Complex division, imag part

Percentage Accurate: 61.6% → 82.9%

Time: 9.1s

Alternatives: 11

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: 61.6% 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: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := b \cdot c - d \cdot a\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.52 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{t\_1}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{\frac{b}{d}}{d}, 0 - \frac{a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))) (t_1 (- (* b c) (* d a))))
   (if (<= d -1.3e+77)
     (/ (- (/ b (/ d c)) a) d)
     (if (<= d -1.52e-111)
       (/ 1.0 (/ t_0 t_1))
       (if (<= d 1.8e-121)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 2.7e+66)
           (/ t_1 t_0)
           (fma c (/ (/ b d) d) (- 0.0 (/ a d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double t_1 = (b * c) - (d * a);
	double tmp;
	if (d <= -1.3e+77) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (d <= -1.52e-111) {
		tmp = 1.0 / (t_0 / t_1);
	} else if (d <= 1.8e-121) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.7e+66) {
		tmp = t_1 / t_0;
	} else {
		tmp = fma(c, ((b / d) / d), (0.0 - (a / d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	t_1 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (d <= -1.3e+77)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (d <= -1.52e-111)
		tmp = Float64(1.0 / Float64(t_0 / t_1));
	elseif (d <= 1.8e-121)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.7e+66)
		tmp = Float64(t_1 / t_0);
	else
		tmp = fma(c, Float64(Float64(b / d) / d), Float64(0.0 - Float64(a / d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.3e+77], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.52e-111], N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e-121], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.7e+66], N[(t$95$1 / t$95$0), $MachinePrecision], N[(c * N[(N[(b / d), $MachinePrecision] / d), $MachinePrecision] + N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.3000000000000001e77

   1. Initial program 44.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 84.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 84.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 86.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \frac{c}{d} - a\right), \color{blue}{d}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{1}{\frac{d}{c}}\right), a\right), d\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b}{\frac{d}{c}}\right), a\right), d\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{d}{c}\right)\right), a\right), d\right) \]

      6. /-lowering-/.f64 86.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(d, c\right)\right), a\right), d\right) \]

   9. Applied egg-rr 86.8%

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

   if -1.3000000000000001e77 < d < -1.51999999999999998e-111

   1. Initial program 93.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(c \cdot c + d \cdot d\right), \color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b \cdot c} - a \cdot d\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b} \cdot c - a \cdot d\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \left(b \cdot \color{blue}{c} - a \cdot d\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\left(b \cdot c\right), \color{blue}{\left(a \cdot d\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \left(\color{blue}{a} \cdot d\right)\right)\right)\right) \]

      9. *-lowering-*.f64 94.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \mathsf{*.f64}\left(a, \color{blue}{d}\right)\right)\right)\right) \]

   4. Applied egg-rr 94.1%

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

   if -1.51999999999999998e-111 < d < 1.79999999999999992e-121

   1. Initial program 61.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 90.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 90.3%

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

   if 1.79999999999999992e-121 < d < 2.7e66

   1. Initial program 74.6%

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

   if 2.7e66 < d

   1. Initial program 39.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 75.4%

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. associate-/l* N/A

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

      4. fmm-def N/A

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

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(c, \color{blue}{\left(\frac{\frac{b}{d}}{d}\right)}, \left(\mathsf{neg}\left(\frac{a}{d}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(c, \mathsf{/.f64}\left(\left(\frac{b}{d}\right), \color{blue}{d}\right), \left(\mathsf{neg}\left(\frac{a}{d}\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(b, d\right), d\right), \left(\mathsf{neg}\left(\frac{a}{d}\right)\right)\right) \]

      8. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(b, d\right), d\right), \mathsf{neg.f64}\left(\left(\frac{a}{d}\right)\right)\right) \]

      9. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{fma.f64}\left(c, \mathsf{/.f64}\left(\mathsf{/.f64}\left(b, d\right), d\right), \mathsf{neg.f64}\left(\mathsf{/.f64}\left(a, d\right)\right)\right) \]

   7. Applied egg-rr 83.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{\frac{b}{d}}{d}, -\frac{a}{d}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.52 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - d \cdot a}}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+66}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{\frac{b}{d}}{d}, 0 - \frac{a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ t_1 := b \cdot c - d \cdot a\\ \mathbf{if}\;d \leq -2.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{t\_1}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))) (t_1 (- (* b c) (* d a))))
   (if (<= d -2.2e+77)
     (/ (- (/ b (/ d c)) a) d)
     (if (<= d -1.42e-111)
       (/ 1.0 (/ t_0 t_1))
       (if (<= d 2.7e-121)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 2.1e+67) (/ t_1 t_0) (/ (- (* b (/ c d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double t_1 = (b * c) - (d * a);
	double tmp;
	if (d <= -2.2e+77) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (d <= -1.42e-111) {
		tmp = 1.0 / (t_0 / t_1);
	} else if (d <= 2.7e-121) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.1e+67) {
		tmp = t_1 / 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) :: t_1
    real(8) :: tmp
    t_0 = (c * c) + (d * d)
    t_1 = (b * c) - (d * a)
    if (d <= (-2.2d+77)) then
        tmp = ((b / (d / c)) - a) / d
    else if (d <= (-1.42d-111)) then
        tmp = 1.0d0 / (t_0 / t_1)
    else if (d <= 2.7d-121) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 2.1d+67) then
        tmp = t_1 / 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 * c) + (d * d);
	double t_1 = (b * c) - (d * a);
	double tmp;
	if (d <= -2.2e+77) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (d <= -1.42e-111) {
		tmp = 1.0 / (t_0 / t_1);
	} else if (d <= 2.7e-121) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.1e+67) {
		tmp = t_1 / t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * c) + (d * d)
	t_1 = (b * c) - (d * a)
	tmp = 0
	if d <= -2.2e+77:
		tmp = ((b / (d / c)) - a) / d
	elif d <= -1.42e-111:
		tmp = 1.0 / (t_0 / t_1)
	elif d <= 2.7e-121:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 2.1e+67:
		tmp = t_1 / t_0
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	t_1 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (d <= -2.2e+77)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (d <= -1.42e-111)
		tmp = Float64(1.0 / Float64(t_0 / t_1));
	elseif (d <= 2.7e-121)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.1e+67)
		tmp = Float64(t_1 / 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 * c) + (d * d);
	t_1 = (b * c) - (d * a);
	tmp = 0.0;
	if (d <= -2.2e+77)
		tmp = ((b / (d / c)) - a) / d;
	elseif (d <= -1.42e-111)
		tmp = 1.0 / (t_0 / t_1);
	elseif (d <= 2.7e-121)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 2.1e+67)
		tmp = t_1 / 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[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.2e+77], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.42e-111], N[(1.0 / N[(t$95$0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.7e-121], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.1e+67], N[(t$95$1 / t$95$0), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.2e77

   1. Initial program 44.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 84.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 84.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 86.7%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \frac{c}{d} - a\right), \color{blue}{d}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{1}{\frac{d}{c}}\right), a\right), d\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b}{\frac{d}{c}}\right), a\right), d\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{d}{c}\right)\right), a\right), d\right) \]

      6. /-lowering-/.f64 86.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(d, c\right)\right), a\right), d\right) \]

   9. Applied egg-rr 86.8%

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

   if -2.2e77 < d < -1.41999999999999991e-111

   1. Initial program 93.9%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(c \cdot c + d \cdot d\right), \color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b \cdot c} - a \cdot d\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b} \cdot c - a \cdot d\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \left(b \cdot \color{blue}{c} - a \cdot d\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\left(b \cdot c\right), \color{blue}{\left(a \cdot d\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \left(\color{blue}{a} \cdot d\right)\right)\right)\right) \]

      9. *-lowering-*.f64 94.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \mathsf{*.f64}\left(a, \color{blue}{d}\right)\right)\right)\right) \]

   4. Applied egg-rr 94.1%

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

   if -1.41999999999999991e-111 < d < 2.7000000000000002e-121

   1. Initial program 61.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 90.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 90.3%

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

   if 2.7000000000000002e-121 < d < 2.1000000000000001e67

   1. Initial program 74.6%

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

   if 2.1000000000000001e67 < d

   1. Initial program 39.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 75.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 82.2%

      \[\leadsto \frac{\color{blue}{b \cdot \frac{c}{d}} - a}{d} \]
3. Recombined 5 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.42 \cdot 10^{-111}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - d \cdot a}}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+67}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{if}\;d \leq -6 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* b (/ c d)) a) d)))
   (if (<= d -6e+114)
     t_1
     (if (<= d -3.2e-111)
       t_0
       (if (<= d 1.45e-121)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.5e+64) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double t_1 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -6e+114) {
		tmp = t_1;
	} else if (d <= -3.2e-111) {
		tmp = t_0;
	} else if (d <= 1.45e-121) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.5e+64) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
    t_1 = ((b * (c / d)) - a) / d
    if (d <= (-6d+114)) then
        tmp = t_1
    else if (d <= (-3.2d-111)) then
        tmp = t_0
    else if (d <= 1.45d-121) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 1.5d+64) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double t_1 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -6e+114) {
		tmp = t_1;
	} else if (d <= -3.2e-111) {
		tmp = t_0;
	} else if (d <= 1.45e-121) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.5e+64) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
	t_1 = ((b * (c / d)) - a) / d
	tmp = 0
	if d <= -6e+114:
		tmp = t_1
	elif d <= -3.2e-111:
		tmp = t_0
	elif d <= 1.45e-121:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 1.5e+64:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(b * Float64(c / d)) - a) / d)
	tmp = 0.0
	if (d <= -6e+114)
		tmp = t_1;
	elseif (d <= -3.2e-111)
		tmp = t_0;
	elseif (d <= 1.45e-121)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.5e+64)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	t_1 = ((b * (c / d)) - a) / d;
	tmp = 0.0;
	if (d <= -6e+114)
		tmp = t_1;
	elseif (d <= -3.2e-111)
		tmp = t_0;
	elseif (d <= 1.45e-121)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 1.5e+64)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6e+114], t$95$1, If[LessEqual[d, -3.2e-111], t$95$0, If[LessEqual[d, 1.45e-121], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.5e+64], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.0000000000000001e114 or 1.5000000000000001e64 < d

   1. Initial program 39.3%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 80.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 80.4%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 85.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 85.6%

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

   if -6.0000000000000001e114 < d < -3.1999999999999998e-111 or 1.45e-121 < d < 1.5000000000000001e64

   1. Initial program 82.1%

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

   if -3.1999999999999998e-111 < d < 1.45e-121

   1. Initial program 61.8%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 90.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 90.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6 \cdot 10^{+114}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -3.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.02e+86)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 5.4e+36) (/ (- (/ b (/ d c)) a) d) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.02e+86) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 5.4e+36) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / 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 <= (-1.02d+86)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 5.4d+36) then
        tmp = ((b / (d / c)) - a) / d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.02e+86) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 5.4e+36) {
		tmp = ((b / (d / c)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.02e+86:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 5.4e+36:
		tmp = ((b / (d / c)) - a) / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.02e+86)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 5.4e+36)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.02e+86)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 5.4e+36)
		tmp = ((b / (d / c)) - a) / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.02e+86], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 5.4e+36], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.01999999999999996e86

   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 c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 86.1%

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

   if -1.01999999999999996e86 < c < 5.4000000000000002e36

   1. Initial program 72.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 77.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 79.6%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b \cdot \frac{c}{d} - a\right), \color{blue}{d}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{1}{\frac{d}{c}}\right), a\right), d\right) \]

      4. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b}{\frac{d}{c}}\right), a\right), d\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \left(\frac{d}{c}\right)\right), a\right), d\right) \]

      6. /-lowering-/.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(b, \mathsf{/.f64}\left(d, c\right)\right), a\right), d\right) \]

   9. Applied egg-rr 79.6%

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

   if 5.4000000000000002e36 < c

   1. Initial program 50.4%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 75.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(a \cdot \frac{d}{c}\right)\right), c\right) \]

      4. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(a \cdot \frac{1}{\frac{1}{\frac{d}{c}}}\right)\right), c\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a}{\frac{1}{\frac{d}{c}}}\right)\right), c\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{d}{c}}\right)\right)\right), c\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), c\right) \]

      8. /-lowering-/.f64 82.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), c\right) \]

   7. Applied egg-rr 82.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1e+86)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 3.6e+36) (/ (- (* b (/ c d)) a) d) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+86) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 3.6e+36) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / 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+86)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 3.6d+36) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a / (c / d))) / 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+86) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 3.6e+36) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1e+86:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 3.6e+36:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1e+86)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 3.6e+36)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1e+86)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 3.6e+36)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1e+86], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.6e+36], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1e86

   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 c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 86.1%

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

   if -1e86 < c < 3.5999999999999997e36

   1. Initial program 72.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

         \[\leadsto \frac{b \cdot c}{{d}^{2}} + \left(\mathsf{neg}\left(\frac{a}{d}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{b \cdot c}{d} - a\right), \color{blue}{d}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(b \cdot c\right), d\right), a\right), d\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(c \cdot b\right), d\right), a\right), d\right) \]

      11. *-lowering-*.f64 77.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(c, b\right), d\right), a\right), d\right) \]

   5. Simplified 77.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{b \cdot c}{d}\right), a\right), d\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(b \cdot \frac{c}{d}\right), a\right), d\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \left(\frac{c}{d}\right)\right), a\right), d\right) \]

      4. /-lowering-/.f64 79.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, \mathsf{/.f64}\left(c, d\right)\right), a\right), d\right) \]

   7. Applied egg-rr 79.6%

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

   if 3.5999999999999997e36 < c

   1. Initial program 50.4%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 75.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 75.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(a \cdot \frac{d}{c}\right)\right), c\right) \]

      4. remove-double-div N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(a \cdot \frac{1}{\frac{1}{\frac{d}{c}}}\right)\right), c\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a}{\frac{1}{\frac{d}{c}}}\right)\right), c\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{1}{\frac{d}{c}}\right)\right)\right), c\right) \]

      7. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \left(\frac{c}{d}\right)\right)\right), c\right) \]

      8. /-lowering-/.f64 82.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(a, \mathsf{/.f64}\left(c, d\right)\right)\right), c\right) \]

   7. Applied egg-rr 82.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+86}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.6 \cdot 10^{+36}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -8.8e+23) t_0 (if (<= d 9.8e+82) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -8.8e+23) {
		tmp = t_0;
	} else if (d <= 9.8e+82) {
		tmp = (b - ((d * a) / c)) / c;
	} 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 = 0.0d0 - (a / d)
    if (d <= (-8.8d+23)) then
        tmp = t_0
    else if (d <= 9.8d+82) then
        tmp = (b - ((d * a) / c)) / c
    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 = 0.0 - (a / d);
	double tmp;
	if (d <= -8.8e+23) {
		tmp = t_0;
	} else if (d <= 9.8e+82) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -8.8e+23:
		tmp = t_0
	elif d <= 9.8e+82:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -8.8e+23)
		tmp = t_0;
	elseif (d <= 9.8e+82)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -8.8e+23)
		tmp = t_0;
	elseif (d <= 9.8e+82)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -8.8e+23], t$95$0, If[LessEqual[d, 9.8e+82], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -8.80000000000000034e23 or 9.8000000000000001e82 < d

   1. Initial program 48.9%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a}{d}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{a}{d}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{a}{d}\right)}\right) \]

      4. /-lowering-/.f64 68.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right) \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{0 - \frac{a}{d}} \]

   if -8.80000000000000034e23 < d < 9.8000000000000001e82

   1. Initial program 69.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\left(a \cdot d\right), c\right)\right), c\right) \]

      6. *-lowering-*.f64 72.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, d\right), c\right)\right), c\right) \]

   5. Simplified 72.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.8 \cdot 10^{+23}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq 9.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.46 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -1.46e+22)
     t_0
     (if (<= d 2.45e+82) (/ (- b (* (/ d c) a)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.46e+22) {
		tmp = t_0;
	} else if (d <= 2.45e+82) {
		tmp = (b - ((d / c) * a)) / c;
	} 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 = 0.0d0 - (a / d)
    if (d <= (-1.46d+22)) then
        tmp = t_0
    else if (d <= 2.45d+82) then
        tmp = (b - ((d / c) * a)) / c
    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 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.46e+22) {
		tmp = t_0;
	} else if (d <= 2.45e+82) {
		tmp = (b - ((d / c) * a)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -1.46e+22:
		tmp = t_0
	elif d <= 2.45e+82:
		tmp = (b - ((d / c) * a)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -1.46e+22)
		tmp = t_0;
	elseif (d <= 2.45e+82)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -1.46e+22)
		tmp = t_0;
	elseif (d <= 2.45e+82)
		tmp = (b - ((d / c) * a)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.46e+22], t$95$0, If[LessEqual[d, 2.45e+82], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.46e22 or 2.45e82 < d

   1. Initial program 48.9%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a}{d}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{a}{d}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{a}{d}\right)}\right) \]

      4. /-lowering-/.f64 68.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right) \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{0 - \frac{a}{d}} \]

   if -1.46e22 < d < 2.45e82

   1. Initial program 69.1%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(c \cdot c + d \cdot d\right), \color{blue}{\left(b \cdot c - a \cdot d\right)}\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(c \cdot c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b \cdot c} - a \cdot d\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \left(d \cdot d\right)\right), \left(\color{blue}{b} \cdot c - a \cdot d\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \left(b \cdot \color{blue}{c} - a \cdot d\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\left(b \cdot c\right), \color{blue}{\left(a \cdot d\right)}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \left(\color{blue}{a} \cdot d\right)\right)\right)\right) \]

      9. *-lowering-*.f64 69.1%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(c, c\right), \mathsf{*.f64}\left(d, d\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(b, c\right), \mathsf{*.f64}\left(a, \color{blue}{d}\right)\right)\right)\right) \]

   4. Applied egg-rr 69.1%

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

   5. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + -1 \cdot \frac{a \cdot d}{c}\right), \color{blue}{c}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b + \left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right), c\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(b - \frac{a \cdot d}{c}\right), c\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(\frac{a \cdot d}{c}\right)\right), c\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \left(a \cdot \frac{d}{c}\right)\right), c\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(a, \left(\frac{d}{c}\right)\right)\right), c\right) \]

      7. /-lowering-/.f64 72.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(b, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(d, c\right)\right)\right), c\right) \]

   7. Simplified 72.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.46 \cdot 10^{+22}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq 2.45 \cdot 10^{+82}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{1}{\frac{c}{b}}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{+43}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e+31)
   (/ 1.0 (/ c b))
   (if (<= c 1.45e+43) (- 0.0 (/ a d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e+31) {
		tmp = 1.0 / (c / b);
	} else if (c <= 1.45e+43) {
		tmp = 0.0 - (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 <= (-5.6d+31)) then
        tmp = 1.0d0 / (c / b)
    else if (c <= 1.45d+43) then
        tmp = 0.0d0 - (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 <= -5.6e+31) {
		tmp = 1.0 / (c / b);
	} else if (c <= 1.45e+43) {
		tmp = 0.0 - (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e+31:
		tmp = 1.0 / (c / b)
	elif c <= 1.45e+43:
		tmp = 0.0 - (a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e+31)
		tmp = Float64(1.0 / Float64(c / b));
	elseif (c <= 1.45e+43)
		tmp = Float64(0.0 - 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 <= -5.6e+31)
		tmp = 1.0 / (c / b);
	elseif (c <= 1.45e+43)
		tmp = 0.0 - (a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e+31], N[(1.0 / N[(c / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.45e+43], N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.60000000000000034e31

   1. Initial program 44.6%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 73.8%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{c}\right) \]

   5. Simplified 73.8%

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

      1. clear-num N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{c}{b}\right)}\right) \]

      3. /-lowering-/.f64 74.4%

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(c, \color{blue}{b}\right)\right) \]

   7. Applied egg-rr 74.4%

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

   if -5.60000000000000034e31 < c < 1.4500000000000001e43

   1. Initial program 72.1%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a}{d}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{a}{d}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{a}{d}\right)}\right) \]

      4. /-lowering-/.f64 59.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right) \]

   5. Simplified 59.0%

      \[\leadsto \color{blue}{0 - \frac{a}{d}} \]

   if 1.4500000000000001e43 < c

   1. Initial program 50.5%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 65.4%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{c}\right) \]

   5. Simplified 65.4%

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

Alternative 9: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.42 \cdot 10^{+32}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.42e+32) (/ b c) (if (<= c 1.35e+42) (- 0.0 (/ a d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.42e+32) {
		tmp = b / c;
	} else if (c <= 1.35e+42) {
		tmp = 0.0 - (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 <= (-1.42d+32)) then
        tmp = b / c
    else if (c <= 1.35d+42) then
        tmp = 0.0d0 - (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 <= -1.42e+32) {
		tmp = b / c;
	} else if (c <= 1.35e+42) {
		tmp = 0.0 - (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.42e+32:
		tmp = b / c
	elif c <= 1.35e+42:
		tmp = 0.0 - (a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.42e+32)
		tmp = Float64(b / c);
	elseif (c <= 1.35e+42)
		tmp = Float64(0.0 - 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 <= -1.42e+32)
		tmp = b / c;
	elseif (c <= 1.35e+42)
		tmp = 0.0 - (a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.42e+32], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.35e+42], N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.41999999999999992e32 or 1.35e42 < c

   1. Initial program 47.3%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 69.9%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{c}\right) \]

   5. Simplified 69.9%

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

   if -1.41999999999999992e32 < c < 1.35e42

   1. Initial program 72.1%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{a}{d}\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{a}{d}} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{a}{d}\right)}\right) \]

      4. /-lowering-/.f64 59.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(a, \color{blue}{d}\right)\right) \]

   5. Simplified 59.0%

      \[\leadsto \color{blue}{0 - \frac{a}{d}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;a \cdot \frac{-1}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e+31) (/ b c) (if (<= c 1.1e+31) (* a (/ -1.0 d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e+31) {
		tmp = b / c;
	} else if (c <= 1.1e+31) {
		tmp = a * (-1.0 / 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 <= (-5.6d+31)) then
        tmp = b / c
    else if (c <= 1.1d+31) then
        tmp = a * ((-1.0d0) / 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 <= -5.6e+31) {
		tmp = b / c;
	} else if (c <= 1.1e+31) {
		tmp = a * (-1.0 / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e+31:
		tmp = b / c
	elif c <= 1.1e+31:
		tmp = a * (-1.0 / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e+31)
		tmp = Float64(b / c);
	elseif (c <= 1.1e+31)
		tmp = Float64(a * Float64(-1.0 / d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.6e+31)
		tmp = b / c;
	elseif (c <= 1.1e+31)
		tmp = a * (-1.0 / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e+31], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.1e+31], N[(a * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.60000000000000034e31 or 1.10000000000000005e31 < c

   1. Initial program 47.4%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 69.6%

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{c}\right) \]

   5. Simplified 69.6%

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

   if -5.60000000000000034e31 < c < 1.10000000000000005e31

   1. Initial program 72.4%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto a \cdot \left(\left(0 - \frac{d}{{c}^{2} + {d}^{2}}\right) + \frac{\color{blue}{b \cdot c}}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto a \cdot \left(0 - \color{blue}{\left(\frac{d}{{c}^{2} + {d}^{2}} - \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto a \cdot \left(0 - \left(\frac{d}{{c}^{2} + {d}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto a \cdot \left(0 - \left(\frac{d}{{c}^{2} + {d}^{2}} + -1 \cdot \color{blue}{\frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)}}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto a \cdot \left(0 - \left(-1 \cdot \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)} + \color{blue}{\frac{d}{{c}^{2} + {d}^{2}}}\right)\right) \]

      7. neg-sub0 N/A

         \[\leadsto a \cdot \left(\mathsf{neg}\left(\left(-1 \cdot \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)} + \frac{d}{{c}^{2} + {d}^{2}}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)} + \frac{d}{{c}^{2} + {d}^{2}}\right)\right)\right)}\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(a, \left(0 - \color{blue}{\left(-1 \cdot \frac{b \cdot c}{a \cdot \left({c}^{2} + {d}^{2}\right)} + \frac{d}{{c}^{2} + {d}^{2}}\right)}\right)\right) \]

   5. Simplified 66.7%

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

   6. Taylor expanded in c around 0

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{d}\right)}\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 58.9%

         \[\leadsto \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(-1, \color{blue}{d}\right)\right) \]

   8. Simplified 58.9%

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

Alternative 11: 43.5% 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 61.8%

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

3. Taylor expanded in c around inf

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

   1. /-lowering-/.f64 42.5%

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{c}\right) \]

5. Simplified 42.5%

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

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024139 
(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))))