Complex division, imag part

Percentage Accurate: 61.4% → 83.7%

Time: 8.2s

Alternatives: 11

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, -\frac{d \cdot a}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -3.2 \cdot 10^{+81}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq -4.3 \cdot 10^{-91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-82}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+39}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma (/ c t_0) b (- (/ (* d a) t_0))))
        (t_2 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -3.2e+81)
     t_2
     (if (<= d -4.3e-91)
       t_1
       (if (<= d 1.85e-82)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 7e+39) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / t_0), b, -((d * a) / t_0));
	double t_2 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -3.2e+81) {
		tmp = t_2;
	} else if (d <= -4.3e-91) {
		tmp = t_1;
	} else if (d <= 1.85e-82) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 7e+39) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), b, Float64(-Float64(Float64(d * a) / t_0)))
	t_2 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -3.2e+81)
		tmp = t_2;
	elseif (d <= -4.3e-91)
		tmp = t_1;
	elseif (d <= 1.85e-82)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 7e+39)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / t$95$0), $MachinePrecision] * b + (-N[(N[(d * a), $MachinePrecision] / t$95$0), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -3.2e+81], t$95$2, If[LessEqual[d, -4.3e-91], t$95$1, If[LessEqual[d, 1.85e-82], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7e+39], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -3.2e81 or 7.0000000000000003e39 < d

   1. Initial program 43.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. 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. lower-fma.f64 N/A

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

      13. lower-/.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. lower-neg.f64 85.8

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

   5. Applied rewrites 85.8%

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

   if -3.2e81 < d < -4.3e-91 or 1.85e-82 < d < 7.0000000000000003e39

   1. Initial program 80.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 N/A

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

   4. Applied rewrites 85.7%

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

   if -4.3e-91 < d < 1.85e-82

   1. Initial program 67.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 98.1

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

   5. Applied rewrites 98.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -4.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, -\frac{d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-82}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 7 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, -\frac{d \cdot a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{t\_0}, \frac{b \cdot c}{t\_0}\right)\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -9e+70)
     t_1
     (if (<= d -1e-85)
       (fma (- d) (/ a t_0) (/ (* b c) t_0))
       (if (<= d 5800000.0) (/ (- b (/ (* d a) c)) c) t_1)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -9e+70) {
		tmp = t_1;
	} else if (d <= -1e-85) {
		tmp = fma(-d, (a / t_0), ((b * c) / t_0));
	} else if (d <= 5800000.0) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -9e+70)
		tmp = t_1;
	elseif (d <= -1e-85)
		tmp = fma(Float64(-d), Float64(a / t_0), Float64(Float64(b * c) / t_0));
	elseif (d <= 5800000.0)
		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[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -9e+70], t$95$1, If[LessEqual[d, -1e-85], N[((-d) * N[(a / t$95$0), $MachinePrecision] + N[(N[(b * c), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5800000.0], 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 < -8.9999999999999999e70 or 5.8e6 < d

   1. Initial program 45.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. 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. lower-fma.f64 N/A

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

      13. lower-/.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. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -8.9999999999999999e70 < d < -9.9999999999999998e-86

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lower-/.f64 82.3

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

   4. Applied rewrites 82.3%

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

   if -9.9999999999999998e-86 < d < 5.8e6

   1. Initial program 69.3%

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

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

      7. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(-d, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{b \cdot c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -6.5e+70)
     t_0
     (if (<= d -1e-85)
       (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))
       (if (<= d 5800000.0) (/ (- 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), -a) / d;
	double tmp;
	if (d <= -6.5e+70) {
		tmp = t_0;
	} else if (d <= -1e-85) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 5800000.0) {
		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(-a)) / d)
	tmp = 0.0
	if (d <= -6.5e+70)
		tmp = t_0;
	elseif (d <= -1e-85)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 5800000.0)
		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] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6.5e+70], t$95$0, If[LessEqual[d, -1e-85], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5800000.0], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.49999999999999978e70 or 5.8e6 < d

   1. Initial program 45.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. 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. lower-fma.f64 N/A

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

      13. lower-/.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. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -6.49999999999999978e70 < d < -9.9999999999999998e-86

   1. Initial program 82.2%

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

   if -9.9999999999999998e-86 < d < 5.8e6

   1. Initial program 69.3%

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

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

      7. lower-*.f64 92.9

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

   5. Applied rewrites 92.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-85}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 15500000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -1.3e+108)
     t_0
     (if (<= d -5e+30)
       (/ (- (* b c) (* d a)) (* d d))
       (if (<= d 15500000.0) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.3e+108) {
		tmp = t_0;
	} else if (d <= -5e+30) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 15500000.0) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-1.3d+108)) then
        tmp = t_0
    else if (d <= (-5d+30)) then
        tmp = ((b * c) - (d * a)) / (d * d)
    else if (d <= 15500000.0d0) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.3e+108) {
		tmp = t_0;
	} else if (d <= -5e+30) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= 15500000.0) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -1.3e+108:
		tmp = t_0
	elif d <= -5e+30:
		tmp = ((b * c) - (d * a)) / (d * d)
	elif d <= 15500000.0:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.3e+108)
		tmp = t_0;
	elseif (d <= -5e+30)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (d <= 15500000.0)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -1.3e+108)
		tmp = t_0;
	elseif (d <= -5e+30)
		tmp = ((b * c) - (d * a)) / (d * d);
	elseif (d <= 15500000.0)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -1.3e+108], t$95$0, If[LessEqual[d, -5e+30], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 15500000.0], 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 < -1.3000000000000001e108 or 1.55e7 < d

   1. Initial program 42.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.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. lower-neg.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.3000000000000001e108 < d < -4.9999999999999998e30

   1. Initial program 92.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   if -4.9999999999999998e30 < d < 1.55e7

   1. Initial program 70.3%

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

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq 15500000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.3 \cdot 10^{+108}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{-a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -1.3e+108)
     t_0
     (if (<= d -4.8e+30)
       (/ (- (* b c) (* d a)) (* d d))
       (if (<= d -1.35e-118)
         (* d (/ (- a) (fma d d (* c c))))
         (if (<= d 4800000.0) (/ b c) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.3e+108) {
		tmp = t_0;
	} else if (d <= -4.8e+30) {
		tmp = ((b * c) - (d * a)) / (d * d);
	} else if (d <= -1.35e-118) {
		tmp = d * (-a / fma(d, d, (c * c)));
	} else if (d <= 4800000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.3e+108)
		tmp = t_0;
	elseif (d <= -4.8e+30)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(d * d));
	elseif (d <= -1.35e-118)
		tmp = Float64(d * Float64(Float64(-a) / fma(d, d, Float64(c * c))));
	elseif (d <= 4800000.0)
		tmp = Float64(b / c);
	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, -1.3e+108], t$95$0, If[LessEqual[d, -4.8e+30], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -1.35e-118], N[(d * N[((-a) / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4800000.0], N[(b / c), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.3000000000000001e108 or 4.8e6 < d

   1. Initial program 42.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.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. lower-neg.f64 73.2

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

   5. Applied rewrites 73.2%

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

   if -1.3000000000000001e108 < d < -4.7999999999999999e30

   1. Initial program 92.2%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 85.6

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

   5. Applied rewrites 85.6%

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

   if -4.7999999999999999e30 < d < -1.34999999999999997e-118

   1. Initial program 82.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   7. Applied rewrites 64.5%

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

   if -1.34999999999999997e-118 < d < 4.8e6

   1. Initial program 67.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -4.8 \cdot 10^{+30}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{-a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -5e+30) t_0 (if (<= d 5800000.0) (/ (- 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), -a) / d;
	double tmp;
	if (d <= -5e+30) {
		tmp = t_0;
	} else if (d <= 5800000.0) {
		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(-a)) / d)
	tmp = 0.0
	if (d <= -5e+30)
		tmp = t_0;
	elseif (d <= 5800000.0)
		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] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -5e+30], t$95$0, If[LessEqual[d, 5800000.0], 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 < -4.9999999999999998e30 or 5.8e6 < d

   1. Initial program 48.8%

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

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

      13. lower-/.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. lower-neg.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if -4.9999999999999998e30 < d < 5.8e6

   1. Initial program 70.3%

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

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 7: 76.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5800000:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ (* b c) d) a) d)))
   (if (<= d -5e+30) t_0 (if (<= d 5800000.0) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (((b * c) / d) - a) / d;
	double tmp;
	if (d <= -5e+30) {
		tmp = t_0;
	} else if (d <= 5800000.0) {
		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 = (((b * c) / d) - a) / d
    if (d <= (-5d+30)) then
        tmp = t_0
    else if (d <= 5800000.0d0) 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 = (((b * c) / d) - a) / d;
	double tmp;
	if (d <= -5e+30) {
		tmp = t_0;
	} else if (d <= 5800000.0) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (((b * c) / d) - a) / d
	tmp = 0
	if d <= -5e+30:
		tmp = t_0
	elif d <= 5800000.0:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(Float64(b * c) / d) - a) / d)
	tmp = 0.0
	if (d <= -5e+30)
		tmp = t_0;
	elseif (d <= 5800000.0)
		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 = (((b * c) / d) - a) / d;
	tmp = 0.0;
	if (d <= -5e+30)
		tmp = t_0;
	elseif (d <= 5800000.0)
		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[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -5e+30], t$95$0, If[LessEqual[d, 5800000.0], 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 < -4.9999999999999998e30 or 5.8e6 < d

   1. Initial program 48.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 19.2

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

   5. Applied rewrites 19.2%

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

   6. Taylor expanded in d around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 79.2

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

   8. Applied rewrites 79.2%

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

   if -4.9999999999999998e30 < d < 5.8e6

   1. Initial program 70.3%

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

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

      7. lower-*.f64 88.2

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

   5. Applied rewrites 88.2%

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

Alternative 8: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.58 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{-a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -1.58e+72)
     t_0
     (if (<= d -1.35e-118)
       (* d (/ (- a) (fma d d (* c c))))
       (if (<= d 4800000.0) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.58e+72) {
		tmp = t_0;
	} else if (d <= -1.35e-118) {
		tmp = d * (-a / fma(d, d, (c * c)));
	} else if (d <= 4800000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.58e+72)
		tmp = t_0;
	elseif (d <= -1.35e-118)
		tmp = Float64(d * Float64(Float64(-a) / fma(d, d, Float64(c * c))));
	elseif (d <= 4800000.0)
		tmp = Float64(b / c);
	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, -1.58e+72], t$95$0, If[LessEqual[d, -1.35e-118], N[(d * N[((-a) / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4800000.0], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.58000000000000006e72 or 4.8e6 < d

   1. Initial program 46.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.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. lower-neg.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -1.58000000000000006e72 < d < -1.34999999999999997e-118

   1. Initial program 83.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   7. Applied rewrites 58.8%

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

   if -1.34999999999999997e-118 < d < 4.8e6

   1. Initial program 67.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.6

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

   5. Applied rewrites 81.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{+72}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -1.35 \cdot 10^{-118}:\\ \;\;\;\;d \cdot \frac{-a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.58 \cdot 10^{+72}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-112}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -1.58e+72)
     t_0
     (if (<= d -4.4e-112)
       (* a (/ (- d) (fma d d (* c c))))
       (if (<= d 4800000.0) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.58e+72) {
		tmp = t_0;
	} else if (d <= -4.4e-112) {
		tmp = a * (-d / fma(d, d, (c * c)));
	} else if (d <= 4800000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.58e+72)
		tmp = t_0;
	elseif (d <= -4.4e-112)
		tmp = Float64(a * Float64(Float64(-d) / fma(d, d, Float64(c * c))));
	elseif (d <= 4800000.0)
		tmp = Float64(b / c);
	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, -1.58e+72], t$95$0, If[LessEqual[d, -4.4e-112], N[(a * N[((-d) / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4800000.0], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.58000000000000006e72 or 4.8e6 < d

   1. Initial program 46.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.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. lower-neg.f64 71.3

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

   5. Applied rewrites 71.3%

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

   if -1.58000000000000006e72 < d < -4.40000000000000042e-112

   1. Initial program 82.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 59.2

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

   5. Applied rewrites 59.2%

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

   if -4.40000000000000042e-112 < d < 4.8e6

   1. Initial program 68.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 81.0

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

   5. Applied rewrites 81.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.58 \cdot 10^{+72}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -4.4 \cdot 10^{-112}:\\ \;\;\;\;a \cdot \frac{-d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -8.8 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4800000:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -8.8e+17) t_0 (if (<= d 4800000.0) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -8.8e+17) {
		tmp = t_0;
	} else if (d <= 4800000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-8.8d+17)) then
        tmp = t_0
    else if (d <= 4800000.0d0) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -8.8e+17) {
		tmp = t_0;
	} else if (d <= 4800000.0) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -8.8e+17:
		tmp = t_0
	elif d <= 4800000.0:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -8.8e+17)
		tmp = t_0;
	elseif (d <= 4800000.0)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -8.8e+17)
		tmp = t_0;
	elseif (d <= 4800000.0)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -8.8e+17], t$95$0, If[LessEqual[d, 4800000.0], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -8.8e17 or 4.8e6 < d

   1. Initial program 49.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-/.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. lower-neg.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -8.8e17 < d < 4.8e6

   1. Initial program 70.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 11: 43.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{c} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 49.5

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

5. Applied rewrites 49.5%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
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 2024232 
(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))))