Complex division, imag part

Percentage Accurate: 61.1% → 96.2%

Time: 10.8s

Alternatives: 9

Speedup: 0.6×

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: 61.1% 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: 96.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)} \cdot \left(-a\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[a_, b_, c_, d_] := N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.8%

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

   1. div-sub 53.6%

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

   2. *-commutative 53.6%

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

   3. add-sqr-sqrt 53.6%

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

   4. times-frac 57.8%

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

   5. fma-neg 57.8%

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

   6. hypot-define 57.8%

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

   7. hypot-define 77.1%

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

   8. associate-/l* 81.9%

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

   9. add-sqr-sqrt 81.9%

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

   10. pow2 81.9%

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

   11. hypot-define 81.9%

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

4. Applied egg-rr 81.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -a \cdot \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} \]
5. Step-by-step derivation

   1. *-un-lft-identity 81.9%

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

   2. unpow2 81.9%

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

   3. times-frac 97.6%

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

   4. add-sqr-sqrt 49.5%

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

   5. sqrt-prod 59.3%

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

   6. sqr-neg 59.3%

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

   7. sqrt-unprod 29.0%

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

   8. add-sqr-sqrt 60.8%

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

   9. hypot-undefine 57.9%

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

   10. +-commutative 57.9%

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

   11. hypot-define 60.8%

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

   12. add-sqr-sqrt 29.0%

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

   13. sqrt-unprod 59.3%

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

   14. sqr-neg 59.3%

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

   15. sqrt-prod 49.5%

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

   16. add-sqr-sqrt 97.6%

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

   17. hypot-undefine 82.0%

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

   18. +-commutative 82.0%

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

   19. hypot-define 97.6%

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

6. Applied egg-rr 97.6%

   \[\leadsto \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -a \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{d}{\mathsf{hypot}\left(d, c\right)}\right)}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 97.6%

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

   2. *-lft-identity 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

   \[\leadsto \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{\frac{d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)} \cdot \left(-a\right)\right) \]
10. Add Preprocessing

Alternative 2: 93.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := \frac{b}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := \mathsf{fma}\left(t\_0, t\_1, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ t_3 := \mathsf{fma}\left(t\_0, t\_1, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -9 \cdot 10^{+153}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+128}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ c (hypot c d)))
        (t_1 (/ b (hypot c d)))
        (t_2 (fma t_0 t_1 (* (/ d (pow (hypot c d) 2.0)) (- a))))
        (t_3 (fma t_0 t_1 (/ a (- d)))))
   (if (<= d -9e+153)
     t_3
     (if (<= d -1e-160)
       t_2
       (if (<= d 3.8e-168)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.3e+128) t_2 t_3))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = c / hypot(c, d);
	double t_1 = b / hypot(c, d);
	double t_2 = fma(t_0, t_1, ((d / pow(hypot(c, d), 2.0)) * -a));
	double t_3 = fma(t_0, t_1, (a / -d));
	double tmp;
	if (d <= -9e+153) {
		tmp = t_3;
	} else if (d <= -1e-160) {
		tmp = t_2;
	} else if (d <= 3.8e-168) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.3e+128) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(c / hypot(c, d))
	t_1 = Float64(b / hypot(c, d))
	t_2 = fma(t_0, t_1, Float64(Float64(d / (hypot(c, d) ^ 2.0)) * Float64(-a)))
	t_3 = fma(t_0, t_1, Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -9e+153)
		tmp = t_3;
	elseif (d <= -1e-160)
		tmp = t_2;
	elseif (d <= 3.8e-168)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.3e+128)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[(N[(d / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$1 + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -9e+153], t$95$3, If[LessEqual[d, -1e-160], t$95$2, If[LessEqual[d, 3.8e-168], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.3e+128], t$95$2, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -9.0000000000000002e153 or 1.3e128 < d

   1. Initial program 30.5%

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

      1. div-sub 30.5%

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

      2. *-commutative 30.5%

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

      3. add-sqr-sqrt 30.5%

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

      4. times-frac 30.8%

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

      5. fma-neg 30.8%

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

      6. hypot-define 30.8%

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

      7. hypot-define 50.2%

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

      8. associate-/l* 51.4%

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

      9. add-sqr-sqrt 51.4%

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

      10. pow2 51.4%

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

      11. hypot-define 51.4%

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

   4. Applied egg-rr 51.4%

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

   5. Taylor expanded in d around inf 94.8%

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

   if -9.0000000000000002e153 < d < -9.9999999999999999e-161 or 3.8e-168 < d < 1.3e128

   1. Initial program 69.1%

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

      1. div-sub 69.1%

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

      2. *-commutative 69.1%

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

      3. add-sqr-sqrt 69.1%

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

      4. times-frac 72.5%

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

      5. fma-neg 72.5%

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

      6. hypot-define 72.6%

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

      7. hypot-define 90.6%

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

      8. associate-/l* 98.0%

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

      9. add-sqr-sqrt 98.0%

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

      10. pow2 98.0%

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

      11. hypot-define 98.0%

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

   4. Applied egg-rr 98.0%

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

   if -9.9999999999999999e-161 < d < 3.8e-168

   1. Initial program 62.0%

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

   3. Taylor expanded in c around inf 95.2%

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

      1. associate-*r/ 95.2%

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

      2. mul-1-neg 95.2%

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

      3. distribute-rgt-neg-out 95.2%

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

   5. Simplified 95.2%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-160}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+277}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* c b) (* d a)) (+ (* c c) (* d d))) 1e+277)
   (* (/ 1.0 (hypot c d)) (/ (fma b c (* a (- d))) (hypot c d)))
   (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ a (- d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((c * b) - (d * a)) / ((c * c) + (d * d))) <= 1e+277) {
		tmp = (1.0 / hypot(c, d)) * (fma(b, c, (a * -d)) / hypot(c, d));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a / -d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d))) <= 1e+277)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(b, c, Float64(a * Float64(-d))) / hypot(c, d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a / Float64(-d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+277], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e277

   1. Initial program 75.9%

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

      1. *-un-lft-identity 75.9%

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

      2. add-sqr-sqrt 75.9%

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

      3. times-frac 75.8%

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

      4. hypot-define 75.9%

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

      5. fma-neg 75.9%

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

      6. distribute-rgt-neg-in 75.9%

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

      7. hypot-define 95.7%

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

   4. Applied egg-rr 95.7%

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

   if 1e277 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.6%

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

      1. div-sub 11.1%

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

      2. *-commutative 11.1%

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

      3. add-sqr-sqrt 11.1%

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

      4. times-frac 19.8%

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

      5. fma-neg 19.8%

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

      6. hypot-define 19.8%

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

      7. hypot-define 54.9%

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

      8. associate-/l* 63.5%

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

      9. add-sqr-sqrt 63.5%

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

      10. pow2 63.5%

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

      11. hypot-define 63.5%

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in d around inf 71.2%

      \[\leadsto \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, -\color{blue}{\frac{a}{d}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 10^{+277}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ t_1 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{if}\;d \leq -7 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-198}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c))
        (t_1 (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ a (- d)))))
   (if (<= d -7e-70)
     t_1
     (if (<= d 9e-198)
       t_0
       (if (<= d 5e-26)
         (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
         (if (<= d 4.4e+22) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double t_1 = fma((c / hypot(c, d)), (b / hypot(c, d)), (a / -d));
	double tmp;
	if (d <= -7e-70) {
		tmp = t_1;
	} else if (d <= 9e-198) {
		tmp = t_0;
	} else if (d <= 5e-26) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 4.4e+22) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	t_1 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a / Float64(-d)))
	tmp = 0.0
	if (d <= -7e-70)
		tmp = t_1;
	elseif (d <= 9e-198)
		tmp = t_0;
	elseif (d <= 5e-26)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 4.4e+22)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a / (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7e-70], t$95$1, If[LessEqual[d, 9e-198], t$95$0, If[LessEqual[d, 5e-26], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.4e+22], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.99999999999999949e-70 or 4.4e22 < d

   1. Initial program 47.2%

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

      1. div-sub 47.2%

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

      2. *-commutative 47.2%

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

      3. add-sqr-sqrt 47.2%

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

      4. times-frac 49.6%

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

      5. fma-neg 49.6%

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

      6. hypot-define 49.6%

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

      7. hypot-define 68.3%

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

      8. associate-/l* 75.2%

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

      9. add-sqr-sqrt 75.2%

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

      10. pow2 75.2%

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

      11. hypot-define 75.2%

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

   4. Applied egg-rr 75.2%

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

   5. Taylor expanded in d around inf 87.5%

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

   if -6.99999999999999949e-70 < d < 8.9999999999999996e-198 or 5.00000000000000019e-26 < d < 4.4e22

   1. Initial program 61.0%

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

   3. Taylor expanded in c around inf 91.9%

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

      1. remove-double-neg 91.9%

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

      2. mul-1-neg 91.9%

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

      3. neg-mul-1 91.9%

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

      4. distribute-lft-in 91.9%

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

      5. distribute-lft-in 91.9%

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

      6. mul-1-neg 91.9%

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

      7. unsub-neg 91.9%

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

      8. neg-mul-1 91.9%

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

      9. mul-1-neg 91.9%

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

      10. remove-double-neg 91.9%

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

      11. associate-/l* 92.0%

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

   5. Simplified 92.0%

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

   if 8.9999999999999996e-198 < d < 5.00000000000000019e-26

   1. Initial program 86.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-198}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-26}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{a}{-d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.55 \cdot 10^{+63}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{b - d \cdot \frac{a}{c}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -4.3e+85)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -1e-123)
       t_0
       (if (<= c 1.6e-59)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 2.55e+63) t_0 (/ 1.0 (/ c (- b (* d (/ a c)))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.3e+85) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -1e-123) {
		tmp = t_0;
	} else if (c <= 1.6e-59) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 2.55e+63) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (c / (b - (d * (a / c))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (c <= (-4.3d+85)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= (-1d-123)) then
        tmp = t_0
    else if (c <= 1.6d-59) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 2.55d+63) then
        tmp = t_0
    else
        tmp = 1.0d0 / (c / (b - (d * (a / c))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.3e+85) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -1e-123) {
		tmp = t_0;
	} else if (c <= 1.6e-59) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 2.55e+63) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (c / (b - (d * (a / c))));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.3e+85:
		tmp = (b - (a * (d / c))) / c
	elif c <= -1e-123:
		tmp = t_0
	elif c <= 1.6e-59:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 2.55e+63:
		tmp = t_0
	else:
		tmp = 1.0 / (c / (b - (d * (a / c))))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -4.3e+85)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -1e-123)
		tmp = t_0;
	elseif (c <= 1.6e-59)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 2.55e+63)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(c / Float64(b - Float64(d * Float64(a / c)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.3e+85)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -1e-123)
		tmp = t_0;
	elseif (c <= 1.6e-59)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 2.55e+63)
		tmp = t_0;
	else
		tmp = 1.0 / (c / (b - (d * (a / c))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.3e+85], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1e-123], t$95$0, If[LessEqual[c, 1.6e-59], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.55e+63], t$95$0, N[(1.0 / N[(c / N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.2999999999999999e85

   1. Initial program 24.9%

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

   3. Taylor expanded in c around inf 77.7%

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

      1. remove-double-neg 77.7%

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

      2. mul-1-neg 77.7%

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

      3. neg-mul-1 77.7%

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

      4. distribute-lft-in 77.7%

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

      5. distribute-lft-in 77.7%

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

      6. mul-1-neg 77.7%

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

      7. unsub-neg 77.7%

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

      8. neg-mul-1 77.7%

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

      9. mul-1-neg 77.7%

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

      10. remove-double-neg 77.7%

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

      11. associate-/l* 81.9%

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

   5. Simplified 81.9%

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

   if -4.2999999999999999e85 < c < -1.0000000000000001e-123 or 1.6e-59 < c < 2.5499999999999999e63

   1. Initial program 80.9%

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

   if -1.0000000000000001e-123 < c < 1.6e-59

   1. Initial program 66.1%

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

   3. Taylor expanded in c around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. mul-1-neg 84.0%

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

      3. unsub-neg 84.0%

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

      4. unpow2 84.0%

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

      5. associate-/r* 92.3%

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

      6. div-sub 93.3%

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

      7. associate-/l* 93.4%

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

   5. Simplified 93.4%

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

   if 2.5499999999999999e63 < c

   1. Initial program 39.7%

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

   3. Taylor expanded in c around inf 86.7%

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

      1. remove-double-neg 86.7%

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

      2. mul-1-neg 86.7%

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

      3. neg-mul-1 86.7%

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

      4. distribute-lft-in 86.7%

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

      5. distribute-lft-in 86.7%

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

      6. mul-1-neg 86.7%

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

      7. unsub-neg 86.7%

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

      8. neg-mul-1 86.7%

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

      9. mul-1-neg 86.7%

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

      10. remove-double-neg 86.7%

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

      11. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

      1. clear-num 88.1%

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

      2. inv-pow 88.1%

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

   7. Applied egg-rr 88.1%

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

      1. unpow-1 88.1%

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

      2. associate-*r/ 86.5%

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

      3. remove-double-neg 86.5%

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

      4. distribute-neg-frac2 86.5%

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

      5. associate-*l/ 90.0%

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

      6. *-commutative 90.0%

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

      7. distribute-rgt-neg-in 90.0%

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

      8. distribute-frac-neg2 90.0%

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

      9. remove-double-neg 90.0%

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

   9. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{c}{b - d \cdot \frac{a}{c}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.55 \cdot 10^{+63}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{c}{b - d \cdot \frac{a}{c}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+103} \lor \neg \left(c \leq -1.1 \cdot 10^{+36} \lor \neg \left(c \leq -8 \cdot 10^{-39}\right) \land c \leq 5.6 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.4e+103)
         (not (or (<= c -1.1e+36) (and (not (<= c -8e-39)) (<= c 5.6e-58)))))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.4e+103) || !((c <= -1.1e+36) || (!(c <= -8e-39) && (c <= 5.6e-58)))) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.4d+103)) .or. (.not. (c <= (-1.1d+36)) .or. (.not. (c <= (-8d-39))) .and. (c <= 5.6d-58))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.4e+103) || !((c <= -1.1e+36) || (!(c <= -8e-39) && (c <= 5.6e-58)))) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.4e+103) or not ((c <= -1.1e+36) or (not (c <= -8e-39) and (c <= 5.6e-58))):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.4e+103) || !((c <= -1.1e+36) || (!(c <= -8e-39) && (c <= 5.6e-58))))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.4e+103) || ~(((c <= -1.1e+36) || (~((c <= -8e-39)) && (c <= 5.6e-58)))))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.4e+103], N[Not[Or[LessEqual[c, -1.1e+36], And[N[Not[LessEqual[c, -8e-39]], $MachinePrecision], LessEqual[c, 5.6e-58]]]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.3999999999999998e103 or -1.1e36 < c < -7.99999999999999943e-39 or 5.6000000000000001e-58 < c

   1. Initial program 49.1%

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

   3. Taylor expanded in c around inf 79.8%

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

      1. remove-double-neg 79.8%

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

      2. mul-1-neg 79.8%

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

      3. neg-mul-1 79.8%

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

      4. distribute-lft-in 79.8%

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

      5. distribute-lft-in 79.8%

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

      6. mul-1-neg 79.8%

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

      7. unsub-neg 79.8%

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

      8. neg-mul-1 79.8%

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

      9. mul-1-neg 79.8%

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

      10. remove-double-neg 79.8%

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

      11. associate-/l* 81.1%

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

   5. Simplified 81.1%

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

   if -3.3999999999999998e103 < c < -1.1e36 or -7.99999999999999943e-39 < c < 5.6000000000000001e-58

   1. Initial program 65.1%

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

   3. Taylor expanded in c around 0 80.3%

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

      1. +-commutative 80.3%

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

      2. mul-1-neg 80.3%

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

      3. unsub-neg 80.3%

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

      4. unpow2 80.3%

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

      5. associate-/r* 86.9%

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

      6. div-sub 87.8%

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

      7. associate-/l* 89.4%

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

   5. Simplified 89.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{+103} \lor \neg \left(c \leq -1.1 \cdot 10^{+36} \lor \neg \left(c \leq -8 \cdot 10^{-39}\right) \land c \leq 5.6 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -0.00021:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+165} \lor \neg \left(d \leq 1.8 \cdot 10^{+184}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{b}{d}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -0.00021)
     t_0
     (if (<= d 7.2e+22)
       (/ (- b (* a (/ d c))) c)
       (if (or (<= d 1e+165) (not (<= d 1.8e+184)))
         t_0
         (/ (* d (/ b d)) c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -0.00021) {
		tmp = t_0;
	} else if (d <= 7.2e+22) {
		tmp = (b - (a * (d / c))) / c;
	} else if ((d <= 1e+165) || !(d <= 1.8e+184)) {
		tmp = t_0;
	} else {
		tmp = (d * (b / d)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-0.00021d0)) then
        tmp = t_0
    else if (d <= 7.2d+22) then
        tmp = (b - (a * (d / c))) / c
    else if ((d <= 1d+165) .or. (.not. (d <= 1.8d+184))) then
        tmp = t_0
    else
        tmp = (d * (b / d)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -0.00021) {
		tmp = t_0;
	} else if (d <= 7.2e+22) {
		tmp = (b - (a * (d / c))) / c;
	} else if ((d <= 1e+165) || !(d <= 1.8e+184)) {
		tmp = t_0;
	} else {
		tmp = (d * (b / d)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -0.00021:
		tmp = t_0
	elif d <= 7.2e+22:
		tmp = (b - (a * (d / c))) / c
	elif (d <= 1e+165) or not (d <= 1.8e+184):
		tmp = t_0
	else:
		tmp = (d * (b / d)) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -0.00021)
		tmp = t_0;
	elseif (d <= 7.2e+22)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif ((d <= 1e+165) || !(d <= 1.8e+184))
		tmp = t_0;
	else
		tmp = Float64(Float64(d * Float64(b / d)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -0.00021)
		tmp = t_0;
	elseif (d <= 7.2e+22)
		tmp = (b - (a * (d / c))) / c;
	elseif ((d <= 1e+165) || ~((d <= 1.8e+184)))
		tmp = t_0;
	else
		tmp = (d * (b / d)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -0.00021], t$95$0, If[LessEqual[d, 7.2e+22], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[d, 1e+165], N[Not[LessEqual[d, 1.8e+184]], $MachinePrecision]], t$95$0, N[(N[(d * N[(b / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.1000000000000001e-4 or 7.2e22 < d < 9.99999999999999899e164 or 1.80000000000000007e184 < d

   1. Initial program 44.1%

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

   3. Taylor expanded in c around 0 68.4%

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

      1. associate-*r/ 68.4%

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

      2. neg-mul-1 68.4%

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

   5. Simplified 68.4%

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

   if -2.1000000000000001e-4 < d < 7.2e22

   1. Initial program 69.5%

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

   3. Taylor expanded in c around inf 81.3%

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

      1. remove-double-neg 81.3%

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

      2. mul-1-neg 81.3%

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

      3. neg-mul-1 81.3%

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

      4. distribute-lft-in 81.3%

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

      5. distribute-lft-in 81.3%

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

      6. mul-1-neg 81.3%

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

      7. unsub-neg 81.3%

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

      8. neg-mul-1 81.3%

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

      9. mul-1-neg 81.3%

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

      10. remove-double-neg 81.3%

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

      11. associate-/l* 81.4%

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

   5. Simplified 81.4%

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

   if 9.99999999999999899e164 < d < 1.80000000000000007e184

   1. Initial program 28.1%

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

   3. Taylor expanded in c around inf 70.4%

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

      1. associate-*r/ 70.4%

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

      2. mul-1-neg 70.4%

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

      3. distribute-rgt-neg-out 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in d around inf 70.1%

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

      1. +-commutative 70.1%

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

      2. mul-1-neg 70.1%

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

      3. unsub-neg 70.1%

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

   8. Simplified 70.1%

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

   9. Taylor expanded in b around inf 84.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.00021:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 7.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 10^{+165} \lor \neg \left(d \leq 1.8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{d \cdot \frac{b}{d}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{+100} \lor \neg \left(c \leq -8 \cdot 10^{+33} \lor \neg \left(c \leq -2 \cdot 10^{-21}\right) \land c \leq 4.6 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2e+100)
         (not (or (<= c -8e+33) (and (not (<= c -2e-21)) (<= c 4.6e-19)))))
   (/ b c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2e+100) || !((c <= -8e+33) || (!(c <= -2e-21) && (c <= 4.6e-19)))) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2d+100)) .or. (.not. (c <= (-8d+33)) .or. (.not. (c <= (-2d-21))) .and. (c <= 4.6d-19))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2e+100) || !((c <= -8e+33) || (!(c <= -2e-21) && (c <= 4.6e-19)))) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2e+100) or not ((c <= -8e+33) or (not (c <= -2e-21) and (c <= 4.6e-19))):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2e+100) || !((c <= -8e+33) || (!(c <= -2e-21) && (c <= 4.6e-19))))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2e+100) || ~(((c <= -8e+33) || (~((c <= -2e-21)) && (c <= 4.6e-19)))))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2e+100], N[Not[Or[LessEqual[c, -8e+33], And[N[Not[LessEqual[c, -2e-21]], $MachinePrecision], LessEqual[c, 4.6e-19]]]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.00000000000000003e100 or -7.9999999999999996e33 < c < -1.99999999999999982e-21 or 4.5999999999999996e-19 < c

   1. Initial program 46.2%

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

   3. Taylor expanded in c around inf 75.6%

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

   if -2.00000000000000003e100 < c < -7.9999999999999996e33 or -1.99999999999999982e-21 < c < 4.5999999999999996e-19

   1. Initial program 66.1%

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

   3. Taylor expanded in c around 0 62.2%

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

      1. associate-*r/ 62.2%

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

      2. neg-mul-1 62.2%

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

   5. Simplified 62.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{+100} \lor \neg \left(c \leq -8 \cdot 10^{+33} \lor \neg \left(c \leq -2 \cdot 10^{-21}\right) \land c \leq 4.6 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 5.0× 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 56.8%

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

3. Taylor expanded in c around inf 45.2%

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

4. Final simplification 45.2%

   \[\leadsto \frac{b}{c} \]
5. Add Preprocessing

Developer target: 99.4% accurate, 0.1× 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 2024075 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))