Complex division, imag part

Percentage Accurate: 61.5% → 80.5%

Time: 9.3s

Alternatives: 12

Speedup: 0.9×

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: 61.5% 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: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{1}{\frac{\frac{c}{d}}{a}}}{c}\\ \mathbf{if}\;c \leq -1.2 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ 1.0 (/ (/ c d) a))) c)))
   (if (<= c -1.2e+108)
     t_0
     (if (<= c -7.5e-126)
       (/ (fma (- d) a (* b c)) (fma d d (* c c)))
       (if (<= c 6000000.0) (/ (fma (/ c d) b (- a)) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (1.0 / ((c / d) / a))) / c;
	double tmp;
	if (c <= -1.2e+108) {
		tmp = t_0;
	} else if (c <= -7.5e-126) {
		tmp = fma(-d, a, (b * c)) / fma(d, d, (c * c));
	} else if (c <= 6000000.0) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(1.0 / Float64(Float64(c / d) / a))) / c)
	tmp = 0.0
	if (c <= -1.2e+108)
		tmp = t_0;
	elseif (c <= -7.5e-126)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / fma(d, d, Float64(c * c)));
	elseif (c <= 6000000.0)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(1.0 / N[(N[(c / d), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.2e+108], t$95$0, If[LessEqual[c, -7.5e-126], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6000000.0], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.20000000000000009e108 or 6e6 < c

   1. Initial program 37.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 81.0%

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

   if -1.20000000000000009e108 < c < -7.49999999999999976e-126

   1. Initial program 80.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 80.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if -7.49999999999999976e-126 < c < 6e6

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

   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. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 89.7

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

   8. Applied rewrites 89.7%

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

Alternative 2: 72.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d}\\ t_1 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (* d d))) (t_1 (/ (- a) d)))
   (if (<= d -4e+121)
     t_1
     (if (<= d -2.9e+59)
       (* (/ b (fma c c (* d d))) c)
       (if (<= d -3.1e-22)
         t_0
         (if (<= d 8e+50)
           (/ (- b (/ (* d a) c)) c)
           (if (<= d 3.8e+76) t_0 (if (<= d 6.1e+88) (/ b c) t_1))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / (d * d);
	double t_1 = -a / d;
	double tmp;
	if (d <= -4e+121) {
		tmp = t_1;
	} else if (d <= -2.9e+59) {
		tmp = (b / fma(c, c, (d * d))) * c;
	} else if (d <= -3.1e-22) {
		tmp = t_0;
	} else if (d <= 8e+50) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 3.8e+76) {
		tmp = t_0;
	} else if (d <= 6.1e+88) {
		tmp = b / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d))
	t_1 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -4e+121)
		tmp = t_1;
	elseif (d <= -2.9e+59)
		tmp = Float64(Float64(b / fma(c, c, Float64(d * d))) * c);
	elseif (d <= -3.1e-22)
		tmp = t_0;
	elseif (d <= 8e+50)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 3.8e+76)
		tmp = t_0;
	elseif (d <= 6.1e+88)
		tmp = Float64(b / c);
	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[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -4e+121], t$95$1, If[LessEqual[d, -2.9e+59], N[(N[(b / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[d, -3.1e-22], t$95$0, If[LessEqual[d, 8e+50], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.8e+76], t$95$0, If[LessEqual[d, 6.1e+88], N[(b / c), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.00000000000000015e121 or 6.0999999999999998e88 < d

   1. Initial program 36.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   if -4.00000000000000015e121 < d < -2.89999999999999991e59

   1. Initial program 62.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

   if -2.89999999999999991e59 < d < -3.10000000000000013e-22 or 8.0000000000000006e50 < d < 3.80000000000000024e76

   1. Initial program 85.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 75.1

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

   5. Applied rewrites 75.1%

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

   if -3.10000000000000013e-22 < d < 8.0000000000000006e50

   1. Initial program 72.6%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 83.8

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

   5. Applied rewrites 83.8%

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

   if 3.80000000000000024e76 < d < 6.0999999999999998e88

   1. Initial program 33.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+121}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.9 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 6.1 \cdot 10^{+88}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ t_1 := \frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)) (t_1 (/ (fma (/ c d) b (- a)) d)))
   (if (<= d -1.45e+100)
     (/ (fma (/ b d) c (- a)) d)
     (if (<= d -1.22e+72)
       t_0
       (if (<= d -3.1e-22) t_1 (if (<= d 7.5e+50) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double t_1 = fma((c / d), b, -a) / d;
	double tmp;
	if (d <= -1.45e+100) {
		tmp = fma((b / d), c, -a) / d;
	} else if (d <= -1.22e+72) {
		tmp = t_0;
	} else if (d <= -3.1e-22) {
		tmp = t_1;
	} else if (d <= 7.5e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	t_1 = Float64(fma(Float64(c / d), b, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.45e+100)
		tmp = Float64(fma(Float64(b / d), c, Float64(-a)) / d);
	elseif (d <= -1.22e+72)
		tmp = t_0;
	elseif (d <= -3.1e-22)
		tmp = t_1;
	elseif (d <= 7.5e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.45e+100], N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.22e+72], t$95$0, If[LessEqual[d, -3.1e-22], t$95$1, If[LessEqual[d, 7.5e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.45e100

   1. Initial program 41.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 11.8

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

   5. Applied rewrites 11.8%

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

   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. unsub-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. unsub-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. *-commutative N/A

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

      17. lower-*.f64 90.6

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

   8. Applied rewrites 90.6%

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

   10. Applied rewrites 92.8%

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

   if -1.45e100 < d < -1.2200000000000001e72 or -3.10000000000000013e-22 < d < 7.4999999999999999e50

   1. Initial program 72.0%

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

   3. Taylor expanded in c around inf

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

      1. 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 84.1

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

   5. Applied rewrites 84.1%

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

   if -1.2200000000000001e72 < d < -3.10000000000000013e-22 or 7.4999999999999999e50 < d

   1. Initial program 51.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 58.6

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

   5. Applied rewrites 58.6%

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

   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. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 74.5

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

   8. Applied rewrites 74.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)) (t_1 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -1.45e+100)
     t_1
     (if (<= d -1.22e+72)
       t_0
       (if (<= d -3.1e-22)
         (/ (- (/ (* b c) d) a) d)
         (if (<= d 7.5e+50) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double t_1 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -1.45e+100) {
		tmp = t_1;
	} else if (d <= -1.22e+72) {
		tmp = t_0;
	} else if (d <= -3.1e-22) {
		tmp = (((b * c) / d) - a) / d;
	} else if (d <= 7.5e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	t_1 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.45e+100)
		tmp = t_1;
	elseif (d <= -1.22e+72)
		tmp = t_0;
	elseif (d <= -3.1e-22)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (d <= 7.5e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.45e+100], t$95$1, If[LessEqual[d, -1.22e+72], t$95$0, If[LessEqual[d, -3.1e-22], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 7.5e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.45e100 or 7.4999999999999999e50 < d

   1. Initial program 40.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 19.9

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

   5. Applied rewrites 19.9%

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

   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. unsub-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. unsub-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. *-commutative N/A

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

      17. lower-*.f64 81.1

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

   8. Applied rewrites 81.1%

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

   10. Applied rewrites 83.3%

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

   if -1.45e100 < d < -1.2200000000000001e72 or -3.10000000000000013e-22 < d < 7.4999999999999999e50

   1. Initial program 72.0%

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

   3. Taylor expanded in c around inf

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

      1. 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 84.1

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

   5. Applied rewrites 84.1%

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

   if -1.2200000000000001e72 < d < -3.10000000000000013e-22

   1. Initial program 81.5%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ t_1 := \frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)) (t_1 (/ (- (/ (* b c) d) a) d)))
   (if (<= d -1.45e+100)
     t_1
     (if (<= d -1.22e+72)
       t_0
       (if (<= d -3.1e-22) t_1 (if (<= d 7.5e+50) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double t_1 = (((b * c) / d) - a) / d;
	double tmp;
	if (d <= -1.45e+100) {
		tmp = t_1;
	} else if (d <= -1.22e+72) {
		tmp = t_0;
	} else if (d <= -3.1e-22) {
		tmp = t_1;
	} else if (d <= 7.5e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b - ((d * a) / c)) / c
    t_1 = (((b * c) / d) - a) / d
    if (d <= (-1.45d+100)) then
        tmp = t_1
    else if (d <= (-1.22d+72)) then
        tmp = t_0
    else if (d <= (-3.1d-22)) then
        tmp = t_1
    else if (d <= 7.5d+50) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double t_1 = (((b * c) / d) - a) / d;
	double tmp;
	if (d <= -1.45e+100) {
		tmp = t_1;
	} else if (d <= -1.22e+72) {
		tmp = t_0;
	} else if (d <= -3.1e-22) {
		tmp = t_1;
	} else if (d <= 7.5e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((d * a) / c)) / c
	t_1 = (((b * c) / d) - a) / d
	tmp = 0
	if d <= -1.45e+100:
		tmp = t_1
	elif d <= -1.22e+72:
		tmp = t_0
	elif d <= -3.1e-22:
		tmp = t_1
	elif d <= 7.5e+50:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	t_1 = Float64(Float64(Float64(Float64(b * c) / d) - a) / d)
	tmp = 0.0
	if (d <= -1.45e+100)
		tmp = t_1;
	elseif (d <= -1.22e+72)
		tmp = t_0;
	elseif (d <= -3.1e-22)
		tmp = t_1;
	elseif (d <= 7.5e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((d * a) / c)) / c;
	t_1 = (((b * c) / d) - a) / d;
	tmp = 0.0;
	if (d <= -1.45e+100)
		tmp = t_1;
	elseif (d <= -1.22e+72)
		tmp = t_0;
	elseif (d <= -3.1e-22)
		tmp = t_1;
	elseif (d <= 7.5e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.45e+100], t$95$1, If[LessEqual[d, -1.22e+72], t$95$0, If[LessEqual[d, -3.1e-22], t$95$1, If[LessEqual[d, 7.5e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.45e100 or -1.2200000000000001e72 < d < -3.10000000000000013e-22 or 7.4999999999999999e50 < d

   1. Initial program 47.3%

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

   3. Taylor expanded in c around 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 79.7

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

   5. Applied rewrites 79.7%

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

   if -1.45e100 < d < -1.2200000000000001e72 or -3.10000000000000013e-22 < d < 7.4999999999999999e50

   1. Initial program 72.0%

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

   3. Taylor expanded in c around inf

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

      1. 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 84.1

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

   5. Applied rewrites 84.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.22 \cdot 10^{+72}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= c -2.1e+107)
     (/ b c)
     (if (<= c -2.1e-127)
       (/ (* b c) (fma d d (* c c)))
       (if (<= c 1.22e-129)
         t_0
         (if (<= c 4.5e-33)
           (/ (- (* b c) (* d a)) (* d d))
           (if (<= c 11500000.0) t_0 (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -2.1e+107) {
		tmp = b / c;
	} else if (c <= -2.1e-127) {
		tmp = (b * c) / fma(d, d, (c * c));
	} else if (c <= 1.22e-129) {
		tmp = t_0;
	} else if (c <= 4.5e-33) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (c <= 11500000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (c <= -2.1e+107)
		tmp = Float64(b / c);
	elseif (c <= -2.1e-127)
		tmp = Float64(Float64(b * c) / fma(d, d, Float64(c * c)));
	elseif (c <= 1.22e-129)
		tmp = t_0;
	elseif (c <= 4.5e-33)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (c <= 11500000.0)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[c, -2.1e+107], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.1e-127], N[(N[(b * c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.22e-129], t$95$0, If[LessEqual[c, 4.5e-33], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 11500000.0], t$95$0, N[(b / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.1e107 or 1.15e7 < c

   1. Initial program 37.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -2.1e107 < c < -2.1000000000000001e-127

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 81.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   if -2.1000000000000001e-127 < c < 1.21999999999999999e-129 or 4.49999999999999991e-33 < c < 1.15e7

   1. Initial program 65.0%

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

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

   5. Applied rewrites 72.4%

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

   if 1.21999999999999999e-129 < c < 4.49999999999999991e-33

   1. Initial program 85.5%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 1.22 \cdot 10^{-129}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.3e+108)
   (/ b c)
   (if (<= c -7.5e-126)
     (/ (fma (- d) a (* b c)) (fma d d (* c c)))
     (if (<= c 6000000.0)
       (/ (fma (/ c d) b (- a)) d)
       (/ (- b (/ (* d a) c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.3e+108) {
		tmp = b / c;
	} else if (c <= -7.5e-126) {
		tmp = fma(-d, a, (b * c)) / fma(d, d, (c * c));
	} else if (c <= 6000000.0) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.3e+108)
		tmp = Float64(b / c);
	elseif (c <= -7.5e-126)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / fma(d, d, Float64(c * c)));
	elseif (c <= 6000000.0)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.3e+108], N[(b / c), $MachinePrecision], If[LessEqual[c, -7.5e-126], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6000000.0], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.2999999999999999e108

   1. Initial program 23.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -2.2999999999999999e108 < c < -7.49999999999999976e-126

   1. Initial program 80.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 80.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 80.9

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

   4. Applied rewrites 80.9%

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

   if -7.49999999999999976e-126 < c < 6e6

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

   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. *-commutative N/A

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

      13. associate-*l/ N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 89.7

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

   8. Applied rewrites 89.7%

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

   if 6e6 < 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

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.6

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

   5. Applied rewrites 79.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -7.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 6000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e+107)
   (/ b c)
   (if (<= c -2.1e-127)
     (/ (* b c) (fma d d (* c c)))
     (if (<= c 11500000.0) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e+107) {
		tmp = b / c;
	} else if (c <= -2.1e-127) {
		tmp = (b * c) / fma(d, d, (c * c));
	} else if (c <= 11500000.0) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e+107)
		tmp = Float64(b / c);
	elseif (c <= -2.1e-127)
		tmp = Float64(Float64(b * c) / fma(d, d, Float64(c * c)));
	elseif (c <= 11500000.0)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e+107], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.1e-127], N[(N[(b * c), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 11500000.0], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.1e107 or 1.15e7 < c

   1. Initial program 37.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -2.1e107 < c < -2.1000000000000001e-127

   1. Initial program 81.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 81.3

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.3

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

   4. Applied rewrites 81.3%

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

   5. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 63.1

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

   7. Applied rewrites 63.1%

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

   if -2.1000000000000001e-127 < c < 1.15e7

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+107}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{+112}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-127}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.2e+112)
   (/ b c)
   (if (<= c -2.1e-127)
     (* (/ c (fma c c (* d d))) b)
     (if (<= c 11500000.0) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e+112) {
		tmp = b / c;
	} else if (c <= -2.1e-127) {
		tmp = (c / fma(c, c, (d * d))) * b;
	} else if (c <= 11500000.0) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.2e+112)
		tmp = Float64(b / c);
	elseif (c <= -2.1e-127)
		tmp = Float64(Float64(c / fma(c, c, Float64(d * d))) * b);
	elseif (c <= 11500000.0)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.2e+112], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.1e-127], N[(N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 11500000.0], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.19999999999999965e112 or 1.15e7 < c

   1. Initial program 36.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -6.19999999999999965e112 < c < -2.1000000000000001e-127

   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 41.2

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

   5. Applied rewrites 41.2%

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

   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 61.9

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

   8. Applied rewrites 61.9%

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

   if -2.1000000000000001e-127 < c < 1.15e7

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

Alternative 10: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot c\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.6e+114)
   (/ b c)
   (if (<= c -2.2e-127)
     (* (/ b (fma c c (* d d))) c)
     (if (<= c 11500000.0) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.6e+114) {
		tmp = b / c;
	} else if (c <= -2.2e-127) {
		tmp = (b / fma(c, c, (d * d))) * c;
	} else if (c <= 11500000.0) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.6e+114)
		tmp = Float64(b / c);
	elseif (c <= -2.2e-127)
		tmp = Float64(Float64(b / fma(c, c, Float64(d * d))) * c);
	elseif (c <= 11500000.0)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.6e+114], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.2e-127], N[(N[(b / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 11500000.0], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.6e114 or 1.15e7 < c

   1. Initial program 35.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -2.6e114 < c < -2.2000000000000001e-127

   1. Initial program 81.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 59.2

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

   5. Applied rewrites 59.2%

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

   if -2.2000000000000001e-127 < c < 1.15e7

   1. Initial program 68.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

Alternative 11: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -4.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 11500000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= c -5.5e+80)
     (/ b c)
     (if (<= c -5.5e+25)
       t_0
       (if (<= c -4.5e-127) (/ b c) (if (<= c 11500000.0) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -5.5e+80) {
		tmp = b / c;
	} else if (c <= -5.5e+25) {
		tmp = t_0;
	} else if (c <= -4.5e-127) {
		tmp = b / c;
	} else if (c <= 11500000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (c <= (-5.5d+80)) then
        tmp = b / c
    else if (c <= (-5.5d+25)) then
        tmp = t_0
    else if (c <= (-4.5d-127)) then
        tmp = b / c
    else if (c <= 11500000.0d0) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (c <= -5.5e+80) {
		tmp = b / c;
	} else if (c <= -5.5e+25) {
		tmp = t_0;
	} else if (c <= -4.5e-127) {
		tmp = b / c;
	} else if (c <= 11500000.0) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if c <= -5.5e+80:
		tmp = b / c
	elif c <= -5.5e+25:
		tmp = t_0
	elif c <= -4.5e-127:
		tmp = b / c
	elif c <= 11500000.0:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (c <= -5.5e+80)
		tmp = Float64(b / c);
	elseif (c <= -5.5e+25)
		tmp = t_0;
	elseif (c <= -4.5e-127)
		tmp = Float64(b / c);
	elseif (c <= 11500000.0)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (c <= -5.5e+80)
		tmp = b / c;
	elseif (c <= -5.5e+25)
		tmp = t_0;
	elseif (c <= -4.5e-127)
		tmp = b / c;
	elseif (c <= 11500000.0)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[c, -5.5e+80], N[(b / c), $MachinePrecision], If[LessEqual[c, -5.5e+25], t$95$0, If[LessEqual[c, -4.5e-127], N[(b / c), $MachinePrecision], If[LessEqual[c, 11500000.0], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if c < -5.49999999999999967e80 or -5.50000000000000018e25 < c < -4.4999999999999999e-127 or 1.15e7 < c

   1. Initial program 51.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if -5.49999999999999967e80 < c < -5.50000000000000018e25 or -4.4999999999999999e-127 < c < 1.15e7

   1. Initial program 68.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

Alternative 12: 42.8% 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 59.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 41.3

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

5. Applied rewrites 41.3%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024259 
(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))))