Complex division, imag part

Percentage Accurate: 61.5% → 80.2%

Time: 10.0s

Alternatives: 12

Speedup: 1.4×

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: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{if}\;d \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(d, \frac{0 - a}{t\_0}, \frac{c \cdot b}{t\_0}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d))) (t_1 (/ (fma c (/ b d) (- 0.0 a)) d)))
   (if (<= d -1.2e+68)
     t_1
     (if (<= d -1.8e-85)
       (fma d (/ (- 0.0 a) t_0) (/ (* c b) t_0))
       (if (<= d 1.4e-14) (/ (- b (/ (* d a) 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(c, (b / d), (0.0 - a)) / d;
	double tmp;
	if (d <= -1.2e+68) {
		tmp = t_1;
	} else if (d <= -1.8e-85) {
		tmp = fma(d, ((0.0 - a) / t_0), ((c * b) / t_0));
	} else if (d <= 1.4e-14) {
		tmp = (b - ((d * a) / 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 = Float64(fma(c, Float64(b / d), Float64(0.0 - a)) / d)
	tmp = 0.0
	if (d <= -1.2e+68)
		tmp = t_1;
	elseif (d <= -1.8e-85)
		tmp = fma(d, Float64(Float64(0.0 - a) / t_0), Float64(Float64(c * b) / t_0));
	elseif (d <= 1.4e-14)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / 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[(c * N[(b / d), $MachinePrecision] + N[(0.0 - a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.2e+68], t$95$1, If[LessEqual[d, -1.8e-85], N[(d * N[(N[(0.0 - a), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-14], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.20000000000000004e68 or 1.4e-14 < d

   1. Initial program 54.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

      16. --lowering--.f64 85.5

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

   5. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 85.5

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

   7. Applied egg-rr 85.5%

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

   if -1.20000000000000004e68 < d < -1.7999999999999999e-85

   1. Initial program 78.6%

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

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

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. neg-lowering-neg.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 N/A

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

      15. accelerator-lowering-fma.f64 N/A

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

      16. *-lowering-*.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   if -1.7999999999999999e-85 < d < 1.4e-14

   1. Initial program 65.9%

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.5

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

   5. Simplified 87.5%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(d, \frac{0 - a}{\mathsf{fma}\left(c, c, d \cdot d\right)}, \frac{c \cdot b}{\mathsf{fma}\left(c, c, d \cdot d\right)}\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;c \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-128}:\\ \;\;\;\;\frac{c \cdot b}{t\_0}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{b}{t\_0}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= c -1.12e+104)
     (/ b c)
     (if (<= c -2.35e-128)
       (/ (* c b) t_0)
       (if (<= c 2.7e-11)
         (- 0.0 (/ a d))
         (if (<= c 1.85e+90) (fma c (/ b t_0) 0.0) (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (c <= -1.12e+104) {
		tmp = b / c;
	} else if (c <= -2.35e-128) {
		tmp = (c * b) / t_0;
	} else if (c <= 2.7e-11) {
		tmp = 0.0 - (a / d);
	} else if (c <= 1.85e+90) {
		tmp = fma(c, (b / t_0), 0.0);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (c <= -1.12e+104)
		tmp = Float64(b / c);
	elseif (c <= -2.35e-128)
		tmp = Float64(Float64(c * b) / t_0);
	elseif (c <= 2.7e-11)
		tmp = Float64(0.0 - Float64(a / d));
	elseif (c <= 1.85e+90)
		tmp = fma(c, Float64(b / t_0), 0.0);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.12e+104], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.35e-128], N[(N[(c * b), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[c, 2.7e-11], N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.85e+90], N[(c * N[(b / t$95$0), $MachinePrecision] + 0.0), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.12000000000000003e104 or 1.85e90 < c

   1. Initial program 39.4%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 73.5

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

   5. Simplified 73.5%

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

   if -1.12000000000000003e104 < c < -2.3500000000000002e-128

   1. Initial program 87.4%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   5. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.6

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

   7. Simplified 66.6%

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

   if -2.3500000000000002e-128 < c < 2.70000000000000005e-11

   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 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 72.7

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 72.7

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

   7. Applied egg-rr 72.7%

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

   if 2.70000000000000005e-11 < c < 1.85e90

   1. Initial program 75.6%

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

         \[\leadsto \color{blue}{\frac{b \cdot c}{{c}^{2} + {d}^{2}} + 0} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot b}}{{c}^{2} + {d}^{2}} + 0 \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{c \cdot \frac{b}{{c}^{2} + {d}^{2}}} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{b}{{c}^{2} + {d}^{2}}, 0\right)} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{b}{{c}^{2} + {d}^{2}}}, 0\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(c, \frac{b}{\color{blue}{{d}^{2} + {c}^{2}}}, 0\right) \]

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \frac{b}{\color{blue}{\mathsf{fma}\left(d, d, {c}^{2}\right)}}, 0\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(c, \frac{b}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}, 0\right) \]

      10. *-lowering-*.f64 68.2

          \[\leadsto \mathsf{fma}\left(c, \frac{b}{\mathsf{fma}\left(d, d, \color{blue}{c \cdot c}\right)}, 0\right) \]

   5. Simplified 68.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}, 0\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.12 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.35 \cdot 10^{-128}:\\ \;\;\;\;\frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-11}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{b}{\mathsf{fma}\left(d, d, c \cdot c\right)}, 0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{if}\;d \leq -1.4 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-89}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- 0.0 a)) d)))
   (if (<= d -1.4e+68)
     t_0
     (if (<= d -5.7e-89)
       (/ (- (* c b) (* d a)) (fma d d (* c c)))
       (if (<= d 3.9e-16) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), (0.0 - a)) / d;
	double tmp;
	if (d <= -1.4e+68) {
		tmp = t_0;
	} else if (d <= -5.7e-89) {
		tmp = ((c * b) - (d * a)) / fma(d, d, (c * c));
	} else if (d <= 3.9e-16) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(0.0 - a)) / d)
	tmp = 0.0
	if (d <= -1.4e+68)
		tmp = t_0;
	elseif (d <= -5.7e-89)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(d, d, Float64(c * c)));
	elseif (d <= 3.9e-16)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + N[(0.0 - a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.4e+68], t$95$0, If[LessEqual[d, -5.7e-89], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.9e-16], 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 < -1.4e68 or 3.89999999999999977e-16 < d

   1. Initial program 54.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

      16. --lowering--.f64 85.5

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

   5. Simplified 85.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 85.5

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

   7. Applied egg-rr 85.5%

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

   if -1.4e68 < d < -5.7000000000000002e-89

   1. Initial program 78.6%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 78.6

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

   4. Applied egg-rr 78.6%

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

   if -5.7000000000000002e-89 < d < 3.89999999999999977e-16

   1. Initial program 65.9%

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 87.5

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

   5. Simplified 87.5%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{elif}\;d \leq -5.7 \cdot 10^{-89}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 3.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+165}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -4e+165)
     t_0
     (if (<= d -1.6e-9)
       (/ (- (* c b) (* d a)) (* d d))
       (if (<= d 7.5e-29) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -4e+165) {
		tmp = t_0;
	} else if (d <= -1.6e-9) {
		tmp = ((c * b) - (d * a)) / (d * d);
	} else if (d <= 7.5e-29) {
		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 = 0.0d0 - (a / d)
    if (d <= (-4d+165)) then
        tmp = t_0
    else if (d <= (-1.6d-9)) then
        tmp = ((c * b) - (d * a)) / (d * d)
    else if (d <= 7.5d-29) 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 = 0.0 - (a / d);
	double tmp;
	if (d <= -4e+165) {
		tmp = t_0;
	} else if (d <= -1.6e-9) {
		tmp = ((c * b) - (d * a)) / (d * d);
	} else if (d <= 7.5e-29) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -4e+165:
		tmp = t_0
	elif d <= -1.6e-9:
		tmp = ((c * b) - (d * a)) / (d * d)
	elif d <= 7.5e-29:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -4e+165)
		tmp = t_0;
	elseif (d <= -1.6e-9)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(d * d));
	elseif (d <= 7.5e-29)
		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 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -4e+165)
		tmp = t_0;
	elseif (d <= -1.6e-9)
		tmp = ((c * b) - (d * a)) / (d * d);
	elseif (d <= 7.5e-29)
		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[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4e+165], t$95$0, If[LessEqual[d, -1.6e-9], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e-29], 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 < -3.9999999999999996e165 or 7.50000000000000006e-29 < d

   1. Initial program 53.5%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 76.3

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 76.3

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

   7. Applied egg-rr 76.3%

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

   if -3.9999999999999996e165 < d < -1.60000000000000006e-9

   1. Initial program 69.1%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.6

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

   5. Simplified 57.6%

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

   if -1.60000000000000006e-9 < d < 7.50000000000000006e-29

   1. Initial program 67.3%

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.5

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

   5. Simplified 85.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+165}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- 0.0 a)) d)))
   (if (<= d -1.7e-9) t_0 (if (<= d 3.6e-18) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), (0.0 - a)) / d;
	double tmp;
	if (d <= -1.7e-9) {
		tmp = t_0;
	} else if (d <= 3.6e-18) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(0.0 - a)) / d)
	tmp = 0.0
	if (d <= -1.7e-9)
		tmp = t_0;
	elseif (d <= 3.6e-18)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + N[(0.0 - a), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.7e-9], t$95$0, If[LessEqual[d, 3.6e-18], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.6999999999999999e-9 or 3.6000000000000001e-18 < d

   1. Initial program 57.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. /-lowering-/.f64 N/A

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

      8. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. neg-sub0 N/A

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

      16. --lowering--.f64 81.2

          \[\leadsto \frac{\mathsf{fma}\left(c, \frac{b}{d}, \color{blue}{0 - a}\right)}{d} \]

   5. Simplified 81.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 81.2

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

   7. Applied egg-rr 81.2%

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

   if -1.6999999999999999e-9 < d < 3.6000000000000001e-18

   1. Initial program 67.0%

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 85.0

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

   5. Simplified 85.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \mathbf{elif}\;d \leq 3.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, 0 - a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -8.5 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.95 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -8.5e-61)
     t_0
     (if (<= c 1.95e-10) (/ (- (/ (* c b) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -8.5e-61) {
		tmp = t_0;
	} else if (c <= 1.95e-10) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - (a * (d / c))) / c
    if (c <= (-8.5d-61)) then
        tmp = t_0
    else if (c <= 1.95d-10) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -8.5e-61) {
		tmp = t_0;
	} else if (c <= 1.95e-10) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -8.5e-61:
		tmp = t_0
	elif c <= 1.95e-10:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -8.5e-61)
		tmp = t_0;
	elseif (c <= 1.95e-10)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -8.5e-61)
		tmp = t_0;
	elseif (c <= 1.95e-10)
		tmp = (((c * b) / 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[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -8.5e-61], t$95$0, If[LessEqual[c, 1.95e-10], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -8.50000000000000016e-61 or 1.95e-10 < c

   1. Initial program 52.4%

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

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 72.5

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

   5. Simplified 72.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 73.4

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

   7. Applied egg-rr 73.4%

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

   if -8.50000000000000016e-61 < c < 1.95e-10

   1. Initial program 74.6%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 74.6

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

   4. Applied egg-rr 74.6%

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

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   6. 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. /-lowering-/.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 87.3

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

   7. Simplified 87.3%

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

Alternative 7: 70.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -1.7e+68) t_0 (if (<= d 7.5e-29) (/ (- b (* a (/ d c))) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.7e+68) {
		tmp = t_0;
	} else if (d <= 7.5e-29) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (a / d)
    if (d <= (-1.7d+68)) then
        tmp = t_0
    else if (d <= 7.5d-29) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.7e+68) {
		tmp = t_0;
	} else if (d <= 7.5e-29) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -1.7e+68:
		tmp = t_0
	elif d <= 7.5e-29:
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -1.7e+68)
		tmp = t_0;
	elseif (d <= 7.5e-29)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -1.7e+68)
		tmp = t_0;
	elseif (d <= 7.5e-29)
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.7e+68], t$95$0, If[LessEqual[d, 7.5e-29], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.70000000000000008e68 or 7.50000000000000006e-29 < d

   1. Initial program 55.0%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 72.2

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 72.2

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

   7. Applied egg-rr 72.2%

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

   if -1.70000000000000008e68 < d < 7.50000000000000006e-29

   1. Initial program 68.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.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. --lowering--.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 79.6

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

   5. Simplified 79.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 79.7

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

   7. Applied egg-rr 79.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -4 \cdot 10^{+165}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -4e+165)
     t_0
     (if (<= d -2.4e-10)
       (/ (- (* c b) (* d a)) (* d d))
       (if (<= d 1.4e-31) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -4e+165) {
		tmp = t_0;
	} else if (d <= -2.4e-10) {
		tmp = ((c * b) - (d * a)) / (d * d);
	} else if (d <= 1.4e-31) {
		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 = 0.0d0 - (a / d)
    if (d <= (-4d+165)) then
        tmp = t_0
    else if (d <= (-2.4d-10)) then
        tmp = ((c * b) - (d * a)) / (d * d)
    else if (d <= 1.4d-31) 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 = 0.0 - (a / d);
	double tmp;
	if (d <= -4e+165) {
		tmp = t_0;
	} else if (d <= -2.4e-10) {
		tmp = ((c * b) - (d * a)) / (d * d);
	} else if (d <= 1.4e-31) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -4e+165:
		tmp = t_0
	elif d <= -2.4e-10:
		tmp = ((c * b) - (d * a)) / (d * d)
	elif d <= 1.4e-31:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -4e+165)
		tmp = t_0;
	elseif (d <= -2.4e-10)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(d * d));
	elseif (d <= 1.4e-31)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -4e+165)
		tmp = t_0;
	elseif (d <= -2.4e-10)
		tmp = ((c * b) - (d * a)) / (d * d);
	elseif (d <= 1.4e-31)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4e+165], t$95$0, If[LessEqual[d, -2.4e-10], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-31], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.9999999999999996e165 or 1.3999999999999999e-31 < d

   1. Initial program 54.4%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 75.8

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 75.8

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

   7. Applied egg-rr 75.8%

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

   if -3.9999999999999996e165 < d < -2.4e-10

   1. Initial program 69.1%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.6

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

   5. Simplified 57.6%

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

   if -2.4e-10 < d < 1.3999999999999999e-31

   1. Initial program 66.7%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 70.0

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

   5. Simplified 70.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+165}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.1e+103)
   (/ b c)
   (if (<= c -3.4e-129)
     (/ (* c b) (fma d d (* c c)))
     (if (<= c 4.8e-9) (- 0.0 (/ a d)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.1e+103) {
		tmp = b / c;
	} else if (c <= -3.4e-129) {
		tmp = (c * b) / fma(d, d, (c * c));
	} else if (c <= 4.8e-9) {
		tmp = 0.0 - (a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.1e+103)
		tmp = Float64(b / c);
	elseif (c <= -3.4e-129)
		tmp = Float64(Float64(c * b) / fma(d, d, Float64(c * c)));
	elseif (c <= 4.8e-9)
		tmp = Float64(0.0 - Float64(a / d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.1e+103], N[(b / c), $MachinePrecision], If[LessEqual[c, -3.4e-129], N[(N[(c * b), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e-9], N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.09999999999999996e103 or 4.8e-9 < c

   1. Initial program 45.7%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 69.0

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

   5. Simplified 69.0%

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

   if -1.09999999999999996e103 < c < -3.40000000000000013e-129

   1. Initial program 87.4%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 87.4

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

   4. Applied egg-rr 87.4%

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

   5. Taylor expanded in b around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 66.6

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

   7. Simplified 66.6%

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

   if -3.40000000000000013e-129 < c < 4.8e-9

   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 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 72.7

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 72.7

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

   7. Applied egg-rr 72.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -3.4 \cdot 10^{-129}:\\ \;\;\;\;\frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ a d))))
   (if (<= d -1.7e+68) t_0 (if (<= d 1.4e-31) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.7e+68) {
		tmp = t_0;
	} else if (d <= 1.4e-31) {
		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 = 0.0d0 - (a / d)
    if (d <= (-1.7d+68)) then
        tmp = t_0
    else if (d <= 1.4d-31) 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 = 0.0 - (a / d);
	double tmp;
	if (d <= -1.7e+68) {
		tmp = t_0;
	} else if (d <= 1.4e-31) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 0.0 - (a / d)
	tmp = 0
	if d <= -1.7e+68:
		tmp = t_0
	elif d <= 1.4e-31:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(0.0 - Float64(a / d))
	tmp = 0.0
	if (d <= -1.7e+68)
		tmp = t_0;
	elseif (d <= 1.4e-31)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 0.0 - (a / d);
	tmp = 0.0;
	if (d <= -1.7e+68)
		tmp = t_0;
	elseif (d <= 1.4e-31)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(0.0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.7e+68], t$95$0, If[LessEqual[d, 1.4e-31], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.70000000000000008e68 or 1.3999999999999999e-31 < d

   1. Initial program 55.8%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 71.9

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 71.9%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 71.9

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

   7. Applied egg-rr 71.9%

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

   if -1.70000000000000008e68 < d < 1.3999999999999999e-31

   1. Initial program 67.6%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 65.3

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

   5. Simplified 65.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+68}:\\ \;\;\;\;0 - \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.9 \cdot 10^{+177}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+171}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.9e+177) (/ a d) (if (<= d 3.2e+171) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.9e+177) {
		tmp = a / d;
	} else if (d <= 3.2e+171) {
		tmp = b / c;
	} else {
		tmp = a / 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 (d <= (-6.9d+177)) then
        tmp = a / d
    else if (d <= 3.2d+171) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.9e+177) {
		tmp = a / d;
	} else if (d <= 3.2e+171) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6.9e+177:
		tmp = a / d
	elif d <= 3.2e+171:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6.9e+177)
		tmp = Float64(a / d);
	elseif (d <= 3.2e+171)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6.9e+177)
		tmp = a / d;
	elseif (d <= 3.2e+171)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6.9e+177], N[(a / d), $MachinePrecision], If[LessEqual[d, 3.2e+171], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.89999999999999971e177 or 3.20000000000000011e171 < d

   1. Initial program 40.3%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

      7. --lowering--.f64 83.1

         \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 83.1

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

   7. Applied egg-rr 83.1%

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

      1. unpow1 N/A

         \[\leadsto \frac{\color{blue}{{a}^{1}}}{\mathsf{neg}\left(d\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{{a}^{\color{blue}{\left(3 - 2\right)}}}{\mathsf{neg}\left(d\right)} \]

      3. pow-div N/A

         \[\leadsto \frac{\color{blue}{\frac{{a}^{3}}{{a}^{2}}}}{\mathsf{neg}\left(d\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{a \cdot a}}}{\mathsf{neg}\left(d\right)} \]

      5. +-rgt-identity N/A

         \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{a \cdot a + 0}}}{\mathsf{neg}\left(d\right)} \]

      6. mul0-lft N/A

         \[\leadsto \frac{\frac{{a}^{3}}{a \cdot a + \color{blue}{0 \cdot a}}}{\mathsf{neg}\left(d\right)} \]

      7. +-lft-identity N/A

         \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{0 + \left(a \cdot a + 0 \cdot a\right)}}}{\mathsf{neg}\left(d\right)} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{0 \cdot 0} + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      9. sqr-pow N/A

         \[\leadsto \frac{\frac{\color{blue}{{a}^{\left(\frac{3}{2}\right)} \cdot {a}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      10. unpow-prod-down N/A

          \[\leadsto \frac{\frac{\color{blue}{{\left(a \cdot a\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      11. sqr-neg N/A

          \[\leadsto \frac{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      12. sub0-neg N/A

          \[\leadsto \frac{\frac{{\left(\color{blue}{\left(0 - a\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      13. sub0-neg N/A

          \[\leadsto \frac{\frac{{\left(\left(0 - a\right) \cdot \color{blue}{\left(0 - a\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      14. unpow-prod-down N/A

          \[\leadsto \frac{\frac{\color{blue}{{\left(0 - a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - a\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      15. sqr-pow N/A

          \[\leadsto \frac{\frac{\color{blue}{{\left(0 - a\right)}^{3}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      16. sub0-neg N/A

          \[\leadsto \frac{\frac{{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      17. cube-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left({a}^{3}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      18. sub0-neg N/A

          \[\leadsto \frac{\frac{\color{blue}{0 - {a}^{3}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      19. metadata-eval N/A

          \[\leadsto \frac{\frac{\color{blue}{{0}^{3}} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

      20. flip3-- N/A

          \[\leadsto \frac{\color{blue}{0 - a}}{\mathsf{neg}\left(d\right)} \]

      21. sub0-neg N/A

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

      22. frac-2neg N/A

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

      23. /-lowering-/.f64 41.3

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

   9. Applied egg-rr 41.3%

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

   if -6.89999999999999971e177 < d < 3.20000000000000011e171

   1. Initial program 68.1%

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

   3. Taylor expanded in c around inf

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

      1. /-lowering-/.f64 51.0

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

   5. Simplified 51.0%

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

Alternative 12: 10.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{a}{d} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (/ a d))

C

double code(double a, double b, double c, double d) {
	return a / 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 = a / d
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_] := N[(a / d), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in c around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. /-lowering-/.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

      \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

   7. --lowering--.f64 42.7

      \[\leadsto \frac{a}{\color{blue}{0 - d}} \]

5. Simplified 42.7%

   \[\leadsto \color{blue}{\frac{a}{0 - d}} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 42.7

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

7. Applied egg-rr 42.7%

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

   1. unpow1 N/A

      \[\leadsto \frac{\color{blue}{{a}^{1}}}{\mathsf{neg}\left(d\right)} \]

   2. metadata-eval N/A

      \[\leadsto \frac{{a}^{\color{blue}{\left(3 - 2\right)}}}{\mathsf{neg}\left(d\right)} \]

   3. pow-div N/A

      \[\leadsto \frac{\color{blue}{\frac{{a}^{3}}{{a}^{2}}}}{\mathsf{neg}\left(d\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{a \cdot a}}}{\mathsf{neg}\left(d\right)} \]

   5. +-rgt-identity N/A

      \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{a \cdot a + 0}}}{\mathsf{neg}\left(d\right)} \]

   6. mul0-lft N/A

      \[\leadsto \frac{\frac{{a}^{3}}{a \cdot a + \color{blue}{0 \cdot a}}}{\mathsf{neg}\left(d\right)} \]

   7. +-lft-identity N/A

      \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{0 + \left(a \cdot a + 0 \cdot a\right)}}}{\mathsf{neg}\left(d\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{{a}^{3}}{\color{blue}{0 \cdot 0} + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   9. sqr-pow N/A

      \[\leadsto \frac{\frac{\color{blue}{{a}^{\left(\frac{3}{2}\right)} \cdot {a}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   10. unpow-prod-down N/A

       \[\leadsto \frac{\frac{\color{blue}{{\left(a \cdot a\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   11. sqr-neg N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(\left(\mathsf{neg}\left(a\right)\right) \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   12. sub0-neg N/A

       \[\leadsto \frac{\frac{{\left(\color{blue}{\left(0 - a\right)} \cdot \left(\mathsf{neg}\left(a\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   13. sub0-neg N/A

       \[\leadsto \frac{\frac{{\left(\left(0 - a\right) \cdot \color{blue}{\left(0 - a\right)}\right)}^{\left(\frac{3}{2}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   14. unpow-prod-down N/A

       \[\leadsto \frac{\frac{\color{blue}{{\left(0 - a\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(0 - a\right)}^{\left(\frac{3}{2}\right)}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   15. sqr-pow N/A

       \[\leadsto \frac{\frac{\color{blue}{{\left(0 - a\right)}^{3}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   16. sub0-neg N/A

       \[\leadsto \frac{\frac{{\color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   17. cube-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{neg}\left({a}^{3}\right)}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   18. sub0-neg N/A

       \[\leadsto \frac{\frac{\color{blue}{0 - {a}^{3}}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   19. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{{0}^{3}} - {a}^{3}}{0 \cdot 0 + \left(a \cdot a + 0 \cdot a\right)}}{\mathsf{neg}\left(d\right)} \]

   20. flip3-- N/A

       \[\leadsto \frac{\color{blue}{0 - a}}{\mathsf{neg}\left(d\right)} \]

   21. sub0-neg N/A

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

   22. frac-2neg N/A

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

   23. /-lowering-/.f64 12.2

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

9. Applied egg-rr 12.2%

   \[\leadsto \color{blue}{\frac{a}{d}} \]
10. 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 2024195 
(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))))