Complex division, imag part

Percentage Accurate: 61.1% → 85.1%

Time: 9.0s

Alternatives: 11

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: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(-a, \frac{d}{t\_0}, b \cdot \frac{c}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+153}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{b + \frac{d}{c} \cdot \frac{1}{\frac{-1}{a}}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma (- a) (/ d t_0) (* b (/ c t_0))))
        (t_2 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -4.2e+153)
     t_2
     (if (<= d -2.1e-141)
       t_1
       (if (<= d 2.7e-138)
         (/ (+ b (* (/ d c) (/ 1.0 (/ -1.0 a)))) c)
         (if (<= d 4.5e+104) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(-a, (d / t_0), (b * (c / t_0)));
	double t_2 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -4.2e+153) {
		tmp = t_2;
	} else if (d <= -2.1e-141) {
		tmp = t_1;
	} else if (d <= 2.7e-138) {
		tmp = (b + ((d / c) * (1.0 / (-1.0 / a)))) / c;
	} else if (d <= 4.5e+104) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(-a), Float64(d / t_0), Float64(b * Float64(c / t_0)))
	t_2 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -4.2e+153)
		tmp = t_2;
	elseif (d <= -2.1e-141)
		tmp = t_1;
	elseif (d <= 2.7e-138)
		tmp = Float64(Float64(b + Float64(Float64(d / c) * Float64(1.0 / Float64(-1.0 / a)))) / c);
	elseif (d <= 4.5e+104)
		tmp = t_1;
	else
		tmp = t_2;
	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[((-a) * N[(d / t$95$0), $MachinePrecision] + N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.2e+153], t$95$2, If[LessEqual[d, -2.1e-141], t$95$1, If[LessEqual[d, 2.7e-138], N[(N[(b + N[(N[(d / c), $MachinePrecision] * N[(1.0 / N[(-1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+104], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.20000000000000033e153 or 4.4999999999999998e104 < d

   1. Initial program 35.4%

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -4.20000000000000033e153 < d < -2.0999999999999999e-141 or 2.70000000000000029e-138 < d < 4.4999999999999998e104

   1. Initial program 78.4%

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

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 84.2

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

   4. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 87.7

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-fma.f64 87.7

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

   6. Applied rewrites 87.7%

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

   if -2.0999999999999999e-141 < d < 2.70000000000000029e-138

   1. Initial program 71.7%

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

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

      7. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

   7. Applied rewrites 92.7%

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

   9. Applied rewrites 94.0%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.1 \cdot 10^{-141}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-138}:\\ \;\;\;\;\frac{b + \frac{d}{c} \cdot \frac{1}{\frac{-1}{a}}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{t\_0}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ (- b (* d (/ a c))) c)))
   (if (<= c -6.2e+59)
     t_1
     (if (<= c -3.55e-103)
       (/ (fma c b (* d (- a))) t_0)
       (if (<= c 5e-161)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 7.5e+106) (/ (- (* b c) (* d a)) t_0) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6.2e+59) {
		tmp = t_1;
	} else if (c <= -3.55e-103) {
		tmp = fma(c, b, (d * -a)) / t_0;
	} else if (c <= 5e-161) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 7.5e+106) {
		tmp = ((b * c) - (d * a)) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -6.2e+59)
		tmp = t_1;
	elseif (c <= -3.55e-103)
		tmp = Float64(fma(c, b, Float64(d * Float64(-a))) / t_0);
	elseif (c <= 5e-161)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 7.5e+106)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.2e+59], t$95$1, If[LessEqual[c, -3.55e-103], N[(N[(c * b + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[c, 5e-161], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.5e+106], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.20000000000000029e59 or 7.50000000000000058e106 < c

   1. Initial program 38.9%

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

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

      7. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 82.1%

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

   if -6.20000000000000029e59 < c < -3.55000000000000023e-103

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. lift-*.f64 N/A

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

      9. lower-neg.f64 84.1

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

   6. Applied rewrites 84.1%

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

   if -3.55000000000000023e-103 < c < 4.9999999999999999e-161

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   7. 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. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 94.9

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

   8. Applied rewrites 94.9%

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

   if 4.9999999999999999e-161 < c < 7.50000000000000058e106

   1. Initial program 86.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.3

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

   4. Applied rewrites 86.3%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (fma c c (* d d))))
        (t_1 (/ (- b (* d (/ a c))) c)))
   (if (<= c -6.2e+59)
     t_1
     (if (<= c -3.55e-103)
       t_0
       (if (<= c 5e-161)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 7.5e+106) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / fma(c, c, (d * d));
	double t_1 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6.2e+59) {
		tmp = t_1;
	} else if (c <= -3.55e-103) {
		tmp = t_0;
	} else if (c <= 5e-161) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 7.5e+106) {
		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)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(Float64(b - Float64(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -6.2e+59)
		tmp = t_1;
	elseif (c <= -3.55e-103)
		tmp = t_0;
	elseif (c <= 5e-161)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 7.5e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.2e+59], t$95$1, If[LessEqual[c, -3.55e-103], t$95$0, If[LessEqual[c, 5e-161], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.5e+106], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.20000000000000029e59 or 7.50000000000000058e106 < c

   1. Initial program 38.9%

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

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

      7. lower-*.f64 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 82.1%

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

   if -6.20000000000000029e59 < c < -3.55000000000000023e-103 or 4.9999999999999999e-161 < c < 7.50000000000000058e106

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.4

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

   4. Applied rewrites 85.4%

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

   if -3.55000000000000023e-103 < c < 4.9999999999999999e-161

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 18.1

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   7. 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. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 94.9

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

   8. Applied rewrites 94.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -3.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - d \cdot a\\ \mathbf{if}\;c \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{t\_0}{d \cdot d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{t\_0}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* d a))))
   (if (<= c -1.32e+154)
     (/ b c)
     (if (<= c -3.1e-86)
       (* b (/ c (fma c c (* d d))))
       (if (<= c 1.55e-108)
         (/ t_0 (* d d))
         (if (<= c 6e+116) (/ 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 <= -1.32e+154) {
		tmp = b / c;
	} else if (c <= -3.1e-86) {
		tmp = b * (c / fma(c, c, (d * d)));
	} else if (c <= 1.55e-108) {
		tmp = t_0 / (d * d);
	} else if (c <= 6e+116) {
		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 <= -1.32e+154)
		tmp = Float64(b / c);
	elseif (c <= -3.1e-86)
		tmp = Float64(b * Float64(c / fma(c, c, Float64(d * d))));
	elseif (c <= 1.55e-108)
		tmp = Float64(t_0 / Float64(d * d));
	elseif (c <= 6e+116)
		tmp = Float64(t_0 / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return 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, -1.32e+154], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.1e-86], N[(b * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.55e-108], N[(t$95$0 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6e+116], N[(t$95$0 / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.31999999999999998e154 or 5.9999999999999997e116 < c

   1. Initial program 26.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -1.31999999999999998e154 < c < -3.09999999999999989e-86

   1. Initial program 75.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 44.9

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

   5. Applied rewrites 44.9%

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

   6. Taylor expanded in b around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 59.5

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

   8. Applied rewrites 59.5%

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

   if -3.09999999999999989e-86 < c < 1.55000000000000007e-108

   1. Initial program 76.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   if 1.55000000000000007e-108 < c < 5.9999999999999997e116

   1. Initial program 84.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 59.0

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

   5. Applied rewrites 59.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-86}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.55 \cdot 10^{-108}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+116}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;a \cdot \frac{-d}{t\_0}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+23}:\\ \;\;\;\;b \cdot \frac{c}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ a (- d))))
   (if (<= d -4.8e+153)
     t_1
     (if (<= d -6.6e-123)
       (* a (/ (- d) t_0))
       (if (<= d 1.2e-142)
         (/ b c)
         (if (<= d 2.25e+23) (* b (/ c t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = a / -d;
	double tmp;
	if (d <= -4.8e+153) {
		tmp = t_1;
	} else if (d <= -6.6e-123) {
		tmp = a * (-d / t_0);
	} else if (d <= 1.2e-142) {
		tmp = b / c;
	} else if (d <= 2.25e+23) {
		tmp = b * (c / t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -4.8e+153)
		tmp = t_1;
	elseif (d <= -6.6e-123)
		tmp = Float64(a * Float64(Float64(-d) / t_0));
	elseif (d <= 1.2e-142)
		tmp = Float64(b / c);
	elseif (d <= 2.25e+23)
		tmp = Float64(b * Float64(c / t_0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -4.8e+153], t$95$1, If[LessEqual[d, -6.6e-123], N[(a * N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e-142], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.25e+23], N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.79999999999999985e153 or 2.2499999999999999e23 < d

   1. Initial program 45.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 72.4

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

   5. Applied rewrites 72.4%

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

   if -4.79999999999999985e153 < d < -6.6000000000000005e-123

   1. Initial program 75.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 57.0

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

   8. Applied rewrites 57.0%

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

   if -6.6000000000000005e-123 < d < 1.19999999999999994e-142

   1. Initial program 71.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if 1.19999999999999994e-142 < d < 2.2499999999999999e23

   1. Initial program 81.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in b around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.9

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

   8. Applied rewrites 51.9%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -6.6 \cdot 10^{-123}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+23}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 14200000000000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -4.8e+153)
     t_0
     (if (<= d -1.2e-41)
       (* a (/ (- d) (fma c c (* d d))))
       (if (<= d 14200000000000.0) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -4.8e+153) {
		tmp = t_0;
	} else if (d <= -1.2e-41) {
		tmp = a * (-d / fma(c, c, (d * d)));
	} else if (d <= 14200000000000.0) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -4.8e+153)
		tmp = t_0;
	elseif (d <= -1.2e-41)
		tmp = Float64(a * Float64(Float64(-d) / fma(c, c, Float64(d * d))));
	elseif (d <= 14200000000000.0)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -4.8e+153], t$95$0, If[LessEqual[d, -1.2e-41], N[(a * N[((-d) / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 14200000000000.0], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.79999999999999985e153 or 1.42e13 < d

   1. Initial program 46.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   if -4.79999999999999985e153 < d < -1.20000000000000011e-41

   1. Initial program 76.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 23.1

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

   5. Applied rewrites 23.1%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.9

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

   8. Applied rewrites 60.9%

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

   if -1.20000000000000011e-41 < d < 1.42e13

   1. Initial program 73.8%

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

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

      7. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -1.2 \cdot 10^{-41}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 14200000000000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 14200000000000:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -4.8e+153)
     t_0
     (if (<= d -2.2e-28)
       (* a (/ (- d) (fma c c (* d d))))
       (if (<= d 14200000000000.0) (/ (- b (* d (/ a c))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -4.8e+153) {
		tmp = t_0;
	} else if (d <= -2.2e-28) {
		tmp = a * (-d / fma(c, c, (d * d)));
	} else if (d <= 14200000000000.0) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -4.8e+153)
		tmp = t_0;
	elseif (d <= -2.2e-28)
		tmp = Float64(a * Float64(Float64(-d) / fma(c, c, Float64(d * d))));
	elseif (d <= 14200000000000.0)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -4.8e+153], t$95$0, If[LessEqual[d, -2.2e-28], N[(a * N[((-d) / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 14200000000000.0], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.79999999999999985e153 or 1.42e13 < d

   1. Initial program 46.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 71.8

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

   5. Applied rewrites 71.8%

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

   if -4.79999999999999985e153 < d < -2.19999999999999996e-28

   1. Initial program 75.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 24.0

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

   5. Applied rewrites 24.0%

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

   6. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   if -2.19999999999999996e-28 < d < 1.42e13

   1. Initial program 74.2%

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

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

      7. lower-*.f64 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 77.2%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 14200000000000:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{-142}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+23}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -2.1e-6)
     t_0
     (if (<= d 1.2e-142)
       (/ b c)
       (if (<= d 2.25e+23) (* b (/ c (fma c c (* d d)))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -2.1e-6) {
		tmp = t_0;
	} else if (d <= 1.2e-142) {
		tmp = b / c;
	} else if (d <= 2.25e+23) {
		tmp = b * (c / fma(c, c, (d * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -2.1e-6)
		tmp = t_0;
	elseif (d <= 1.2e-142)
		tmp = Float64(b / c);
	elseif (d <= 2.25e+23)
		tmp = Float64(b * Float64(c / fma(c, c, Float64(d * d))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -2.1e-6], t$95$0, If[LessEqual[d, 1.2e-142], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.25e+23], N[(b * N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.0999999999999998e-6 or 2.2499999999999999e23 < d

   1. Initial program 55.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 63.1

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

   5. Applied rewrites 63.1%

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

   if -2.0999999999999998e-6 < d < 1.19999999999999994e-142

   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 c around inf

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

      1. lower-/.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if 1.19999999999999994e-142 < d < 2.2499999999999999e23

   1. Initial program 81.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 34.7

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in b around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.9

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

   8. Applied rewrites 51.9%

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

Alternative 9: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -6.7 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* d (/ a c))) c)))
   (if (<= c -6.7e+24)
     t_0
     (if (<= c 1.45e-30) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (d * (a / c))) / c;
	double tmp;
	if (c <= -6.7e+24) {
		tmp = t_0;
	} else if (c <= 1.45e-30) {
		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 - (d * (a / c))) / c
    if (c <= (-6.7d+24)) then
        tmp = t_0
    else if (c <= 1.45d-30) 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 - (d * (a / c))) / c;
	double tmp;
	if (c <= -6.7e+24) {
		tmp = t_0;
	} else if (c <= 1.45e-30) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (d * (a / c))) / c
	tmp = 0
	if c <= -6.7e+24:
		tmp = t_0
	elif c <= 1.45e-30:
		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(d * Float64(a / c))) / c)
	tmp = 0.0
	if (c <= -6.7e+24)
		tmp = t_0;
	elseif (c <= 1.45e-30)
		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 - (d * (a / c))) / c;
	tmp = 0.0;
	if (c <= -6.7e+24)
		tmp = t_0;
	elseif (c <= 1.45e-30)
		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[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -6.7e+24], t$95$0, If[LessEqual[c, 1.45e-30], 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 < -6.6999999999999999e24 or 1.44999999999999995e-30 < c

   1. Initial program 52.2%

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

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

      7. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 76.4%

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

   if -6.6999999999999999e24 < c < 1.44999999999999995e-30

   1. Initial program 79.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 20.3

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

   5. Applied rewrites 20.3%

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

   6. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   7. 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. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 79.5

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

   8. Applied rewrites 79.5%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 14200000000000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -2.1e-6) t_0 (if (<= d 14200000000000.0) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -2.1e-6) {
		tmp = t_0;
	} else if (d <= 14200000000000.0) {
		tmp = b / 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 = a / -d
    if (d <= (-2.1d-6)) then
        tmp = t_0
    else if (d <= 14200000000000.0d0) then
        tmp = b / 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 = a / -d;
	double tmp;
	if (d <= -2.1e-6) {
		tmp = t_0;
	} else if (d <= 14200000000000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -2.1e-6:
		tmp = t_0
	elif d <= 14200000000000.0:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -2.1e-6)
		tmp = t_0;
	elseif (d <= 14200000000000.0)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -2.1e-6)
		tmp = t_0;
	elseif (d <= 14200000000000.0)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -2.1e-6], t$95$0, If[LessEqual[d, 14200000000000.0], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.0999999999999998e-6 or 1.42e13 < d

   1. Initial program 55.9%

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

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

      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 62.9

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

   5. Applied rewrites 62.9%

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

   if -2.0999999999999998e-6 < d < 1.42e13

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 59.8

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

   5. Applied rewrites 59.8%

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

Alternative 11: 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 65.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 40.6

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

5. Applied rewrites 40.6%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024221 
(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))))