Complex division, imag part

Percentage Accurate: 61.4% → 80.9%

Time: 7.4s

Alternatives: 12

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 80.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \mathbf{if}\;c \leq -1.15 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{t\_0}}{a}, c, \frac{-d}{t\_0}\right) \cdot a\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d)))
        (t_1
         (fma
          (fma
           (- (* (/ a (pow c 4.0)) d) (/ b (pow c 3.0)))
           d
           (/ (/ (- a) c) c))
          d
          (/ b c))))
   (if (<= c -1.15e+138)
     t_1
     (if (<= c -2.5e-148)
       (* (fma (/ (/ b t_0) a) c (/ (- d) t_0)) a)
       (if (<= c 9e-146)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 1.05e+114)
           (/ (fma (- b) c (* d a)) (- (fma d d (* c c))))
           t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = fma(fma((((a / pow(c, 4.0)) * d) - (b / pow(c, 3.0))), d, ((-a / c) / c)), d, (b / c));
	double tmp;
	if (c <= -1.15e+138) {
		tmp = t_1;
	} else if (c <= -2.5e-148) {
		tmp = fma(((b / t_0) / a), c, (-d / t_0)) * a;
	} else if (c <= 9e-146) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.05e+114) {
		tmp = fma(-b, c, (d * a)) / -fma(d, d, (c * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = fma(fma(Float64(Float64(Float64(a / (c ^ 4.0)) * d) - Float64(b / (c ^ 3.0))), d, Float64(Float64(Float64(-a) / c) / c)), d, Float64(b / c))
	tmp = 0.0
	if (c <= -1.15e+138)
		tmp = t_1;
	elseif (c <= -2.5e-148)
		tmp = Float64(fma(Float64(Float64(b / t_0) / a), c, Float64(Float64(-d) / t_0)) * a);
	elseif (c <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 1.05e+114)
		tmp = Float64(fma(Float64(-b), c, Float64(d * a)) / Float64(-fma(d, d, Float64(c * c))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(N[(a / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] - N[(b / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d + N[(N[((-a) / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * d + N[(b / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.15e+138], t$95$1, If[LessEqual[c, -2.5e-148], N[(N[(N[(N[(b / t$95$0), $MachinePrecision] / a), $MachinePrecision] * c + N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[c, 9e-146], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.05e+114], N[(N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision] / (-N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.15000000000000004e138 or 1.05e114 < c

   1. Initial program 32.9%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) + \frac{b}{c}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) \cdot d} + \frac{b}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right), d, \frac{b}{c}\right)} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)} \]

   if -1.15000000000000004e138 < c < -2.4999999999999999e-148

   1. Initial program 70.8%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-+l- N/A

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

      4. unsub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. neg-sub0 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   if -2.4999999999999999e-148 < c < 9.0000000000000001e-146

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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 92.9

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

   5. Applied rewrites 92.9%

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

   if 9.0000000000000001e-146 < c < 1.05e114

   1. Initial program 79.8%

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

      1. lift-/.f64 N/A

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

      2. 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 79.8

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

      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 79.9

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

   4. Applied rewrites 79.9%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)}}{a}, c, \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\right) \cdot a\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{d}, \frac{b}{d}, \frac{-a}{d}\right)\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-30}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{t\_0}}{a}, c, \frac{-d}{t\_0}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (fma (/ c d) (/ b d) (/ (- a) d))))
   (if (<= d -2.8e+53)
     t_1
     (if (<= d -1.3e-144)
       (/ (fma (- b) c (* d a)) (- (fma d d (* c c))))
       (if (<= d 3e-30)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 3.25e+124)
           (* (fma (/ (/ b t_0) a) c (/ (- d) t_0)) a)
           t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = fma((c / d), (b / d), (-a / d));
	double tmp;
	if (d <= -2.8e+53) {
		tmp = t_1;
	} else if (d <= -1.3e-144) {
		tmp = fma(-b, c, (d * a)) / -fma(d, d, (c * c));
	} else if (d <= 3e-30) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 3.25e+124) {
		tmp = fma(((b / t_0) / a), c, (-d / t_0)) * a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = fma(Float64(c / d), Float64(b / d), Float64(Float64(-a) / d))
	tmp = 0.0
	if (d <= -2.8e+53)
		tmp = t_1;
	elseif (d <= -1.3e-144)
		tmp = Float64(fma(Float64(-b), c, Float64(d * a)) / Float64(-fma(d, d, Float64(c * c))));
	elseif (d <= 3e-30)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 3.25e+124)
		tmp = Float64(fma(Float64(Float64(b / t_0) / a), c, Float64(Float64(-d) / t_0)) * a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.8e+53], t$95$1, If[LessEqual[d, -1.3e-144], N[(N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision] / (-N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 3e-30], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.25e+124], N[(N[(N[(N[(b / t$95$0), $MachinePrecision] / a), $MachinePrecision] * c + N[((-d) / t$95$0), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.8e53 or 3.25000000000000004e124 < d

   1. Initial program 40.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 22.0

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

   5. Applied rewrites 22.0%

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

   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. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if -2.8e53 < d < -1.3e-144

   1. Initial program 83.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 83.6

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

      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 83.6

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

   4. Applied rewrites 83.6%

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

   if -1.3e-144 < d < 2.9999999999999999e-30

   1. Initial program 73.3%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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 88.7

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

   5. Applied rewrites 88.7%

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

   if 2.9999999999999999e-30 < d < 3.25000000000000004e124

   1. Initial program 59.6%

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

   3. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-+l- N/A

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

      4. unsub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. neg-sub0 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

   5. Applied rewrites 75.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{b}{d}, \frac{-a}{d}\right)\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-144}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{-30}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.25 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{b}{\mathsf{fma}\left(c, c, d \cdot d\right)}}{a}, c, \frac{-d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{b}{d}, \frac{-a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := b \cdot c - d \cdot a\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{t\_1}{d \cdot d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-149}:\\ \;\;\;\;\frac{t\_1}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (- (* b c) (* d a))))
   (if (<= d -6.8e+53)
     t_0
     (if (<= d -2.8e-60)
       (/ t_1 (* d d))
       (if (<= d -1.45e-149)
         (/ t_1 (* c c))
         (if (<= d 1.9e-114)
           (/ b c)
           (if (<= d 2.7e+124) (* (/ d (fma c c (* d d))) (- a)) t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (b * c) - (d * a);
	double tmp;
	if (d <= -6.8e+53) {
		tmp = t_0;
	} else if (d <= -2.8e-60) {
		tmp = t_1 / (d * d);
	} else if (d <= -1.45e-149) {
		tmp = t_1 / (c * c);
	} else if (d <= 1.9e-114) {
		tmp = b / c;
	} else if (d <= 2.7e+124) {
		tmp = (d / fma(c, c, (d * d))) * -a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (d <= -6.8e+53)
		tmp = t_0;
	elseif (d <= -2.8e-60)
		tmp = Float64(t_1 / Float64(d * d));
	elseif (d <= -1.45e-149)
		tmp = Float64(t_1 / Float64(c * c));
	elseif (d <= 1.9e-114)
		tmp = Float64(b / c);
	elseif (d <= 2.7e+124)
		tmp = Float64(Float64(d / fma(c, c, Float64(d * d))) * Float64(-a));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.8e+53], t$95$0, If[LessEqual[d, -2.8e-60], N[(t$95$1 / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.45e-149], N[(t$95$1 / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-114], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.7e+124], N[(N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -6.79999999999999995e53 or 2.69999999999999978e124 < d

   1. Initial program 40.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 76.3

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

   5. Applied rewrites 76.3%

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

   if -6.79999999999999995e53 < d < -2.8000000000000002e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -2.8000000000000002e-60 < d < -1.45e-149

   1. Initial program 91.1%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.45e-149 < d < 1.8999999999999999e-114

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 1.8999999999999999e-114 < d < 2.69999999999999978e124

   1. Initial program 62.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 53.8

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

   5. Applied rewrites 53.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.8 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-149}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -3.1e-35)
     t_0
     (if (<= c 9e-146)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 6.7e+137)
         (/ (fma (- b) c (* d a)) (- (fma d d (* c c))))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.1e-35) {
		tmp = t_0;
	} else if (c <= 9e-146) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 6.7e+137) {
		tmp = fma(-b, c, (d * a)) / -fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -3.1e-35)
		tmp = t_0;
	elseif (c <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 6.7e+137)
		tmp = Float64(fma(Float64(-b), c, Float64(d * a)) / 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[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.1e-35], t$95$0, If[LessEqual[c, 9e-146], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.7e+137], N[(N[((-b) * c + N[(d * a), $MachinePrecision]), $MachinePrecision] / (-N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.10000000000000012e-35 or 6.6999999999999999e137 < c

   1. Initial program 45.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   if -3.10000000000000012e-35 < c < 9.0000000000000001e-146

   1. Initial program 70.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 9.0000000000000001e-146 < c < 6.6999999999999999e137

   1. Initial program 78.0%

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

      1. lift-/.f64 N/A

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

      2. 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 78.0

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

      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 78.0

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

   4. Applied rewrites 78.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-b, c, d \cdot a\right)}{-\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -3.1e-35)
     t_0
     (if (<= c 9e-146)
       (/ (- (/ (* b c) d) a) d)
       (if (<= c 6.7e+137) (/ (- (* b c) (* d a)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.1e-35) {
		tmp = t_0;
	} else if (c <= 9e-146) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 6.7e+137) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((d * a) / c)) / c
    if (c <= (-3.1d-35)) then
        tmp = t_0
    else if (c <= 9d-146) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 6.7d+137) then
        tmp = ((b * c) - (d * a)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.1e-35) {
		tmp = t_0;
	} else if (c <= 9e-146) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 6.7e+137) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((d * a) / c)) / c
	tmp = 0
	if c <= -3.1e-35:
		tmp = t_0
	elif c <= 9e-146:
		tmp = (((b * c) / d) - a) / d
	elif c <= 6.7e+137:
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -3.1e-35)
		tmp = t_0;
	elseif (c <= 9e-146)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 6.7e+137)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((d * a) / c)) / c;
	tmp = 0.0;
	if (c <= -3.1e-35)
		tmp = t_0;
	elseif (c <= 9e-146)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 6.7e+137)
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.1e-35], t$95$0, If[LessEqual[c, 9e-146], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.7e+137], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.10000000000000012e-35 or 6.6999999999999999e137 < c

   1. Initial program 45.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 76.4

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

   5. Applied rewrites 76.4%

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

   if -3.10000000000000012e-35 < c < 9.0000000000000001e-146

   1. Initial program 70.4%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

   if 9.0000000000000001e-146 < c < 6.6999999999999999e137

   1. Initial program 78.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6.7 \cdot 10^{+137}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-149}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -4e-54)
     t_0
     (if (<= d -1.45e-149)
       (/ (- (* b c) (* d a)) (* c c))
       (if (<= d 1.9e-114)
         (/ b c)
         (if (<= d 2.7e+124) (* (/ d (fma c c (* d d))) (- a)) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -4e-54) {
		tmp = t_0;
	} else if (d <= -1.45e-149) {
		tmp = ((b * c) - (d * a)) / (c * c);
	} else if (d <= 1.9e-114) {
		tmp = b / c;
	} else if (d <= 2.7e+124) {
		tmp = (d / fma(c, c, (d * d))) * -a;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -4e-54)
		tmp = t_0;
	elseif (d <= -1.45e-149)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c));
	elseif (d <= 1.9e-114)
		tmp = Float64(b / c);
	elseif (d <= 2.7e+124)
		tmp = Float64(Float64(d / fma(c, c, Float64(d * d))) * Float64(-a));
	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, -4e-54], t$95$0, If[LessEqual[d, -1.45e-149], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e-114], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.7e+124], N[(N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.0000000000000001e-54 or 2.69999999999999978e124 < d

   1. Initial program 48.7%

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

   3. Taylor expanded in c around 0

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

      1. 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 68.4

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

   5. Applied rewrites 68.4%

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

   if -4.0000000000000001e-54 < d < -1.45e-149

   1. Initial program 91.9%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -1.45e-149 < d < 1.8999999999999999e-114

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if 1.8999999999999999e-114 < d < 2.69999999999999978e124

   1. Initial program 62.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 53.8

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

   5. Applied rewrites 53.8%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-149}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (* (/ d (fma c c (* d d))) (- a))))
   (if (<= d -1.3e+147)
     t_0
     (if (<= d -8.6e-98)
       t_1
       (if (<= d 1.9e-114) (/ b c) (if (<= d 2.7e+124) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = (d / fma(c, c, (d * d))) * -a;
	double tmp;
	if (d <= -1.3e+147) {
		tmp = t_0;
	} else if (d <= -8.6e-98) {
		tmp = t_1;
	} else if (d <= 1.9e-114) {
		tmp = b / c;
	} else if (d <= 2.7e+124) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(d / fma(c, c, Float64(d * d))) * Float64(-a))
	tmp = 0.0
	if (d <= -1.3e+147)
		tmp = t_0;
	elseif (d <= -8.6e-98)
		tmp = t_1;
	elseif (d <= 1.9e-114)
		tmp = Float64(b / c);
	elseif (d <= 2.7e+124)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, If[LessEqual[d, -1.3e+147], t$95$0, If[LessEqual[d, -8.6e-98], t$95$1, If[LessEqual[d, 1.9e-114], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.7e+124], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.2999999999999999e147 or 2.69999999999999978e124 < d

   1. Initial program 37.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 81.4

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

   5. Applied rewrites 81.4%

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

   if -1.2999999999999999e147 < d < -8.59999999999999977e-98 or 1.8999999999999999e-114 < d < 2.69999999999999978e124

   1. Initial program 66.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 56.7

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

   5. Applied rewrites 56.7%

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

   if -8.59999999999999977e-98 < d < 1.8999999999999999e-114

   1. Initial program 77.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.4

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

   5. Applied rewrites 72.4%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+147}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+124}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{if}\;c \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ c (fma c c (* d d))) b)))
   (if (<= c -1e+141)
     (/ b c)
     (if (<= c -2.7e-35)
       t_0
       (if (<= c 1.3e-114) (/ (- a) d) (if (<= c 1.02e+114) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c / fma(c, c, (d * d))) * b;
	double tmp;
	if (c <= -1e+141) {
		tmp = b / c;
	} else if (c <= -2.7e-35) {
		tmp = t_0;
	} else if (c <= 1.3e-114) {
		tmp = -a / d;
	} else if (c <= 1.02e+114) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c / fma(c, c, Float64(d * d))) * b)
	tmp = 0.0
	if (c <= -1e+141)
		tmp = Float64(b / c);
	elseif (c <= -2.7e-35)
		tmp = t_0;
	elseif (c <= 1.3e-114)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1.02e+114)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[c, -1e+141], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.7e-35], t$95$0, If[LessEqual[c, 1.3e-114], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1.02e+114], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.00000000000000002e141 or 1.01999999999999999e114 < c

   1. Initial program 33.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -1.00000000000000002e141 < c < -2.6999999999999997e-35 or 1.30000000000000007e-114 < c < 1.01999999999999999e114

   1. Initial program 73.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 36.1

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

   5. Applied rewrites 36.1%

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

   6. Taylor expanded in b around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 55.4

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

   8. Applied rewrites 55.4%

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

   if -2.6999999999999997e-35 < c < 1.30000000000000007e-114

   1. Initial program 71.8%

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

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

   5. Applied rewrites 67.4%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.7 \cdot 10^{-35}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+114}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -6.8e+53)
     t_0
     (if (<= d -3.1e-60)
       (/ (- (* b c) (* d a)) (* d d))
       (if (<= d 2.1e+122) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -6.8e+53) {
		tmp = t_0;
	} else if (d <= -3.1e-60) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 2.1e+122) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-6.8d+53)) then
        tmp = t_0
    else if (d <= (-3.1d-60)) then
        tmp = ((b * c) - (d * a)) / (d * d)
    else if (d <= 2.1d+122) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -6.8e+53) {
		tmp = t_0;
	} else if (d <= -3.1e-60) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 2.1e+122) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -6.8e+53:
		tmp = t_0
	elif d <= -3.1e-60:
		tmp = ((b * c) - (d * a)) / (d * d)
	elif d <= 2.1e+122:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -6.8e+53)
		tmp = t_0;
	elseif (d <= -3.1e-60)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (d <= 2.1e+122)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -6.8e+53)
		tmp = t_0;
	elseif (d <= -3.1e-60)
		tmp = ((b * c) - (d * a)) / (d * d);
	elseif (d <= 2.1e+122)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -6.8e+53], t$95$0, If[LessEqual[d, -3.1e-60], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.1e+122], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.79999999999999995e53 or 2.10000000000000016e122 < d

   1. Initial program 40.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 75.5

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

   5. Applied rewrites 75.5%

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

   if -6.79999999999999995e53 < d < -3.09999999999999988e-60

   1. Initial program 78.0%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 64.1

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

   5. Applied rewrites 64.1%

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

   if -3.09999999999999988e-60 < d < 2.10000000000000016e122

   1. Initial program 72.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -3.1 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+122}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -3.1e-35) t_0 (if (<= c 5e-52) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.1e-35) {
		tmp = t_0;
	} else if (c <= 5e-52) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((d * a) / c)) / c
    if (c <= (-3.1d-35)) then
        tmp = t_0
    else if (c <= 5d-52) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -3.1e-35) {
		tmp = t_0;
	} else if (c <= 5e-52) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((d * a) / c)) / c
	tmp = 0
	if c <= -3.1e-35:
		tmp = t_0
	elif c <= 5e-52:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -3.1e-35)
		tmp = t_0;
	elseif (c <= 5e-52)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((d * a) / c)) / c;
	tmp = 0.0;
	if (c <= -3.1e-35)
		tmp = t_0;
	elseif (c <= 5e-52)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.1e-35], t$95$0, If[LessEqual[c, 5e-52], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.10000000000000012e-35 or 5e-52 < c

   1. Initial program 52.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   if -3.10000000000000012e-35 < c < 5e-52

   1. Initial program 73.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 83.9

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

   5. Applied rewrites 83.9%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{-35}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.8 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.8e-39) t_0 (if (<= d 3.1e-30) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.8e-39) {
		tmp = t_0;
	} else if (d <= 3.1e-30) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-1.8d-39)) then
        tmp = t_0
    else if (d <= 3.1d-30) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.8e-39) {
		tmp = t_0;
	} else if (d <= 3.1e-30) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -1.8e-39:
		tmp = t_0
	elif d <= 3.1e-30:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.8e-39)
		tmp = t_0;
	elseif (d <= 3.1e-30)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -1.8e-39)
		tmp = t_0;
	elseif (d <= 3.1e-30)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.8e-39], t$95$0, If[LessEqual[d, 3.1e-30], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.8e-39 or 3.09999999999999991e-30 < 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 62.0

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

   5. Applied rewrites 62.0%

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

   if -1.8e-39 < d < 3.09999999999999991e-30

   1. Initial program 75.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 63.9

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

   5. Applied rewrites 63.9%

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

Alternative 12: 42.9% 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 62.2%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.4

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

5. Applied rewrites 42.4%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024249 
(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))))