      2. associate-/l* 81.8%

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

      3. add-sqr-sqrt 81.8%

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

      4. pow2 81.8%

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

      5. hypot-define 81.8%

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

      6. associate-/l* 85.2%

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

      7. add-sqr-sqrt 85.2%

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

      8. pow2 85.2%

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

      9. hypot-define 85.2%

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

   4. Applied egg-rr 85.2%

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

   if 7.8e158 < c

   1. Initial program 28.4%

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

   3. Taylor expanded in d around 0 78.9%

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

      1. *-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. times-frac 92.5%

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

   5. Applied egg-rr 92.5%

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

      1. associate-*l/ 92.5%

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

      2. clear-num 92.5%

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

      3. un-div-inv 92.5%

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

   7. Applied egg-rr 92.5%

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

Alternative 4: 80.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -10000000000000:\\ \;\;\;\;-1 \cdot \left(\frac{d}{c} \cdot \frac{a}{c}\right) + \frac{b}{c}\\ \mathbf{elif}\;c \leq 2.75 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+97}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{d}{\frac{c}{a}}}{c} + \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -10000000000000.0)
   (+ (* -1.0 (* (/ d c) (/ a c))) (/ b c))
   (if (<= c 2.75e-136)
     (/ (- (/ b (/ d c)) a) d)
     (if (<= c 4e+97)
       (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
       (+ (* -1.0 (/ (/ d (/ c a)) c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -10000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 2.75e-136) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 4e+97) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-10000000000000.0d0)) then
        tmp = ((-1.0d0) * ((d / c) * (a / c))) + (b / c)
    else if (c <= 2.75d-136) then
        tmp = ((b / (d / c)) - a) / d
    else if (c <= 4d+97) then
        tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
    else
        tmp = ((-1.0d0) * ((d / (c / a)) / c)) + (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -10000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 2.75e-136) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 4e+97) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -10000000000000.0:
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c)
	elif c <= 2.75e-136:
		tmp = ((b / (d / c)) - a) / d
	elif c <= 4e+97:
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
	else:
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -10000000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / c) * Float64(a / c))) + Float64(b / c));
	elseif (c <= 2.75e-136)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 4e+97)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / Float64(c / a)) / c)) + Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -10000000000000.0)
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	elseif (c <= 2.75e-136)
		tmp = ((b / (d / c)) - a) / d;
	elseif (c <= 4e+97)
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	else
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -10000000000000.0], N[(N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.75e-136], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4e+97], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1e13

   1. Initial program 40.0%

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

   3. Taylor expanded in d around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. times-frac 76.2%

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

   5. Applied egg-rr 76.2%

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

   if -1e13 < c < 2.75e-136

   1. Initial program 70.4%

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

   3. Taylor expanded in d around inf 92.1%

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

      1. +-commutative 92.1%

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

      2. mul-1-neg 92.1%

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

      3. unsub-neg 92.1%

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

      4. associate-/l* 93.0%

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

   5. Applied egg-rr 93.0%

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

      1. clear-num 93.0%

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

      2. un-div-inv 93.0%

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

   7. Applied egg-rr 93.0%

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

   if 2.75e-136 < c < 4.0000000000000003e97

   1. Initial program 84.0%

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

   if 4.0000000000000003e97 < c

   1. Initial program 30.4%

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

   3. Taylor expanded in d around 0 78.5%

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

      1. *-commutative 78.5%

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

      2. unpow2 78.5%

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

      3. times-frac 89.8%

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

   5. Applied egg-rr 89.8%

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

      1. associate-*l/ 89.9%

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

      2. clear-num 89.9%

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

      3. un-div-inv 89.9%

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

   7. Applied egg-rr 89.9%

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

Alternative 5: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -14000000000000:\\ \;\;\;\;-1 \cdot \left(\frac{d}{c} \cdot \frac{a}{c}\right) + \frac{b}{c}\\ \mathbf{elif}\;c \leq 1950:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{\frac{d}{\frac{c}{a}}}{c} + \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -14000000000000.0)
   (+ (* -1.0 (* (/ d c) (/ a c))) (/ b c))
   (if (<= c 1950.0)
     (/ (- (/ b (/ d c)) a) d)
     (+ (* -1.0 (/ (/ d (/ c a)) c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -14000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 1950.0) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-14000000000000.0d0)) then
        tmp = ((-1.0d0) * ((d / c) * (a / c))) + (b / c)
    else if (c <= 1950.0d0) then
        tmp = ((b / (d / c)) - a) / d
    else
        tmp = ((-1.0d0) * ((d / (c / a)) / c)) + (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -14000000000000.0) {
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	} else if (c <= 1950.0) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -14000000000000.0:
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c)
	elif c <= 1950.0:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -14000000000000.0)
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / c) * Float64(a / c))) + Float64(b / c));
	elseif (c <= 1950.0)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = Float64(Float64(-1.0 * Float64(Float64(d / Float64(c / a)) / c)) + Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -14000000000000.0)
		tmp = (-1.0 * ((d / c) * (a / c))) + (b / c);
	elseif (c <= 1950.0)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = (-1.0 * ((d / (c / a)) / c)) + (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -14000000000000.0], N[(N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1950.0], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(-1.0 * N[(N[(d / N[(c / a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.4e13

   1. Initial program 40.0%

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

   3. Taylor expanded in d around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. times-frac 76.2%

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

   5. Applied egg-rr 76.2%

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

   if -1.4e13 < c < 1950

   1. Initial program 72.7%

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

   3. Taylor expanded in d around inf 85.7%

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

      1. +-commutative 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-/l* 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. clear-num 86.5%

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

      2. un-div-inv 86.5%

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

   7. Applied egg-rr 86.5%

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

   if 1950 < c

   1. Initial program 47.1%

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

   3. Taylor expanded in d around 0 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 82.7%

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

   5. Applied egg-rr 82.7%

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

      1. associate-*l/ 84.2%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

   7. Applied egg-rr 84.2%

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

Alternative 6: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \left(\frac{d}{c} \cdot \frac{a}{c}\right) + \frac{b}{c}\\ \mathbf{if}\;c \leq -13500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 35000:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* -1.0 (* (/ d c) (/ a c))) (/ b c))))
   (if (<= c -13500000000000.0)
     t_0
     (if (<= c 35000.0) (/ (- (/ b (/ d c)) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (-1.0 * ((d / c) * (a / c))) + (b / c);
	double tmp;
	if (c <= -13500000000000.0) {
		tmp = t_0;
	} else if (c <= 35000.0) {
		tmp = ((b / (d / c)) - 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 = ((-1.0d0) * ((d / c) * (a / c))) + (b / c)
    if (c <= (-13500000000000.0d0)) then
        tmp = t_0
    else if (c <= 35000.0d0) then
        tmp = ((b / (d / c)) - 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 = (-1.0 * ((d / c) * (a / c))) + (b / c);
	double tmp;
	if (c <= -13500000000000.0) {
		tmp = t_0;
	} else if (c <= 35000.0) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (-1.0 * ((d / c) * (a / c))) + (b / c)
	tmp = 0
	if c <= -13500000000000.0:
		tmp = t_0
	elif c <= 35000.0:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-1.0 * Float64(Float64(d / c) * Float64(a / c))) + Float64(b / c))
	tmp = 0.0
	if (c <= -13500000000000.0)
		tmp = t_0;
	elseif (c <= 35000.0)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (-1.0 * ((d / c) * (a / c))) + (b / c);
	tmp = 0.0;
	if (c <= -13500000000000.0)
		tmp = t_0;
	elseif (c <= 35000.0)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -13500000000000.0], t$95$0, If[LessEqual[c, 35000.0], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.35e13 or 35000 < c

   1. Initial program 43.8%

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

   3. Taylor expanded in d around 0 72.6%

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

      1. *-commutative 72.6%

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

      2. unpow2 72.6%

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

      3. times-frac 79.7%

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

   5. Applied egg-rr 79.7%

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

   if -1.35e13 < c < 35000

   1. Initial program 72.7%

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

   3. Taylor expanded in d around inf 85.7%

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

      1. +-commutative 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-/l* 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. clear-num 86.5%

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

      2. un-div-inv 86.5%

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

   7. Applied egg-rr 86.5%

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

Alternative 7: 76.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -11500000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 10000:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ b (* -1.0 (/ (* a d) c))) c)))
   (if (<= c -11500000000000.0)
     t_0
     (if (<= c 10000.0) (/ (- (/ b (/ d c)) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b + (-1.0 * ((a * d) / c))) / c;
	double tmp;
	if (c <= -11500000000000.0) {
		tmp = t_0;
	} else if (c <= 10000.0) {
		tmp = ((b / (d / c)) - 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 + ((-1.0d0) * ((a * d) / c))) / c
    if (c <= (-11500000000000.0d0)) then
        tmp = t_0
    else if (c <= 10000.0d0) then
        tmp = ((b / (d / c)) - 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 + (-1.0 * ((a * d) / c))) / c;
	double tmp;
	if (c <= -11500000000000.0) {
		tmp = t_0;
	} else if (c <= 10000.0) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b + (-1.0 * ((a * d) / c))) / c
	tmp = 0
	if c <= -11500000000000.0:
		tmp = t_0
	elif c <= 10000.0:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b + Float64(-1.0 * Float64(Float64(a * d) / c))) / c)
	tmp = 0.0
	if (c <= -11500000000000.0)
		tmp = t_0;
	elseif (c <= 10000.0)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b + (-1.0 * ((a * d) / c))) / c;
	tmp = 0.0;
	if (c <= -11500000000000.0)
		tmp = t_0;
	elseif (c <= 10000.0)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b + N[(-1.0 * N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -11500000000000.0], t$95$0, If[LessEqual[c, 10000.0], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.15e13 or 1e4 < c

   1. Initial program 43.8%

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

   3. Taylor expanded in c around inf 75.1%

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

   if -1.15e13 < c < 1e4

   1. Initial program 72.7%

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

   3. Taylor expanded in d around inf 85.7%

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

      1. +-commutative 85.7%

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

      2. mul-1-neg 85.7%

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

      3. unsub-neg 85.7%

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

      4. associate-/l* 86.4%

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

   5. Applied egg-rr 86.4%

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

      1. clear-num 86.5%

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

      2. un-div-inv 86.5%

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

   7. Applied egg-rr 86.5%

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

Alternative 8: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1e+153)
   (/ b c)
   (if (<= c 3.8e+40) (/ (- (/ b (/ d c)) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+153) {
		tmp = b / c;
	} else if (c <= 3.8e+40) {
		tmp = ((b / (d / c)) - a) / d;
	} 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) :: tmp
    if (c <= (-1d+153)) then
        tmp = b / c
    else if (c <= 3.8d+40) then
        tmp = ((b / (d / c)) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+153) {
		tmp = b / c;
	} else if (c <= 3.8e+40) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1e+153:
		tmp = b / c
	elif c <= 3.8e+40:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1e+153)
		tmp = Float64(b / c);
	elseif (c <= 3.8e+40)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1e+153)
		tmp = b / c;
	elseif (c <= 3.8e+40)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1e+153], N[(b / c), $MachinePrecision], If[LessEqual[c, 3.8e+40], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1e153 or 3.80000000000000004e40 < c

   1. Initial program 30.8%

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

   3. Taylor expanded in c around inf 77.6%

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

   if -1e153 < c < 3.80000000000000004e40

   1. Initial program 73.4%

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

   3. Taylor expanded in d around inf 75.0%

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

      1. +-commutative 75.0%

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

      2. mul-1-neg 75.0%

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

      3. unsub-neg 75.0%

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

      4. associate-/l* 76.2%

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

   5. Applied egg-rr 76.2%

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

      1. clear-num 76.2%

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

      2. un-div-inv 76.2%

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

   7. Applied egg-rr 76.2%

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

Alternative 9: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+153}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1e+153)
   (/ b c)
   (if (<= c 3.6e+40) (/ (- (* b (/ c d)) a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+153) {
		tmp = b / c;
	} else if (c <= 3.6e+40) {
		tmp = ((b * (c / d)) - a) / d;
	} 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) :: tmp
    if (c <= (-1d+153)) then
        tmp = b / c
    else if (c <= 3.6d+40) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+153) {
		tmp = b / c;
	} else if (c <= 3.6e+40) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1e+153:
		tmp = b / c
	elif c <= 3.6e+40:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1e+153)
		tmp = Float64(b / c);
	elseif (c <= 3.6e+40)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1e+153)
		tmp = b / c;
	elseif (c <= 3.6e+40)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1e+153], N[(b / c), $MachinePrecision], If[LessEqual[c, 3.6e+40], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1e153 or 3.59999999999999996e40 < c

   1. Initial program 30.8%

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

   3. Taylor expanded in c around inf 77.6%

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

   if -1e153 < c < 3.59999999999999996e40

   1. Initial program 73.4%

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

   3. Taylor expanded in d around inf 75.0%

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

      1. +-commutative 75.0%

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

      2. mul-1-neg 75.0%

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

      3. unsub-neg 75.0%

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

      4. associate-/l* 76.2%

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

   5. Applied egg-rr 76.2%

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

Alternative 10: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7800000000000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+26}:\\ \;\;\;\;-1 \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7800000000000.0)
   (/ b c)
   (if (<= c 6e+26) (* -1.0 (/ a d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7800000000000.0) {
		tmp = b / c;
	} else if (c <= 6e+26) {
		tmp = -1.0 * (a / d);
	} 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) :: tmp
    if (c <= (-7800000000000.0d0)) then
        tmp = b / c
    else if (c <= 6d+26) then
        tmp = (-1.0d0) * (a / d)
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7800000000000.0) {
		tmp = b / c;
	} else if (c <= 6e+26) {
		tmp = -1.0 * (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7800000000000.0:
		tmp = b / c
	elif c <= 6e+26:
		tmp = -1.0 * (a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7800000000000.0)
		tmp = Float64(b / c);
	elseif (c <= 6e+26)
		tmp = Float64(-1.0 * Float64(a / d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -7800000000000.0)
		tmp = b / c;
	elseif (c <= 6e+26)
		tmp = -1.0 * (a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7800000000000.0], N[(b / c), $MachinePrecision], If[LessEqual[c, 6e+26], N[(-1.0 * N[(a / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.8e12 or 5.99999999999999994e26 < c

   1. Initial program 42.0%

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

   3. Taylor expanded in c around inf 66.7%

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

   if -7.8e12 < c < 5.99999999999999994e26

   1. Initial program 73.5%

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

   3. Taylor expanded in c around 0 68.1%

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

Alternative 11: 46.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+167}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+230}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.2e+167) (/ a d) (if (<= d 5.2e+230) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.2e+167) {
		tmp = a / d;
	} else if (d <= 5.2e+230) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.2d+167)) then
        tmp = a / d
    else if (d <= 5.2d+230) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.2e+167) {
		tmp = a / d;
	} else if (d <= 5.2e+230) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.2e+167:
		tmp = a / d
	elif d <= 5.2e+230:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.2e+167)
		tmp = Float64(a / d);
	elseif (d <= 5.2e+230)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.2e+167)
		tmp = a / d;
	elseif (d <= 5.2e+230)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.2e+167], N[(a / d), $MachinePrecision], If[LessEqual[d, 5.2e+230], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.19999999999999999e167 or 5.1999999999999997e230 < d

   1. Initial program 37.2%

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

      1. fma-neg 37.2%

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

      2. distribute-rgt-neg-out 37.2%

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

      3. *-un-lft-identity 37.2%

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

      4. add-sqr-sqrt 37.2%

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

      5. times-frac 37.2%

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

      6. hypot-define 37.2%

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

      7. add-sqr-sqrt 22.2%

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

      8. sqrt-unprod 34.2%

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

      9. distribute-rgt-neg-out 34.2%

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

      10. distribute-rgt-neg-out 34.2%

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

      11. sqr-neg 34.2%

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

      12. sqrt-unprod 15.0%

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

      13. add-sqr-sqrt 37.2%

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

      14. hypot-define 43.2%

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

   4. Applied egg-rr 43.2%

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

   5. Taylor expanded in c around 0 37.9%

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

   if -1.19999999999999999e167 < d < 5.1999999999999997e230

   1. Initial program 63.0%

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

   3. Taylor expanded in c around inf 48.4%

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

Alternative 12: 15.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+107}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.8e+145) (/ a d) (if (<= d 1.3e+107) (/ a c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.8e+145) {
		tmp = a / d;
	} else if (d <= 1.3e+107) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3.8d+145)) then
        tmp = a / d
    else if (d <= 1.3d+107) then
        tmp = a / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.8e+145) {
		tmp = a / d;
	} else if (d <= 1.3e+107) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.8e+145:
		tmp = a / d
	elif d <= 1.3e+107:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.8e+145)
		tmp = Float64(a / d);
	elseif (d <= 1.3e+107)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.8e+145)
		tmp = a / d;
	elseif (d <= 1.3e+107)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.8e+145], N[(a / d), $MachinePrecision], If[LessEqual[d, 1.3e+107], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.80000000000000012e145 or 1.3000000000000001e107 < d

   1. Initial program 31.0%

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

      1. fma-neg 31.0%

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

      2. distribute-rgt-neg-out 31.0%

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

      3. *-un-lft-identity 31.0%

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

      4. add-sqr-sqrt 31.0%

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

      5. times-frac 31.0%

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

      6. hypot-define 31.0%

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

      7. add-sqr-sqrt 16.7%

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

      8. sqrt-unprod 26.4%

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

      9. distribute-rgt-neg-out 26.4%

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

      10. distribute-rgt-neg-out 26.4%

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

      11. sqr-neg 26.4%

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

      12. sqrt-unprod 13.0%

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

      13. add-sqr-sqrt 28.5%

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

      14. hypot-define 36.7%

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

   4. Applied egg-rr 36.7%

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

   5. Taylor expanded in c around 0 26.1%

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

   if -3.80000000000000012e145 < d < 1.3000000000000001e107

   1. Initial program 70.8%

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

      1. fma-neg 70.9%

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

      2. distribute-rgt-neg-out 70.9%

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

      3. *-un-lft-identity 70.9%

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

      4. add-sqr-sqrt 70.9%

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

      5. times-frac 70.7%

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

      6. hypot-define 70.7%

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

      7. add-sqr-sqrt 48.9%

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

      8. sqrt-unprod 51.8%

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

      9. distribute-rgt-neg-out 51.8%

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

      10. distribute-rgt-neg-out 51.8%

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

      11. sqr-neg 51.8%

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

      12. sqrt-unprod 18.1%

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

      13. add-sqr-sqrt 35.9%

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

      14. hypot-define 43.6%

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

   4. Applied egg-rr 43.6%

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

   5. Taylor expanded in c around -inf 37.8%

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

   6. Taylor expanded in d around -inf 13.3%

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

Alternative 13: 9.8% 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 58.2%

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

   1. fma-neg 58.2%

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

   2. distribute-rgt-neg-out 58.2%

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

   3. *-un-lft-identity 58.2%

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

   4. add-sqr-sqrt 58.2%

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

   5. times-frac 58.1%

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

   6. hypot-define 58.2%

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

   7. add-sqr-sqrt 38.7%

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

   8. sqrt-unprod 43.8%

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

   9. distribute-rgt-neg-out 43.8%

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

   10. distribute-rgt-neg-out 43.8%

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

   11. sqr-neg 43.8%

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

   12. sqrt-unprod 16.5%

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

   13. add-sqr-sqrt 33.5%

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

   14. hypot-define 41.4%

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

4. Applied egg-rr 41.4%

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

5. Taylor expanded in c around -inf 29.6%

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

6. Taylor expanded in d around -inf 10.7%

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

Developer Target 1: 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 2024116 -o generate:simplify
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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