Complex division, imag part

Percentage Accurate: 61.4% → 82.6%

Time: 6.7s

Alternatives: 8

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: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, a \cdot d\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -7e+104)
     t_0
     (if (<= d -9e-133)
       (* (/ -1.0 (fma d d (* c c))) (fma (- b) c (* a d)))
       (if (<= d 1.4e-155)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 3.2e+141)
           (/ (- (* c b) (* a d)) (+ (* d d) (* c c)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -7e+104) {
		tmp = t_0;
	} else if (d <= -9e-133) {
		tmp = (-1.0 / fma(d, d, (c * c))) * fma(-b, c, (a * d));
	} else if (d <= 1.4e-155) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.2e+141) {
		tmp = ((c * b) - (a * d)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7e+104)
		tmp = t_0;
	elseif (d <= -9e-133)
		tmp = Float64(Float64(-1.0 / fma(d, d, Float64(c * c))) * fma(Float64(-b), c, Float64(a * d)));
	elseif (d <= 1.4e-155)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3.2e+141)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7e+104], t$95$0, If[LessEqual[d, -9e-133], N[(N[(-1.0 / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-b) * c + N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.4e-155], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.2e+141], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.0000000000000003e104 or 3.20000000000000019e141 < d

   1. Initial program 27.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 12.7

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 74.1

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 85.3%

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

   if -7.0000000000000003e104 < d < -9.00000000000000019e-133

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      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. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 82.9

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 82.9

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

   4. Applied rewrites 82.9%

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

   if -9.00000000000000019e-133 < d < 1.4e-155

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if 1.4e-155 < d < 3.20000000000000019e141

   1. Initial program 82.9%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \mathsf{fma}\left(-b, c, a \cdot d\right)\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* d d) (* c c))))
        (t_1 (/ (fma (/ b d) c (- a)) d)))
   (if (<= d -7e+104)
     t_1
     (if (<= d -9e-133)
       t_0
       (if (<= d 1.4e-155)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 3.2e+141) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	double t_1 = fma((b / d), c, -a) / d;
	double tmp;
	if (d <= -7e+104) {
		tmp = t_1;
	} else if (d <= -9e-133) {
		tmp = t_0;
	} else if (d <= 1.4e-155) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.2e+141) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(fma(Float64(b / d), c, Float64(-a)) / d)
	tmp = 0.0
	if (d <= -7e+104)
		tmp = t_1;
	elseif (d <= -9e-133)
		tmp = t_0;
	elseif (d <= 1.4e-155)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3.2e+141)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / d), $MachinePrecision] * c + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7e+104], t$95$1, If[LessEqual[d, -9e-133], t$95$0, If[LessEqual[d, 1.4e-155], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.2e+141], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -7.0000000000000003e104 or 3.20000000000000019e141 < d

   1. Initial program 27.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 12.7

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. +-commutative N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 74.1

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

   8. Applied rewrites 74.1%

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

   10. Applied rewrites 85.3%

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

   if -7.0000000000000003e104 < d < -9.00000000000000019e-133 or 1.4e-155 < d < 3.20000000000000019e141

   1. Initial program 82.9%

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

   if -9.00000000000000019e-133 < d < 1.4e-155

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-133}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-155}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{d}, c, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-8}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-d}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.4e-8)
   (/ (- a) d)
   (if (<= d 2.7e-80)
     (/ (- b (/ (* a d) c)) c)
     (if (<= d 6e-8)
       (/ (- (* c b) (* a d)) (* d d))
       (if (<= d 3.7e+141) (/ (- b (* (/ a c) d)) c) (/ 1.0 (/ (- d) a)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.4e-8) {
		tmp = -a / d;
	} else if (d <= 2.7e-80) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 6e-8) {
		tmp = ((c * b) - (a * d)) / (d * d);
	} else if (d <= 3.7e+141) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = 1.0 / (-d / a);
	}
	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 <= (-4.4d-8)) then
        tmp = -a / d
    else if (d <= 2.7d-80) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 6d-8) then
        tmp = ((c * b) - (a * d)) / (d * d)
    else if (d <= 3.7d+141) then
        tmp = (b - ((a / c) * d)) / c
    else
        tmp = 1.0d0 / (-d / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.4e-8) {
		tmp = -a / d;
	} else if (d <= 2.7e-80) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 6e-8) {
		tmp = ((c * b) - (a * d)) / (d * d);
	} else if (d <= 3.7e+141) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = 1.0 / (-d / a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.4e-8:
		tmp = -a / d
	elif d <= 2.7e-80:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 6e-8:
		tmp = ((c * b) - (a * d)) / (d * d)
	elif d <= 3.7e+141:
		tmp = (b - ((a / c) * d)) / c
	else:
		tmp = 1.0 / (-d / a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.4e-8)
		tmp = Float64(Float64(-a) / d);
	elseif (d <= 2.7e-80)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 6e-8)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 3.7e+141)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	else
		tmp = Float64(1.0 / Float64(Float64(-d) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4.4e-8)
		tmp = -a / d;
	elseif (d <= 2.7e-80)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 6e-8)
		tmp = ((c * b) - (a * d)) / (d * d);
	elseif (d <= 3.7e+141)
		tmp = (b - ((a / c) * d)) / c;
	else
		tmp = 1.0 / (-d / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.4e-8], N[((-a) / d), $MachinePrecision], If[LessEqual[d, 2.7e-80], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6e-8], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.7e+141], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(1.0 / N[((-d) / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -4.3999999999999997e-8

   1. Initial program 45.1%

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

   3. Taylor expanded in c around 0

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

      1. 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 67.9

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

   5. Applied rewrites 67.9%

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

   if -4.3999999999999997e-8 < d < 2.7000000000000002e-80

   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 84.0

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

   5. Applied rewrites 84.0%

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

   if 2.7000000000000002e-80 < d < 5.99999999999999946e-8

   1. Initial program 99.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 79.8

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

   5. Applied rewrites 79.8%

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

   if 5.99999999999999946e-8 < d < 3.7000000000000003e141

   1. Initial program 75.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 50.4

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

   5. Applied rewrites 50.4%

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

   7. Applied rewrites 58.3%

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

   if 3.7000000000000003e141 < d

   1. Initial program 31.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 31.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 31.4

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 31.4

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.9

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

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{-d}{a}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-80}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{-8}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-d}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4.4 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -4.4e-18)
     t_0
     (if (<= d 9.6e-200)
       (/ b c)
       (if (<= d 2.3e+154) (* (/ 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 <= -4.4e-18) {
		tmp = t_0;
	} else if (d <= 9.6e-200) {
		tmp = b / c;
	} else if (d <= 2.3e+154) {
		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 <= -4.4e-18)
		tmp = t_0;
	elseif (d <= 9.6e-200)
		tmp = Float64(b / c);
	elseif (d <= 2.3e+154)
		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, -4.4e-18], t$95$0, If[LessEqual[d, 9.6e-200], N[(b / c), $MachinePrecision], If[LessEqual[d, 2.3e+154], N[(N[(d / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.3999999999999997e-18 or 2.3e154 < d

   1. Initial program 37.6%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 66.1

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

   5. Applied rewrites 66.1%

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

   if -4.3999999999999997e-18 < d < 9.60000000000000006e-200

   1. Initial program 70.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if 9.60000000000000006e-200 < d < 2.3e154

   1. Initial program 81.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(c, c, d \cdot d\right)} \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -4.4 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -4.4e+102)
     t_0
     (if (<= c 2.35e+45) (/ (- (/ (* c b) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -4.4e+102) {
		tmp = t_0;
	} else if (c <= 2.35e+45) {
		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 / c) * d)) / c
    if (c <= (-4.4d+102)) then
        tmp = t_0
    else if (c <= 2.35d+45) 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 / c) * d)) / c;
	double tmp;
	if (c <= -4.4e+102) {
		tmp = t_0;
	} else if (c <= 2.35e+45) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a / c) * d)) / c
	tmp = 0
	if c <= -4.4e+102:
		tmp = t_0
	elif c <= 2.35e+45:
		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(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -4.4e+102)
		tmp = t_0;
	elseif (c <= 2.35e+45)
		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 / c) * d)) / c;
	tmp = 0.0;
	if (c <= -4.4e+102)
		tmp = t_0;
	elseif (c <= 2.35e+45)
		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[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.4e+102], t$95$0, If[LessEqual[c, 2.35e+45], 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 < -4.40000000000000015e102 or 2.35000000000000001e45 < c

   1. Initial program 46.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 85.5%

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

   if -4.40000000000000015e102 < c < 2.35000000000000001e45

   1. Initial program 74.7%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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 77.2

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

   5. Applied rewrites 77.2%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{+45}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-d}{a}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.4e-8)
   (/ (- a) d)
   (if (<= d 3.7e+141) (/ (- b (* (/ a c) d)) c) (/ 1.0 (/ (- d) a)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.4e-8) {
		tmp = -a / d;
	} else if (d <= 3.7e+141) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = 1.0 / (-d / a);
	}
	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 <= (-4.4d-8)) then
        tmp = -a / d
    else if (d <= 3.7d+141) then
        tmp = (b - ((a / c) * d)) / c
    else
        tmp = 1.0d0 / (-d / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.4e-8) {
		tmp = -a / d;
	} else if (d <= 3.7e+141) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = 1.0 / (-d / a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4.4e-8:
		tmp = -a / d
	elif d <= 3.7e+141:
		tmp = (b - ((a / c) * d)) / c
	else:
		tmp = 1.0 / (-d / a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.4e-8)
		tmp = Float64(Float64(-a) / d);
	elseif (d <= 3.7e+141)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	else
		tmp = Float64(1.0 / Float64(Float64(-d) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4.4e-8)
		tmp = -a / d;
	elseif (d <= 3.7e+141)
		tmp = (b - ((a / c) * d)) / c;
	else
		tmp = 1.0 / (-d / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.4e-8], N[((-a) / d), $MachinePrecision], If[LessEqual[d, 3.7e+141], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(1.0 / N[((-d) / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.3999999999999997e-8

   1. Initial program 45.1%

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

   3. Taylor expanded in c around 0

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

      1. 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 67.9

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

   5. Applied rewrites 67.9%

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

   if -4.3999999999999997e-8 < d < 3.7000000000000003e141

   1. Initial program 76.4%

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

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

   5. Applied rewrites 71.8%

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

   7. Applied rewrites 72.9%

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

   if 3.7000000000000003e141 < d

   1. Initial program 31.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 31.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 31.4

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

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

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

      15. lower-fma.f64 N/A

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

      16. lower-neg.f64 31.4

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

   4. Applied rewrites 31.4%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 67.9

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

   7. Applied rewrites 67.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{-d}{a}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{+141}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-d}{a}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+132}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.7e+104) (/ b c) (if (<= c 6.5e+132) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.7e+104) {
		tmp = b / c;
	} else if (c <= 6.5e+132) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.7d+104)) then
        tmp = b / c
    else if (c <= 6.5d+132) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.7e+104) {
		tmp = b / c;
	} else if (c <= 6.5e+132) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.7e+104:
		tmp = b / c
	elif c <= 6.5e+132:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.7e+104)
		tmp = Float64(b / c);
	elseif (c <= 6.5e+132)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.7e+104)
		tmp = b / c;
	elseif (c <= 6.5e+132)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.7e+104], N[(b / c), $MachinePrecision], If[LessEqual[c, 6.5e+132], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.6999999999999998e104 or 6.4999999999999994e132 < 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. lower-/.f64 72.7

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

   5. Applied rewrites 72.7%

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

   if -3.6999999999999998e104 < c < 6.4999999999999994e132

   1. Initial program 72.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

Alternative 8: 43.2% 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 63.1%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 40.6

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

5. Applied rewrites 40.6%

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

Developer Target 1: 99.4% 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 2024271 
(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))))