Complex division, imag part

Percentage Accurate: 62.3% → 81.8%

Time: 7.3s

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: 62.3% 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: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ t_1 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_1}, b, \frac{a}{t\_1} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)) (t_1 (fma d d (* c c))))
   (if (<= d -9.2e+14)
     t_0
     (if (<= d 7e-145)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 7.8e+127) (fma (/ c t_1) b (* (/ a t_1) (- d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double t_1 = fma(d, d, (c * c));
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 7e-145) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 7.8e+127) {
		tmp = fma((c / t_1), b, ((a / t_1) * -d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	t_1 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 7e-145)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 7.8e+127)
		tmp = fma(Float64(c / t_1), b, Float64(Float64(a / t_1) * Float64(-d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, Block[{t$95$1 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9.2e+14], t$95$0, If[LessEqual[d, 7e-145], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.8e+127], N[(N[(c / t$95$1), $MachinePrecision] * b + N[(N[(a / t$95$1), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.2e14 or 7.79999999999999962e127 < d

   1. Initial program 39.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 83.6

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 83.6%

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

   if -9.2e14 < d < 6.99999999999999994e-145

   1. Initial program 69.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if 6.99999999999999994e-145 < d < 7.79999999999999962e127

   1. Initial program 83.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 87.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{-145}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{-145}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, a \cdot d\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -9.2e+14)
     t_0
     (if (<= d 1.48e-145)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 7.8e+127)
         (/ (fma (- b) c (* a d)) (- (fma d d (* c c))))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 1.48e-145) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 7.8e+127) {
		tmp = fma(-b, c, (a * d)) / -fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 1.48e-145)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 7.8e+127)
		tmp = Float64(fma(Float64(-b), c, Float64(a * d)) / Float64(-fma(d, d, Float64(c * c))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.2e+14], t$95$0, If[LessEqual[d, 1.48e-145], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.8e+127], N[(N[((-b) * c + N[(a * d), $MachinePrecision]), $MachinePrecision] / (-N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.2e14 or 7.79999999999999962e127 < d

   1. Initial program 39.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 83.6

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 83.6%

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

   if -9.2e14 < d < 1.47999999999999995e-145

   1. Initial program 69.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if 1.47999999999999995e-145 < d < 7.79999999999999962e127

   1. Initial program 83.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. frac-2neg N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-neg-in N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-neg.f64 83.5

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 83.5

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

   4. Applied rewrites 83.5%

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

Alternative 3: 81.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{-145}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -9.2e+14)
     t_0
     (if (<= d 1.48e-145)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 7.8e+127) (/ (- (* c b) (* a d)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 1.48e-145) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 7.8e+127) {
		tmp = ((c * b) - (a * d)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 1.48e-145)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 7.8e+127)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.2e+14], t$95$0, If[LessEqual[d, 1.48e-145], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.8e+127], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.2e14 or 7.79999999999999962e127 < d

   1. Initial program 39.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 83.6

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

   8. Applied rewrites 83.6%

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

   10. Applied rewrites 83.6%

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

   if -9.2e14 < d < 1.47999999999999995e-145

   1. Initial program 69.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if 1.47999999999999995e-145 < d < 7.79999999999999962e127

   1. Initial program 83.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq 1.48 \cdot 10^{-145}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 7.8 \cdot 10^{+127}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{d}{t\_0} \cdot \left(-a\right)\\ \mathbf{elif}\;c \leq 10^{+137}:\\ \;\;\;\;\frac{c}{t\_0} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))))
   (if (<= c -5.5e-36)
     (/ b c)
     (if (<= c 8.5e-173)
       (/ (- a) d)
       (if (<= c 1.4e-28)
         (* (/ d t_0) (- a))
         (if (<= c 1e+137) (* (/ c t_0) b) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double tmp;
	if (c <= -5.5e-36) {
		tmp = b / c;
	} else if (c <= 8.5e-173) {
		tmp = -a / d;
	} else if (c <= 1.4e-28) {
		tmp = (d / t_0) * -a;
	} else if (c <= 1e+137) {
		tmp = (c / t_0) * b;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	tmp = 0.0
	if (c <= -5.5e-36)
		tmp = Float64(b / c);
	elseif (c <= 8.5e-173)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1.4e-28)
		tmp = Float64(Float64(d / t_0) * Float64(-a));
	elseif (c <= 1e+137)
		tmp = Float64(Float64(c / t_0) * b);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.5e-36], N[(b / c), $MachinePrecision], If[LessEqual[c, 8.5e-173], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1.4e-28], N[(N[(d / t$95$0), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[c, 1e+137], N[(N[(c / t$95$0), $MachinePrecision] * b), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.49999999999999984e-36 or 1e137 < c

   1. Initial program 50.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -5.49999999999999984e-36 < c < 8.4999999999999996e-173

   1. Initial program 67.2%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 8.4999999999999996e-173 < c < 1.3999999999999999e-28

   1. Initial program 85.8%

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

   3. Taylor expanded in b around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if 1.3999999999999999e-28 < c < 1e137

   1. Initial program 60.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in b around inf

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

      1. *-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. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 57.7

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

   8. Applied rewrites 57.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-173}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;c \leq 10^{+137}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -8.2 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 240000000000:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-d\right) \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -8.2e+67)
     t_0
     (if (<= d 240000000000.0)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 5.5e+148) (/ (fma c b (* (- d) a)) (* d d)) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -8.2e+67) {
		tmp = t_0;
	} else if (d <= 240000000000.0) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 5.5e+148) {
		tmp = fma(c, b, (-d * a)) / (d * d);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -8.2e+67)
		tmp = t_0;
	elseif (d <= 240000000000.0)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 5.5e+148)
		tmp = Float64(fma(c, b, Float64(Float64(-d) * a)) / 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, -8.2e+67], t$95$0, If[LessEqual[d, 240000000000.0], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.5e+148], N[(N[(c * b + N[((-d) * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -8.19999999999999959e67 or 5.5e148 < d

   1. Initial program 35.1%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -8.19999999999999959e67 < d < 2.4e11

   1. Initial program 71.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if 2.4e11 < d < 5.5e148

   1. Initial program 80.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 62.8

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

   5. Applied rewrites 62.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. lower-*.f64 62.8

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

   7. Applied rewrites 62.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 240000000000:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-d\right) \cdot a\right)}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 10^{+137}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e-36)
   (/ b c)
   (if (<= c 8e-8)
     (/ (- a) d)
     (if (<= c 1e+137) (* (/ c (fma c c (* d d))) b) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e-36) {
		tmp = b / c;
	} else if (c <= 8e-8) {
		tmp = -a / d;
	} else if (c <= 1e+137) {
		tmp = (c / fma(c, c, (d * d))) * b;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e-36)
		tmp = Float64(b / c);
	elseif (c <= 8e-8)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1e+137)
		tmp = Float64(Float64(c / fma(c, c, Float64(d * d))) * b);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e-36], N[(b / c), $MachinePrecision], If[LessEqual[c, 8e-8], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1e+137], N[(N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.49999999999999984e-36 or 1e137 < c

   1. Initial program 50.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -5.49999999999999984e-36 < c < 8.0000000000000002e-8

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 8.0000000000000002e-8 < c < 1e137

   1. Initial program 62.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 39.0

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

   5. Applied rewrites 39.0%

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

   6. Taylor expanded in b around inf

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

      1. *-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. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

Alternative 7: 63.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+136}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e-36)
   (/ b c)
   (if (<= c 8e-8)
     (/ (- a) d)
     (if (<= c 5.4e+136) (* (/ b (fma c c (* d d))) c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e-36) {
		tmp = b / c;
	} else if (c <= 8e-8) {
		tmp = -a / d;
	} else if (c <= 5.4e+136) {
		tmp = (b / fma(c, c, (d * d))) * c;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e-36)
		tmp = Float64(b / c);
	elseif (c <= 8e-8)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 5.4e+136)
		tmp = Float64(Float64(b / fma(c, c, Float64(d * d))) * c);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e-36], N[(b / c), $MachinePrecision], If[LessEqual[c, 8e-8], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 5.4e+136], N[(N[(b / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.49999999999999984e-36 or 5.4000000000000003e136 < c

   1. Initial program 50.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if -5.49999999999999984e-36 < c < 8.0000000000000002e-8

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 8.0000000000000002e-8 < c < 5.4000000000000003e136

   1. Initial program 62.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 60.0

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

   5. Applied rewrites 60.0%

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

Alternative 8: 78.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 240000000000:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -9.2e+14)
     t_0
     (if (<= d 240000000000.0) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 240000000000.0) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 240000000000.0)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.2e+14], t$95$0, If[LessEqual[d, 240000000000.0], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.2e14 or 2.4e11 < d

   1. Initial program 50.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 81.3

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

   8. Applied rewrites 81.3%

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

   10. Applied rewrites 81.3%

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

   if -9.2e14 < d < 2.4e11

   1. Initial program 71.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

Alternative 9: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 240000000000:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* c b) d) a) d)))
   (if (<= d -9.2e+14)
     t_0
     (if (<= d 240000000000.0) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (((c * b) / d) - a) / d;
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 240000000000.0) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((c * b) / d) - a) / d
    if (d <= (-9.2d+14)) then
        tmp = t_0
    else if (d <= 240000000000.0d0) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (((c * b) / d) - a) / d;
	double tmp;
	if (d <= -9.2e+14) {
		tmp = t_0;
	} else if (d <= 240000000000.0) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (((c * b) / d) - a) / d
	tmp = 0
	if d <= -9.2e+14:
		tmp = t_0
	elif d <= 240000000000.0:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(Float64(c * b) / d) - a) / d)
	tmp = 0.0
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 240000000000.0)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (((c * b) / d) - a) / d;
	tmp = 0.0;
	if (d <= -9.2e+14)
		tmp = t_0;
	elseif (d <= 240000000000.0)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9.2e+14], t$95$0, If[LessEqual[d, 240000000000.0], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.2e14 or 2.4e11 < d

   1. Initial program 50.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      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 76.8

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

   5. Applied rewrites 76.8%

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

   if -9.2e14 < d < 2.4e11

   1. Initial program 71.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 84.8

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

   5. Applied rewrites 84.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 240000000000:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.5e-36) (/ b c) (if (<= c 4.2e+61) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.5e-36) {
		tmp = b / c;
	} else if (c <= 4.2e+61) {
		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 <= (-5.5d-36)) then
        tmp = b / c
    else if (c <= 4.2d+61) 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 <= -5.5e-36) {
		tmp = b / c;
	} else if (c <= 4.2e+61) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.5e-36:
		tmp = b / c
	elif c <= 4.2e+61:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.5e-36)
		tmp = Float64(b / c);
	elseif (c <= 4.2e+61)
		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 <= -5.5e-36)
		tmp = b / c;
	elseif (c <= 4.2e+61)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.5e-36], N[(b / c), $MachinePrecision], If[LessEqual[c, 4.2e+61], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.49999999999999984e-36 or 4.2000000000000002e61 < c

   1. Initial program 52.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -5.49999999999999984e-36 < c < 4.2000000000000002e61

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 62.1

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

   5. Applied rewrites 62.1%

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

Alternative 11: 42.1% 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 60.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 43.9

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

5. Applied rewrites 43.9%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024284 
(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))))