Complex division, imag part

Percentage Accurate: 61.8% → 82.3%

Time: 7.4s

Alternatives: 10

Speedup: 1.5×

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: 61.8% 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: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.32 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a}{t\_0} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -4.8e+126)
     t_1
     (if (<= c -1.05e-160)
       (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
       (if (<= c 2.1e-6)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 1.32e+113) (fma (/ c t_0) b (* (/ a t_0) (- d))) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -4.8e+126) {
		tmp = t_1;
	} else if (c <= -1.05e-160) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else if (c <= 2.1e-6) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.32e+113) {
		tmp = fma((c / t_0), b, ((a / t_0) * -d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -4.8e+126)
		tmp = t_1;
	elseif (c <= -1.05e-160)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (c <= 2.1e-6)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 1.32e+113)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(a / t_0) * Float64(-d)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.8e+126], t$95$1, If[LessEqual[c, -1.05e-160], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.1e-6], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.32e+113], N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a / t$95$0), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.80000000000000024e126 or 1.31999999999999996e113 < c

   1. Initial program 42.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.6

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

   5. Applied rewrites 81.6%

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

   7. Applied rewrites 92.7%

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

   if -4.80000000000000024e126 < c < -1.05e-160

   1. Initial program 84.1%

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

   if -1.05e-160 < c < 2.0999999999999998e-6

   1. Initial program 69.7%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 2.0999999999999998e-6 < c < 1.31999999999999996e113

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 91.0%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.32 \cdot 10^{+113}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.2e+144)
   (/ b c)
   (if (<= c -3.3e-78)
     (* (/ c (fma d d (* c c))) b)
     (if (<= c 2.1e-6)
       (/ (- a) d)
       (if (<= c 1.9e+100) (/ (- (* b c) (* d a)) (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.2e+144) {
		tmp = b / c;
	} else if (c <= -3.3e-78) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else if (c <= 2.1e-6) {
		tmp = -a / d;
	} else if (c <= 1.9e+100) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.2e+144)
		tmp = Float64(b / c);
	elseif (c <= -3.3e-78)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	elseif (c <= 2.1e-6)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1.9e+100)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.2e+144], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.3e-78], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 2.1e-6], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1.9e+100], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.19999999999999988e144 or 1.89999999999999982e100 < c

   1. Initial program 42.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -2.19999999999999988e144 < c < -3.29999999999999982e-78

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -3.29999999999999982e-78 < c < 2.0999999999999998e-6

   1. Initial program 73.1%

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

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 68.3

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

   5. Applied rewrites 68.3%

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

   if 2.0999999999999998e-6 < c < 1.89999999999999982e100

   1. Initial program 90.0%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -4.8e+126)
     t_0
     (if (<= c -1.05e-160)
       (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
       (if (<= c 2.4e+15) (/ (- (/ (* b c) d) a) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -4.8e+126) {
		tmp = t_0;
	} else if (c <= -1.05e-160) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else if (c <= 2.4e+15) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((a / c) * d)) / c
    if (c <= (-4.8d+126)) then
        tmp = t_0
    else if (c <= (-1.05d-160)) then
        tmp = ((b * c) - (d * a)) / ((d * d) + (c * c))
    else if (c <= 2.4d+15) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -4.8e+126) {
		tmp = t_0;
	} else if (c <= -1.05e-160) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else if (c <= 2.4e+15) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a / c) * d)) / c
	tmp = 0
	if c <= -4.8e+126:
		tmp = t_0
	elif c <= -1.05e-160:
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c))
	elif c <= 2.4e+15:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -4.8e+126)
		tmp = t_0;
	elseif (c <= -1.05e-160)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (c <= 2.4e+15)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((a / c) * d)) / c;
	tmp = 0.0;
	if (c <= -4.8e+126)
		tmp = t_0;
	elseif (c <= -1.05e-160)
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	elseif (c <= 2.4e+15)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.8e+126], t$95$0, If[LessEqual[c, -1.05e-160], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.4e+15], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.80000000000000024e126 or 2.4e15 < c

   1. Initial program 53.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 89.0%

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

   if -4.80000000000000024e126 < c < -1.05e-160

   1. Initial program 84.1%

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

   if -1.05e-160 < c < 2.4e15

   1. Initial program 70.6%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+126}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.65)
   (/ b c)
   (if (<= c 2.4e+15)
     (/ (fma (/ c d) b (- a)) d)
     (if (<= c 1.9e+100) (/ (- (* b c) (* d a)) (* c c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.65) {
		tmp = b / c;
	} else if (c <= 2.4e+15) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 1.9e+100) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.65)
		tmp = Float64(b / c);
	elseif (c <= 2.4e+15)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 1.9e+100)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.65], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.4e+15], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.9e+100], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.64999999999999991 or 1.89999999999999982e100 < c

   1. Initial program 51.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -3.64999999999999991 < c < 2.4e15

   1. Initial program 75.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{d}} \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{d}, \frac{c}{d}, -1\right), a, b \cdot \left(\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}\right)\right)}{d}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 81.0%

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

   if 2.4e15 < c < 1.89999999999999982e100

   1. Initial program 91.5%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - d \cdot a\\ \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* d a))))
   (if (<= c -3.65)
     (/ b c)
     (if (<= c 3.5e-95)
       (/ t_0 (* d d))
       (if (<= c 1.9e+100) (/ t_0 (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if (c <= -3.65) {
		tmp = b / c;
	} else if (c <= 3.5e-95) {
		tmp = t_0 / (d * d);
	} else if (c <= 1.9e+100) {
		tmp = t_0 / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b * c) - (d * a)
    if (c <= (-3.65d0)) then
        tmp = b / c
    else if (c <= 3.5d-95) then
        tmp = t_0 / (d * d)
    else if (c <= 1.9d+100) then
        tmp = t_0 / (c * c)
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if (c <= -3.65) {
		tmp = b / c;
	} else if (c <= 3.5e-95) {
		tmp = t_0 / (d * d);
	} else if (c <= 1.9e+100) {
		tmp = t_0 / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (d * a)
	tmp = 0
	if c <= -3.65:
		tmp = b / c
	elif c <= 3.5e-95:
		tmp = t_0 / (d * d)
	elif c <= 1.9e+100:
		tmp = t_0 / (c * c)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (c <= -3.65)
		tmp = Float64(b / c);
	elseif (c <= 3.5e-95)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 1.9e+100)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (d * a);
	tmp = 0.0;
	if (c <= -3.65)
		tmp = b / c;
	elseif (c <= 3.5e-95)
		tmp = t_0 / (d * d);
	elseif (c <= 1.9e+100)
		tmp = t_0 / (c * c);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.65], N[(b / c), $MachinePrecision], If[LessEqual[c, 3.5e-95], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.9e+100], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.64999999999999991 or 1.89999999999999982e100 < c

   1. Initial program 51.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if -3.64999999999999991 < c < 3.4999999999999997e-95

   1. Initial program 77.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 3.4999999999999997e-95 < c < 1.89999999999999982e100

   1. Initial program 79.1%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+100}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -3.65) t_0 (if (<= c 2.4e+15) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.65) {
		tmp = t_0;
	} else if (c <= 2.4e+15) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((a / c) * d)) / c
    if (c <= (-3.65d0)) then
        tmp = t_0
    else if (c <= 2.4d+15) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.65) {
		tmp = t_0;
	} else if (c <= 2.4e+15) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a / c) * d)) / c
	tmp = 0
	if c <= -3.65:
		tmp = t_0
	elif c <= 2.4e+15:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -3.65)
		tmp = t_0;
	elseif (c <= 2.4e+15)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((a / c) * d)) / c;
	tmp = 0.0;
	if (c <= -3.65)
		tmp = t_0;
	elseif (c <= 2.4e+15)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.65], t$95$0, If[LessEqual[c, 2.4e+15], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.64999999999999991 or 2.4e15 < c

   1. Initial program 58.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 82.5%

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

   if -3.64999999999999991 < c < 2.4e15

   1. Initial program 75.4%

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -3.65) t_0 (if (<= c 2.4e+15) (/ (fma (/ c d) b (- a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.65) {
		tmp = t_0;
	} else if (c <= 2.4e+15) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -3.65)
		tmp = t_0;
	elseif (c <= 2.4e+15)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.65], t$95$0, If[LessEqual[c, 2.4e+15], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.64999999999999991 or 2.4e15 < c

   1. Initial program 58.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   7. Applied rewrites 82.5%

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

   if -3.64999999999999991 < c < 2.4e15

   1. Initial program 75.4%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{d}} \]

   5. Applied rewrites 77.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{d}, \frac{c}{d}, -1\right), a, b \cdot \left(\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}\right)\right)}{d}} \]

   6. Taylor expanded in c around 0

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

   8. Applied rewrites 81.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.65:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.2e+144)
   (/ b c)
   (if (<= c -3.3e-78)
     (* (/ c (fma d d (* c c))) b)
     (if (<= c 2.4e+15) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.2e+144) {
		tmp = b / c;
	} else if (c <= -3.3e-78) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else if (c <= 2.4e+15) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.2e+144)
		tmp = Float64(b / c);
	elseif (c <= -3.3e-78)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	elseif (c <= 2.4e+15)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.2e+144], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.3e-78], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 2.4e+15], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.19999999999999988e144 or 2.4e15 < c

   1. Initial program 54.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.2

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

   5. Applied rewrites 76.2%

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

   if -2.19999999999999988e144 < c < -3.29999999999999982e-78

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 78.4%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 56.6

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

   7. Applied rewrites 56.6%

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

   if -3.29999999999999982e-78 < c < 2.4e15

   1. Initial program 73.7%

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

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.8e-77) (/ b c) (if (<= c 2.4e+15) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.8e-77) {
		tmp = b / c;
	} else if (c <= 2.4e+15) {
		tmp = -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 <= (-7.8d-77)) then
        tmp = b / c
    else if (c <= 2.4d+15) then
        tmp = -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 <= -7.8e-77) {
		tmp = b / c;
	} else if (c <= 2.4e+15) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -7.8e-77:
		tmp = b / c
	elif c <= 2.4e+15:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.8e-77)
		tmp = Float64(b / c);
	elseif (c <= 2.4e+15)
		tmp = Float64(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 <= -7.8e-77)
		tmp = b / c;
	elseif (c <= 2.4e+15)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.8e-77], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.4e+15], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -7.79999999999999958e-77 or 2.4e15 < c

   1. Initial program 61.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -7.79999999999999958e-77 < c < 2.4e15

   1. Initial program 73.7%

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

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.3% accurate, 3.2× 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 66.7%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 44.4

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

5. Applied rewrites 44.4%

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

Developer Target 1: 99.3% accurate, 0.6× 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 2024296 
(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))))