Complex division, imag part

Percentage Accurate: 62.1% → 95.8%

Time: 10.1s

Alternatives: 11

Speedup: 0.8×

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: 62.1% 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: 95.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c, d)
	return fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(d / hypot(d, c)) / Float64(-hypot(d, c)))))
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[(a * N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.5%

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

   1. div-sub 65.1%

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

   2. *-commutative 65.1%

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

   3. add-sqr-sqrt 65.1%

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

   4. times-frac 67.0%

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

   5. fmm-def 67.0%

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

   6. hypot-define 67.0%

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

   7. hypot-define 82.3%

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

   8. associate-/l* 84.6%

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

   9. add-sqr-sqrt 84.6%

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

   10. pow2 84.6%

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

   11. hypot-define 84.6%

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

4. Applied egg-rr 84.6%

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

   1. *-un-lft-identity 84.6%

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

   2. unpow2 84.6%

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

   3. times-frac 96.7%

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

   4. hypot-undefine 84.6%

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

   5. +-commutative 84.6%

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

   6. hypot-undefine 96.7%

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

   7. hypot-undefine 84.6%

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

   8. +-commutative 84.6%

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

   9. hypot-undefine 96.7%

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

6. Applied egg-rr 96.7%

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

   1. associate-*l/ 96.8%

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

   2. *-lft-identity 96.8%

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 2: 90.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ t_1 := \frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{if}\;d \leq -2.95 \cdot 10^{+180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-175}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (/ c (hypot c d))
          (/ b (hypot c d))
          (* (/ d (pow (hypot c d) 2.0)) (- a))))
        (t_1 (/ (- (/ c (/ d b)) a) d)))
   (if (<= d -2.95e+180)
     t_1
     (if (<= d -1.3e-120)
       t_0
       (if (<= d 4e-175)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 2.2e+152) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), ((d / pow(hypot(c, d), 2.0)) * -a));
	double t_1 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -2.95e+180) {
		tmp = t_1;
	} else if (d <= -1.3e-120) {
		tmp = t_0;
	} else if (d <= 4e-175) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.2e+152) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(d / (hypot(c, d) ^ 2.0)) * Float64(-a)))
	t_1 = Float64(Float64(Float64(c / Float64(d / b)) - a) / d)
	tmp = 0.0
	if (d <= -2.95e+180)
		tmp = t_1;
	elseif (d <= -1.3e-120)
		tmp = t_0;
	elseif (d <= 4e-175)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.2e+152)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(d / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.95e+180], t$95$1, If[LessEqual[d, -1.3e-120], t$95$0, If[LessEqual[d, 4e-175], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.2e+152], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.9500000000000001e180 or 2.1999999999999998e152 < d

   1. Initial program 34.3%

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

   3. Taylor expanded in c around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. mul-1-neg 70.4%

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

      3. unsub-neg 70.4%

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

      4. unpow2 70.4%

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

      5. associate-/r* 74.1%

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

      6. div-sub 74.1%

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

      7. *-commutative 74.1%

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

      8. associate-/l* 89.2%

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

   5. Simplified 89.2%

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

      1. clear-num 91.0%

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

      2. un-div-inv 91.0%

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

   7. Applied egg-rr 91.0%

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

   if -2.9500000000000001e180 < d < -1.3000000000000001e-120 or 4e-175 < d < 2.1999999999999998e152

   1. Initial program 78.0%

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

      1. div-sub 78.0%

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

      2. *-commutative 78.0%

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

      3. add-sqr-sqrt 77.9%

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

      4. times-frac 80.5%

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

      5. fmm-def 80.5%

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

      6. hypot-define 80.6%

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

      7. hypot-define 90.5%

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

      8. associate-/l* 93.9%

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

      9. add-sqr-sqrt 93.9%

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

      10. pow2 93.9%

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

      11. hypot-define 93.9%

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

   4. Applied egg-rr 93.9%

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

   if -1.3000000000000001e-120 < d < 4e-175

   1. Initial program 71.2%

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

      1. div-sub 63.8%

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

      2. *-commutative 63.8%

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

      3. add-sqr-sqrt 63.8%

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

      4. times-frac 66.4%

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

      5. fmm-def 66.4%

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

      6. hypot-define 66.4%

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

      7. hypot-define 89.1%

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

      8. associate-/l* 89.2%

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

      9. add-sqr-sqrt 89.2%

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

      10. pow2 89.2%

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

      11. hypot-define 89.2%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in c around inf 96.3%

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

      1. mul-1-neg 96.3%

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

      2. unsub-neg 96.3%

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

      3. *-commutative 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.95 \cdot 10^{+180}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \mathbf{elif}\;d \leq 4 \cdot 10^{-175}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}} \cdot \left(-a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{if}\;d \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ c (/ d b)) a) d)))
   (if (<= d -2.1e+118)
     t_0
     (if (<= d -2.5e-114)
       (/ (fma b c (* d (- a))) (fma d d (* c c)))
       (if (<= d 5.4e-158)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.45e+75)
           (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -2.1e+118) {
		tmp = t_0;
	} else if (d <= -2.5e-114) {
		tmp = fma(b, c, (d * -a)) / fma(d, d, (c * c));
	} else if (d <= 5.4e-158) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.45e+75) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c / Float64(d / b)) - a) / d)
	tmp = 0.0
	if (d <= -2.1e+118)
		tmp = t_0;
	elseif (d <= -2.5e-114)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(d, d, Float64(c * c)));
	elseif (d <= 5.4e-158)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.45e+75)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.1e+118], t$95$0, If[LessEqual[d, -2.5e-114], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.4e-158], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.45e+75], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.1e118 or 1.4499999999999999e75 < d

   1. Initial program 41.3%

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

   3. Taylor expanded in c around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

      4. unpow2 66.6%

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

      5. associate-/r* 69.1%

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

      6. div-sub 69.1%

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

      7. *-commutative 69.1%

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

      8. associate-/l* 80.4%

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

   5. Simplified 80.4%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.8%

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

   7. Applied egg-rr 82.8%

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

   if -2.1e118 < d < -2.49999999999999995e-114

   1. Initial program 82.8%

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

      1. fmm-def 82.8%

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

      2. distribute-rgt-neg-out 82.8%

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

      3. +-commutative 82.8%

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

      4. fma-define 82.8%

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

   3. Simplified 82.8%

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

   if -2.49999999999999995e-114 < d < 5.3999999999999997e-158

   1. Initial program 71.0%

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

      1. div-sub 64.0%

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

      2. *-commutative 64.0%

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

      3. add-sqr-sqrt 64.0%

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

      4. times-frac 66.4%

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

      5. fmm-def 66.4%

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

      6. hypot-define 66.4%

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

      7. hypot-define 88.8%

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

      8. associate-/l* 88.9%

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

      9. add-sqr-sqrt 88.9%

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

      10. pow2 88.9%

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

      11. hypot-define 88.9%

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

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in c around inf 95.5%

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

      1. mul-1-neg 95.5%

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

      2. unsub-neg 95.5%

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

      3. *-commutative 95.5%

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

   7. Simplified 95.5%

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

   if 5.3999999999999997e-158 < d < 1.4499999999999999e75

   1. Initial program 88.1%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{+75}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- (/ c (/ d b)) a) d)))
   (if (<= d -1.9e+118)
     t_1
     (if (<= d -1.45e-114)
       t_0
       (if (<= d 1.65e-158)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 1.8e+75) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -1.9e+118) {
		tmp = t_1;
	} else if (d <= -1.45e-114) {
		tmp = t_0;
	} else if (d <= 1.65e-158) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.8e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    t_1 = ((c / (d / b)) - a) / d
    if (d <= (-1.9d+118)) then
        tmp = t_1
    else if (d <= (-1.45d-114)) then
        tmp = t_0
    else if (d <= 1.65d-158) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 1.8d+75) then
        tmp = t_0
    else
        tmp = t_1
    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 t_1 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -1.9e+118) {
		tmp = t_1;
	} else if (d <= -1.45e-114) {
		tmp = t_0;
	} else if (d <= 1.65e-158) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 1.8e+75) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = ((c / (d / b)) - a) / d
	tmp = 0
	if d <= -1.9e+118:
		tmp = t_1
	elif d <= -1.45e-114:
		tmp = t_0
	elif d <= 1.65e-158:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 1.8e+75:
		tmp = t_0
	else:
		tmp = t_1
	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)))
	t_1 = Float64(Float64(Float64(c / Float64(d / b)) - a) / d)
	tmp = 0.0
	if (d <= -1.9e+118)
		tmp = t_1;
	elseif (d <= -1.45e-114)
		tmp = t_0;
	elseif (d <= 1.65e-158)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 1.8e+75)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = ((c / (d / b)) - a) / d;
	tmp = 0.0;
	if (d <= -1.9e+118)
		tmp = t_1;
	elseif (d <= -1.45e-114)
		tmp = t_0;
	elseif (d <= 1.65e-158)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 1.8e+75)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.9e+118], t$95$1, If[LessEqual[d, -1.45e-114], t$95$0, If[LessEqual[d, 1.65e-158], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.8e+75], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.90000000000000008e118 or 1.8e75 < d

   1. Initial program 41.3%

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

   3. Taylor expanded in c around 0 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

      4. unpow2 66.6%

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

      5. associate-/r* 69.1%

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

      6. div-sub 69.1%

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

      7. *-commutative 69.1%

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

      8. associate-/l* 80.4%

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

   5. Simplified 80.4%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.8%

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

   7. Applied egg-rr 82.8%

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

   if -1.90000000000000008e118 < d < -1.44999999999999998e-114 or 1.6500000000000001e-158 < d < 1.8e75

   1. Initial program 85.2%

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

   if -1.44999999999999998e-114 < d < 1.6500000000000001e-158

   1. Initial program 71.0%

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

      1. div-sub 64.0%

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

      2. *-commutative 64.0%

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

      3. add-sqr-sqrt 64.0%

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

      4. times-frac 66.4%

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

      5. fmm-def 66.4%

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

      6. hypot-define 66.4%

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

      7. hypot-define 88.8%

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

      8. associate-/l* 88.9%

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

      9. add-sqr-sqrt 88.9%

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

      10. pow2 88.9%

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

      11. hypot-define 88.9%

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

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in c around inf 95.5%

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

      1. mul-1-neg 95.5%

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

      2. unsub-neg 95.5%

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

      3. *-commutative 95.5%

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

   7. Simplified 95.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-114}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.65 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3600:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.6e-108)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 7.5e-113)
     (/ (- (/ (* c b) d) a) d)
     (if (<= c 1.05e-30)
       (/ (* c b) (+ (* c c) (* d d)))
       (if (<= c 3600.0)
         (/ (- (* b (/ c d)) a) d)
         (/ (- b (* d (/ a c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-108) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 7.5e-113) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1.05e-30) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= 3600.0) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-5.6d-108)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 7.5d-113) then
        tmp = (((c * b) / d) - a) / d
    else if (c <= 1.05d-30) then
        tmp = (c * b) / ((c * c) + (d * d))
    else if (c <= 3600.0d0) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.6e-108) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 7.5e-113) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1.05e-30) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= 3600.0) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.6e-108:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 7.5e-113:
		tmp = (((c * b) / d) - a) / d
	elif c <= 1.05e-30:
		tmp = (c * b) / ((c * c) + (d * d))
	elif c <= 3600.0:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.6e-108)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 7.5e-113)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 1.05e-30)
		tmp = Float64(Float64(c * b) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 3600.0)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.6e-108)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 7.5e-113)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 1.05e-30)
		tmp = (c * b) / ((c * c) + (d * d));
	elseif (c <= 3600.0)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.6e-108], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.5e-113], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.05e-30], N[(N[(c * b), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3600.0], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -5.6e-108

   1. Initial program 67.6%

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

      1. div-sub 67.6%

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

      2. *-commutative 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. times-frac 70.4%

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

      5. fmm-def 70.4%

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

      6. hypot-define 70.4%

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

      7. hypot-define 89.6%

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

      8. associate-/l* 90.9%

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

      9. add-sqr-sqrt 90.9%

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

      10. pow2 90.9%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -5.6e-108 < c < 7.5000000000000002e-113

   1. Initial program 82.0%

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

   3. Taylor expanded in c around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. unpow2 84.5%

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

      5. associate-/r* 91.0%

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

      6. div-sub 93.5%

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

      7. *-commutative 93.5%

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

      8. associate-/l* 92.1%

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

   5. Simplified 92.1%

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

      1. associate-*r/ 93.5%

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

   7. Applied egg-rr 93.5%

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

   if 7.5000000000000002e-113 < c < 1.0500000000000001e-30

   1. Initial program 99.7%

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

   3. Taylor expanded in b around inf 79.1%

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

      1. *-commutative 79.1%

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

   5. Simplified 79.1%

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

   if 1.0500000000000001e-30 < c < 3600

   1. Initial program 43.7%

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

   3. Taylor expanded in c around 0 86.0%

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

      1. +-commutative 86.0%

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

      2. mul-1-neg 86.0%

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

      3. unsub-neg 86.0%

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

      4. unpow2 86.0%

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

      5. associate-/r* 86.2%

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

      6. div-sub 86.2%

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

      7. *-commutative 86.2%

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

      8. associate-/l* 86.0%

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

   5. Simplified 86.0%

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

      1. clear-num 86.0%

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

      2. un-div-inv 86.0%

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

   7. Applied egg-rr 86.0%

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

      1. associate-/r/ 86.2%

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

   9. Applied egg-rr 86.2%

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

   if 3600 < c

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. *-commutative 48.2%

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

      3. add-sqr-sqrt 48.2%

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

      4. times-frac 52.8%

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

      5. fmm-def 52.8%

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

      6. hypot-define 52.8%

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

      7. hypot-define 80.8%

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

      8. associate-/l* 83.4%

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

      9. add-sqr-sqrt 83.4%

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

      10. pow2 83.4%

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

      11. hypot-define 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in c around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unsub-neg 68.4%

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

      3. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. associate-/l* 74.9%

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

   9. Applied egg-rr 74.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3600:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+79} \lor \neg \left(d \leq 4.7 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.2e+79) (not (<= d 4.7e+33)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.2e+79) || !(d <= 4.7e+33)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.2d+79)) .or. (.not. (d <= 4.7d+33))) then
        tmp = a / -d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.2e+79) || !(d <= 4.7e+33)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.2e+79) or not (d <= 4.7e+33):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.2e+79) || !(d <= 4.7e+33))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.2e+79) || ~((d <= 4.7e+33)))
		tmp = a / -d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.2e+79], N[Not[LessEqual[d, 4.7e+33]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.19999999999999993e79 or 4.6999999999999998e33 < d

   1. Initial program 52.0%

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

   3. Taylor expanded in c around 0 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   5. Simplified 67.0%

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

   if -1.19999999999999993e79 < d < 4.6999999999999998e33

   1. Initial program 76.5%

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

   3. Taylor expanded in c around inf 75.6%

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

      1. mul-1-neg 75.6%

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

      2. unsub-neg 75.6%

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

      3. unsub-neg 75.6%

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

      4. mul-1-neg 75.6%

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

      5. remove-double-neg 75.6%

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

      6. mul-1-neg 75.6%

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

      7. neg-mul-1 75.6%

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

      8. distribute-lft-in 75.6%

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

      9. distribute-lft-in 75.6%

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

      10. mul-1-neg 75.6%

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

      11. unsub-neg 75.6%

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

      12. neg-mul-1 75.6%

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

      13. mul-1-neg 75.6%

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

      14. remove-double-neg 75.6%

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

      15. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+79} \lor \neg \left(d \leq 4.7 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 3000:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e-105)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 3000.0) (/ (- (* c (/ b d)) a) d) (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e-105) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 3000.0) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.35d-105)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 3000.0d0) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e-105) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 3000.0) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e-105:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 3000.0:
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e-105)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 3000.0)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.35e-105)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 3000.0)
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e-105], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3000.0], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.34999999999999996e-105

   1. Initial program 67.6%

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

      1. div-sub 67.6%

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

      2. *-commutative 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. times-frac 70.4%

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

      5. fmm-def 70.4%

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

      6. hypot-define 70.4%

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

      7. hypot-define 89.6%

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

      8. associate-/l* 90.9%

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

      9. add-sqr-sqrt 90.9%

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

      10. pow2 90.9%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -1.34999999999999996e-105 < c < 3e3

   1. Initial program 81.8%

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

   3. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. unpow2 79.0%

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

      5. associate-/r* 84.1%

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

      6. div-sub 86.2%

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

      7. *-commutative 86.2%

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

      8. associate-/l* 85.1%

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

   5. Simplified 85.1%

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

   if 3e3 < c

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. *-commutative 48.2%

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

      3. add-sqr-sqrt 48.2%

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

      4. times-frac 52.8%

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

      5. fmm-def 52.8%

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

      6. hypot-define 52.8%

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

      7. hypot-define 80.8%

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

      8. associate-/l* 83.4%

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

      9. add-sqr-sqrt 83.4%

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

      10. pow2 83.4%

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

      11. hypot-define 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in c around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unsub-neg 68.4%

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

      3. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. associate-/l* 74.9%

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

   9. Applied egg-rr 74.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 3000:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 2900:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.35e-105)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 2900.0) (/ (- (* b (/ c d)) a) d) (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e-105) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 2900.0) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.35d-105)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 2900.0d0) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.35e-105) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 2900.0) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.35e-105:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 2900.0:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.35e-105)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 2900.0)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.35e-105)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 2900.0)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.35e-105], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2900.0], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.34999999999999996e-105

   1. Initial program 67.6%

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

      1. div-sub 67.6%

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

      2. *-commutative 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. times-frac 70.4%

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

      5. fmm-def 70.4%

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

      6. hypot-define 70.4%

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

      7. hypot-define 89.6%

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

      8. associate-/l* 90.9%

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

      9. add-sqr-sqrt 90.9%

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

      10. pow2 90.9%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -1.34999999999999996e-105 < c < 2900

   1. Initial program 81.8%

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

   3. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. unpow2 79.0%

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

      5. associate-/r* 84.1%

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

      6. div-sub 86.2%

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

      7. *-commutative 86.2%

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

      8. associate-/l* 85.1%

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

   5. Simplified 85.1%

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

      1. clear-num 85.0%

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

      2. un-div-inv 85.1%

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

   7. Applied egg-rr 85.1%

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

      1. associate-/r/ 86.0%

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

   9. Applied egg-rr 86.0%

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

   if 2900 < c

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. *-commutative 48.2%

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

      3. add-sqr-sqrt 48.2%

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

      4. times-frac 52.8%

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

      5. fmm-def 52.8%

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

      6. hypot-define 52.8%

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

      7. hypot-define 80.8%

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

      8. associate-/l* 83.4%

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

      9. add-sqr-sqrt 83.4%

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

      10. pow2 83.4%

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

      11. hypot-define 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in c around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unsub-neg 68.4%

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

      3. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. associate-/l* 74.9%

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

   9. Applied egg-rr 74.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.35 \cdot 10^{-105}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 2900:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 2800:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.9e-108)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 2800.0) (/ (- (/ (* c b) d) a) d) (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.9e-108) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 2800.0) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.9d-108)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 2800.0d0) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.9e-108) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 2800.0) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.9e-108:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 2800.0:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.9e-108)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 2800.0)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.9e-108)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 2800.0)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.9e-108], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2800.0], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.9000000000000001e-108

   1. Initial program 67.6%

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

      1. div-sub 67.6%

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

      2. *-commutative 67.6%

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

      3. add-sqr-sqrt 67.6%

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

      4. times-frac 70.4%

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

      5. fmm-def 70.4%

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

      6. hypot-define 70.4%

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

      7. hypot-define 89.6%

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

      8. associate-/l* 90.9%

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

      9. add-sqr-sqrt 90.9%

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

      10. pow2 90.9%

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

      11. hypot-define 90.8%

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in c around inf 74.9%

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

      1. mul-1-neg 74.9%

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

      2. unsub-neg 74.9%

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

      3. *-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -2.9000000000000001e-108 < c < 2800

   1. Initial program 81.8%

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

   3. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. unpow2 79.0%

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

      5. associate-/r* 84.1%

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

      6. div-sub 86.2%

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

      7. *-commutative 86.2%

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

      8. associate-/l* 85.1%

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

   5. Simplified 85.1%

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

      1. associate-*r/ 86.2%

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

   7. Applied egg-rr 86.2%

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

   if 2800 < c

   1. Initial program 48.2%

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

      1. div-sub 48.2%

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

      2. *-commutative 48.2%

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

      3. add-sqr-sqrt 48.2%

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

      4. times-frac 52.8%

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

      5. fmm-def 52.8%

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

      6. hypot-define 52.8%

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

      7. hypot-define 80.8%

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

      8. associate-/l* 83.4%

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

      9. add-sqr-sqrt 83.4%

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

      10. pow2 83.4%

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

      11. hypot-define 83.4%

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

   4. Applied egg-rr 83.4%

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

   5. Taylor expanded in c around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unsub-neg 68.4%

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

      3. *-commutative 68.4%

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

   7. Simplified 68.4%

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

      1. associate-/l* 74.9%

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

   9. Applied egg-rr 74.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.9 \cdot 10^{-108}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 2800:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-108} \lor \neg \left(c \leq 8.5 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.6e-108) (not (<= c 8.5e-124))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.6e-108) || !(c <= 8.5e-124)) {
		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 <= (-5.6d-108)) .or. (.not. (c <= 8.5d-124))) 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 <= -5.6e-108) || !(c <= 8.5e-124)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.6e-108) or not (c <= 8.5e-124):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.6e-108) || !(c <= 8.5e-124))
		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 <= -5.6e-108) || ~((c <= 8.5e-124)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.6e-108], N[Not[LessEqual[c, 8.5e-124]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.6e-108 or 8.5000000000000002e-124 < c

   1. Initial program 61.7%

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

   3. Taylor expanded in c around inf 60.0%

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

   if -5.6e-108 < c < 8.5000000000000002e-124

   1. Initial program 81.5%

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

   3. Taylor expanded in c around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.6 \cdot 10^{-108} \lor \neg \left(c \leq 8.5 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.0% 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 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 47.4%

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

4. Final simplification 47.4%

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

Developer target: 99.2% 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 2024095 
(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))))