Complex division, imag part

Percentage Accurate: 60.8% → 91.3%

Time: 8.6s

Alternatives: 9

Speedup: 1.8×

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: 60.8% 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: 91.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{if}\;d \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{a - b \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{b}{c} + \frac{\frac{-1}{\frac{c}{d \cdot a}}}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (/ c (hypot c d))
          (/ b (hypot c d))
          (/ (- a) (/ (pow (hypot c d) 2.0) d)))))
   (if (<= d -1.35e+154)
     (/ (- a (* b (/ c d))) (hypot c d))
     (if (<= d -6e-133)
       t_0
       (if (<= d 3.7e-178)
         (+ (/ b c) (/ (/ -1.0 (/ c (* d a))) c))
         (if (<= d 2.3e+149) t_0 (- (* (/ c d) (/ b d)) (/ a d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / (pow(hypot(c, d), 2.0) / d)));
	double tmp;
	if (d <= -1.35e+154) {
		tmp = (a - (b * (c / d))) / hypot(c, d);
	} else if (d <= -6e-133) {
		tmp = t_0;
	} else if (d <= 3.7e-178) {
		tmp = (b / c) + ((-1.0 / (c / (d * a))) / c);
	} else if (d <= 2.3e+149) {
		tmp = t_0;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)))
	tmp = 0.0
	if (d <= -1.35e+154)
		tmp = Float64(Float64(a - Float64(b * Float64(c / d))) / hypot(c, d));
	elseif (d <= -6e-133)
		tmp = t_0;
	elseif (d <= 3.7e-178)
		tmp = Float64(Float64(b / c) + Float64(Float64(-1.0 / Float64(c / Float64(d * a))) / c));
	elseif (d <= 2.3e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	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[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.35e+154], N[(N[(a - N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -6e-133], t$95$0, If[LessEqual[d, 3.7e-178], N[(N[(b / c), $MachinePrecision] + N[(N[(-1.0 / N[(c / N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.3e+149], t$95$0, N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.35000000000000003e154

   1. Initial program 39.3%

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

      1. *-un-lft-identity 39.3%

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

      2. add-sqr-sqrt 39.3%

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

      3. times-frac 39.3%

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

      4. hypot-def 39.3%

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

      5. hypot-def 63.5%

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

   3. Applied egg-rr 63.5%

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

      1. associate-*l/ 63.5%

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

      2. *-un-lft-identity 63.5%

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

      3. *-commutative 63.5%

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

      4. *-commutative 63.5%

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

   5. Applied egg-rr 63.5%

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

   6. Taylor expanded in d around -inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. associate-*r/ 88.3%

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

      3. sub-neg 88.3%

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

   8. Simplified 88.3%

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

   if -1.35000000000000003e154 < d < -6.00000000000000038e-133 or 3.70000000000000004e-178 < d < 2.2999999999999998e149

   1. Initial program 77.4%

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

      1. div-sub 76.6%

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

      2. *-commutative 76.6%

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

      3. add-sqr-sqrt 76.6%

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

      4. times-frac 79.9%

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

      5. fma-neg 79.9%

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

      6. hypot-def 79.9%

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

      7. hypot-def 91.3%

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

      8. associate-/l* 95.2%

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

      9. add-sqr-sqrt 95.2%

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

      10. pow2 95.2%

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

      11. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

   if -6.00000000000000038e-133 < d < 3.70000000000000004e-178

   1. Initial program 64.9%

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

   2. Taylor expanded in c around inf 83.9%

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

      1. +-commutative 83.9%

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

      2. mul-1-neg 83.9%

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

      3. unsub-neg 83.9%

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

      4. associate-/l* 80.9%

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

      5. unpow2 80.9%

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

      6. associate-/l* 92.2%

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

   4. Simplified 92.2%

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

      1. clear-num 92.3%

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

      2. inv-pow 92.3%

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

      3. associate-/r/ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. unpow-1 92.3%

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

      2. associate-/l* 93.3%

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

   8. Simplified 93.3%

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

      1. inv-pow 93.3%

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

      2. associate-/r/ 94.8%

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

      3. unpow-prod-down 94.8%

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

      4. inv-pow 94.8%

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

   10. Applied egg-rr 94.8%

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

       1. associate-*r/ 94.8%

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

       2. *-rgt-identity 94.8%

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

       3. unpow-1 94.8%

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

       4. associate-/l/ 95.4%

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

       5. *-commutative 95.4%

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

   12. Simplified 95.4%

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

   if 2.2999999999999998e149 < d

   1. Initial program 28.5%

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

      1. div-sub 28.5%

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

      2. *-commutative 28.5%

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

      3. add-sqr-sqrt 28.5%

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

      4. times-frac 28.6%

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

      5. fma-neg 28.6%

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

      6. hypot-def 28.6%

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

      7. hypot-def 34.3%

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

      8. associate-/l* 38.6%

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

      9. add-sqr-sqrt 38.6%

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

      10. pow2 38.6%

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

      11. hypot-def 38.6%

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

   3. Applied egg-rr 38.6%

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

   4. Taylor expanded in c around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. mul-1-neg 82.6%

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

      3. unsub-neg 82.6%

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

      4. *-commutative 82.6%

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

      5. unpow2 82.6%

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

      6. times-frac 88.5%

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

   6. Simplified 88.5%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{a - b \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 3.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{b}{c} + \frac{\frac{-1}{\frac{c}{d \cdot a}}}{c}\\ \mathbf{elif}\;d \leq 2.3 \cdot 10^{+149}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 2e+307)
     (/ (/ t_0 (hypot c d)) (hypot c d))
     (- (/ b c) (* (/ a c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+307) {
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+307) {
		tmp = (t_0 / Math.hypot(c, d)) / Math.hypot(c, d);
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 2e+307:
		tmp = (t_0 / math.hypot(c, d)) / math.hypot(c, d)
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+307)
		tmp = Float64(Float64(t_0 / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+307)
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.99999999999999997e307

   1. Initial program 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. add-sqr-sqrt 77.7%

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

      3. times-frac 77.7%

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

      4. hypot-def 77.7%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

      1. associate-*l/ 96.1%

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

      2. *-un-lft-identity 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   5. Applied egg-rr 96.1%

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

   if 1.99999999999999997e307 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 6.1%

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

      1. div-sub 4.1%

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

      2. *-commutative 4.1%

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

      3. add-sqr-sqrt 4.1%

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

      4. times-frac 9.2%

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

      5. fma-neg 9.2%

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

      6. hypot-def 9.2%

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

      7. hypot-def 42.4%

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

      8. associate-/l* 48.0%

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

      9. add-sqr-sqrt 48.0%

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

      10. pow2 48.0%

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

      11. hypot-def 48.0%

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

   3. Applied egg-rr 48.0%

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

   4. Taylor expanded in c around inf 41.5%

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

      1. fma-def 41.5%

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

      2. unpow2 41.5%

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

      3. times-frac 57.1%

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

      4. fma-def 57.1%

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

      5. neg-mul-1 57.1%

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

      6. +-commutative 57.1%

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

      7. sub-neg 57.1%

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

   6. Simplified 57.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{b \cdot c - d \cdot a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 3: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{a}{\frac{c}{d}} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -4.2e+63)
     (/ (- (/ a (/ c d)) b) (hypot c d))
     (if (<= c -9.5e-64)
       t_0
       (if (<= c 8.2e-113)
         (* (- a (* b (/ c d))) (/ -1.0 d))
         (if (<= c 1.75e+46) t_0 (- (/ b c) (* (/ a c) (/ d c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.2e+63) {
		tmp = ((a / (c / d)) - b) / hypot(c, d);
	} else if (c <= -9.5e-64) {
		tmp = t_0;
	} else if (c <= 8.2e-113) {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	} else if (c <= 1.75e+46) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.2e+63) {
		tmp = ((a / (c / d)) - b) / Math.hypot(c, d);
	} else if (c <= -9.5e-64) {
		tmp = t_0;
	} else if (c <= 8.2e-113) {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	} else if (c <= 1.75e+46) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.2e+63:
		tmp = ((a / (c / d)) - b) / math.hypot(c, d)
	elif c <= -9.5e-64:
		tmp = t_0
	elif c <= 8.2e-113:
		tmp = (a - (b * (c / d))) * (-1.0 / d)
	elif c <= 1.75e+46:
		tmp = t_0
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -4.2e+63)
		tmp = Float64(Float64(Float64(a / Float64(c / d)) - b) / hypot(c, d));
	elseif (c <= -9.5e-64)
		tmp = t_0;
	elseif (c <= 8.2e-113)
		tmp = Float64(Float64(a - Float64(b * Float64(c / d))) * Float64(-1.0 / d));
	elseif (c <= 1.75e+46)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.2e+63)
		tmp = ((a / (c / d)) - b) / hypot(c, d);
	elseif (c <= -9.5e-64)
		tmp = t_0;
	elseif (c <= 8.2e-113)
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	elseif (c <= 1.75e+46)
		tmp = t_0;
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.2e+63], N[(N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -9.5e-64], t$95$0, If[LessEqual[c, 8.2e-113], N[(N[(a - N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.75e+46], t$95$0, N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.2000000000000004e63

   1. Initial program 34.5%

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

      1. *-un-lft-identity 34.5%

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

      2. add-sqr-sqrt 34.5%

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

      3. times-frac 34.5%

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

      4. hypot-def 34.5%

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

      5. hypot-def 63.4%

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

   3. Applied egg-rr 63.4%

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

      1. associate-*l/ 63.6%

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

      2. *-un-lft-identity 63.6%

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

      3. *-commutative 63.6%

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

      4. *-commutative 63.6%

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

   5. Applied egg-rr 63.6%

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

   6. Taylor expanded in c around -inf 80.9%

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

      1. +-commutative 80.9%

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

      2. neg-mul-1 80.9%

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

      3. unsub-neg 80.9%

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

      4. associate-/l* 85.9%

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

   8. Simplified 85.9%

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

   if -4.2000000000000004e63 < c < -9.50000000000000043e-64 or 8.1999999999999999e-113 < c < 1.74999999999999992e46

   1. Initial program 84.2%

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

   if -9.50000000000000043e-64 < c < 8.1999999999999999e-113

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. add-sqr-sqrt 71.7%

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

      3. times-frac 71.8%

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

      4. hypot-def 71.8%

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

      5. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in d around -inf 55.2%

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

      1. mul-1-neg 55.2%

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

      2. unsub-neg 55.2%

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

      3. associate-/l* 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in d around -inf 90.1%

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

   8. Taylor expanded in b around 0 90.1%

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

      1. associate-*r/ 90.2%

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

   10. Simplified 90.2%

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

   if 1.74999999999999992e46 < c

   1. Initial program 43.7%

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

      1. div-sub 43.7%

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

      2. *-commutative 43.7%

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

      3. add-sqr-sqrt 43.7%

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

      4. times-frac 48.1%

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

      5. fma-neg 48.1%

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

      6. hypot-def 48.1%

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

      7. hypot-def 77.7%

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

      8. associate-/l* 74.8%

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

      9. add-sqr-sqrt 74.8%

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

      10. pow2 74.8%

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

      11. hypot-def 74.8%

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

   3. Applied egg-rr 74.8%

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

   4. Taylor expanded in c around inf 74.3%

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

      1. fma-def 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 86.7%

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

      4. fma-def 86.7%

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

      5. neg-mul-1 86.7%

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

      6. +-commutative 86.7%

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

      7. sub-neg 86.7%

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

   6. Simplified 86.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{\frac{a}{\frac{c}{d}} - b}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-113}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 4: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{\frac{c}{d}}{\frac{a}{c}}}\\ \mathbf{elif}\;c \leq -7 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-106}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -9.6e+62)
     (+ (/ b c) (/ -1.0 (/ (/ c d) (/ a c))))
     (if (<= c -7e-64)
       t_0
       (if (<= c 1.65e-106)
         (* (- a (* b (/ c d))) (/ -1.0 d))
         (if (<= c 1.75e+46) t_0 (- (/ b c) (* (/ a c) (/ d c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -9.6e+62) {
		tmp = (b / c) + (-1.0 / ((c / d) / (a / c)));
	} else if (c <= -7e-64) {
		tmp = t_0;
	} else if (c <= 1.65e-106) {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	} else if (c <= 1.75e+46) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
    if (c <= (-9.6d+62)) then
        tmp = (b / c) + ((-1.0d0) / ((c / d) / (a / c)))
    else if (c <= (-7d-64)) then
        tmp = t_0
    else if (c <= 1.65d-106) then
        tmp = (a - (b * (c / d))) * ((-1.0d0) / d)
    else if (c <= 1.75d+46) then
        tmp = t_0
    else
        tmp = (b / c) - ((a / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -9.6e+62) {
		tmp = (b / c) + (-1.0 / ((c / d) / (a / c)));
	} else if (c <= -7e-64) {
		tmp = t_0;
	} else if (c <= 1.65e-106) {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	} else if (c <= 1.75e+46) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -9.6e+62:
		tmp = (b / c) + (-1.0 / ((c / d) / (a / c)))
	elif c <= -7e-64:
		tmp = t_0
	elif c <= 1.65e-106:
		tmp = (a - (b * (c / d))) * (-1.0 / d)
	elif c <= 1.75e+46:
		tmp = t_0
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -9.6e+62)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(Float64(c / d) / Float64(a / c))));
	elseif (c <= -7e-64)
		tmp = t_0;
	elseif (c <= 1.65e-106)
		tmp = Float64(Float64(a - Float64(b * Float64(c / d))) * Float64(-1.0 / d));
	elseif (c <= 1.75e+46)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -9.6e+62)
		tmp = (b / c) + (-1.0 / ((c / d) / (a / c)));
	elseif (c <= -7e-64)
		tmp = t_0;
	elseif (c <= 1.65e-106)
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	elseif (c <= 1.75e+46)
		tmp = t_0;
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.6e+62], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(N[(c / d), $MachinePrecision] / N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -7e-64], t$95$0, If[LessEqual[c, 1.65e-106], N[(N[(a - N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.75e+46], t$95$0, N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.6e62

   1. Initial program 34.5%

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

   2. Taylor expanded in c around inf 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. unsub-neg 76.0%

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

      4. associate-/l* 76.3%

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

      5. unpow2 76.3%

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

      6. associate-/l* 81.1%

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

   4. Simplified 81.1%

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

      1. clear-num 81.1%

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

      2. inv-pow 81.1%

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

      3. associate-/r/ 81.0%

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

   6. Applied egg-rr 81.0%

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

      1. unpow-1 81.0%

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

      2. associate-/l* 85.6%

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

   8. Simplified 85.6%

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

   if -9.6e62 < c < -7.0000000000000006e-64 or 1.65000000000000008e-106 < c < 1.74999999999999992e46

   1. Initial program 84.2%

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

   if -7.0000000000000006e-64 < c < 1.65000000000000008e-106

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. add-sqr-sqrt 71.7%

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

      3. times-frac 71.8%

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

      4. hypot-def 71.8%

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

      5. hypot-def 83.5%

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

   3. Applied egg-rr 83.5%

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

   4. Taylor expanded in d around -inf 55.2%

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

      1. mul-1-neg 55.2%

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

      2. unsub-neg 55.2%

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

      3. associate-/l* 56.2%

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

   6. Simplified 56.2%

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

   7. Taylor expanded in d around -inf 90.1%

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

   8. Taylor expanded in b around 0 90.1%

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

      1. associate-*r/ 90.2%

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

   10. Simplified 90.2%

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

   if 1.74999999999999992e46 < c

   1. Initial program 43.7%

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

      1. div-sub 43.7%

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

      2. *-commutative 43.7%

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

      3. add-sqr-sqrt 43.7%

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

      4. times-frac 48.1%

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

      5. fma-neg 48.1%

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

      6. hypot-def 48.1%

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

      7. hypot-def 77.7%

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

      8. associate-/l* 74.8%

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

      9. add-sqr-sqrt 74.8%

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

      10. pow2 74.8%

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

      11. hypot-def 74.8%

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

   3. Applied egg-rr 74.8%

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

   4. Taylor expanded in c around inf 74.3%

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

      1. fma-def 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 86.7%

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

      4. fma-def 86.7%

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

      5. neg-mul-1 86.7%

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

      6. +-commutative 86.7%

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

      7. sub-neg 86.7%

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

   6. Simplified 86.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{\frac{c}{d}}{\frac{a}{c}}}\\ \mathbf{elif}\;c \leq -7 \cdot 10^{-64}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-106}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+46}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 5: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-18} \lor \neg \left(c \leq 1.2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.65e-18) (not (<= c 1.2e+69)))
   (/ b c)
   (* (- a (* b (/ c d))) (/ -1.0 d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.65e-18) || !(c <= 1.2e+69)) {
		tmp = b / c;
	} else {
		tmp = (a - (b * (c / d))) * (-1.0 / 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 <= (-1.65d-18)) .or. (.not. (c <= 1.2d+69))) then
        tmp = b / c
    else
        tmp = (a - (b * (c / d))) * ((-1.0d0) / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.65e-18) || !(c <= 1.2e+69)) {
		tmp = b / c;
	} else {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.65e-18) or not (c <= 1.2e+69):
		tmp = b / c
	else:
		tmp = (a - (b * (c / d))) * (-1.0 / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.65e-18) || !(c <= 1.2e+69))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(a - Float64(b * Float64(c / d))) * Float64(-1.0 / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.65e-18) || ~((c <= 1.2e+69)))
		tmp = b / c;
	else
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.65e-18], N[Not[LessEqual[c, 1.2e+69]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(a - N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.6500000000000001e-18 or 1.2000000000000001e69 < c

   1. Initial program 49.1%

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

   2. Taylor expanded in c around inf 70.9%

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

   if -1.6500000000000001e-18 < c < 1.2000000000000001e69

   1. Initial program 74.0%

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

      1. *-un-lft-identity 74.0%

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

      2. add-sqr-sqrt 74.0%

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

      3. times-frac 74.0%

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

      4. hypot-def 74.1%

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

      5. hypot-def 84.9%

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

   3. Applied egg-rr 84.9%

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

   4. Taylor expanded in d around -inf 49.1%

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

      1. mul-1-neg 49.1%

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

      2. unsub-neg 49.1%

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

      3. associate-/l* 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in d around -inf 81.8%

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

   8. Taylor expanded in b around 0 81.8%

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

      1. associate-*r/ 81.8%

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

   10. Simplified 81.8%

       \[\leadsto \frac{-1}{d} \cdot \left(a - \color{blue}{b \cdot \frac{c}{d}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{-18} \lor \neg \left(c \leq 1.2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \end{array} \]

Alternative 6: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-20} \lor \neg \left(c \leq 1.32 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4e-20) (not (<= c 1.32e+17)))
   (- (/ b c) (* (/ a c) (/ d c)))
   (* (- a (* b (/ c d))) (/ -1.0 d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-20) || !(c <= 1.32e+17)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = (a - (b * (c / d))) * (-1.0 / 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 <= (-4d-20)) .or. (.not. (c <= 1.32d+17))) then
        tmp = (b / c) - ((a / c) * (d / c))
    else
        tmp = (a - (b * (c / d))) * ((-1.0d0) / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4e-20) || !(c <= 1.32e+17)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4e-20) or not (c <= 1.32e+17):
		tmp = (b / c) - ((a / c) * (d / c))
	else:
		tmp = (a - (b * (c / d))) * (-1.0 / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4e-20) || !(c <= 1.32e+17))
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	else
		tmp = Float64(Float64(a - Float64(b * Float64(c / d))) * Float64(-1.0 / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4e-20) || ~((c <= 1.32e+17)))
		tmp = (b / c) - ((a / c) * (d / c));
	else
		tmp = (a - (b * (c / d))) * (-1.0 / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4e-20], N[Not[LessEqual[c, 1.32e+17]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a - N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.99999999999999978e-20 or 1.32e17 < c

   1. Initial program 50.0%

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

      1. div-sub 50.1%

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

      2. *-commutative 50.1%

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

      3. add-sqr-sqrt 50.1%

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

      4. times-frac 54.0%

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

      5. fma-neg 54.0%

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

      6. hypot-def 54.0%

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

      7. hypot-def 81.9%

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

      8. associate-/l* 80.7%

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

      9. add-sqr-sqrt 80.7%

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

      10. pow2 80.7%

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

      11. hypot-def 80.7%

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

   3. Applied egg-rr 80.7%

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

   4. Taylor expanded in c around inf 74.3%

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

      1. fma-def 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 83.1%

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

      4. fma-def 83.1%

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

      5. neg-mul-1 83.1%

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

      6. +-commutative 83.1%

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

      7. sub-neg 83.1%

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

   6. Simplified 83.1%

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

   if -3.99999999999999978e-20 < c < 1.32e17

   1. Initial program 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. add-sqr-sqrt 74.3%

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

      3. times-frac 74.4%

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

      4. hypot-def 74.4%

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

      5. hypot-def 85.0%

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

   3. Applied egg-rr 85.0%

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

   4. Taylor expanded in d around -inf 50.4%

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

      1. mul-1-neg 50.4%

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

      2. unsub-neg 50.4%

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

      3. associate-/l* 51.2%

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

   6. Simplified 51.2%

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

   7. Taylor expanded in d around -inf 83.1%

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

   8. Taylor expanded in b around 0 83.1%

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

      1. associate-*r/ 83.1%

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

   10. Simplified 83.1%

       \[\leadsto \frac{-1}{d} \cdot \left(a - \color{blue}{b \cdot \frac{c}{d}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-20} \lor \neg \left(c \leq 1.32 \cdot 10^{+17}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\left(a - b \cdot \frac{c}{d}\right) \cdot \frac{-1}{d}\\ \end{array} \]

Alternative 7: 64.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+15} \lor \neg \left(d \leq 8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.1e+15) (not (<= d 8e-19))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.1e+15) || !(d <= 8e-19)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.1d+15)) .or. (.not. (d <= 8d-19))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.1e+15) || !(d <= 8e-19)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.1e+15) or not (d <= 8e-19):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.1e+15) || !(d <= 8e-19))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.1e+15) || ~((d <= 8e-19)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.1e+15], N[Not[LessEqual[d, 8e-19]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1e15 or 7.9999999999999998e-19 < d

   1. Initial program 59.8%

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

   2. Taylor expanded in c around 0 67.2%

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

      1. associate-*r/ 67.2%

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

      2. neg-mul-1 67.2%

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

   4. Simplified 67.2%

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

   if -2.1e15 < d < 7.9999999999999998e-19

   1. Initial program 67.0%

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

   2. Taylor expanded in c around inf 66.8%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+15} \lor \neg \left(d \leq 8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8: 46.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -9.5e+158) (/ a d) (if (<= d 9e+89) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.5e+158) {
		tmp = a / d;
	} else if (d <= 9e+89) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-9.5d+158)) then
        tmp = a / d
    else if (d <= 9d+89) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.5e+158) {
		tmp = a / d;
	} else if (d <= 9e+89) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -9.5e+158:
		tmp = a / d
	elif d <= 9e+89:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9.5e+158)
		tmp = Float64(a / d);
	elseif (d <= 9e+89)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -9.5e+158)
		tmp = a / d;
	elseif (d <= 9e+89)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9.5e+158], N[(a / d), $MachinePrecision], If[LessEqual[d, 9e+89], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -9.49999999999999913e158 or 9e89 < d

   1. Initial program 43.3%

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

      1. *-un-lft-identity 43.3%

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

      2. add-sqr-sqrt 43.3%

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

      3. times-frac 43.3%

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

      4. hypot-def 43.3%

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

      5. hypot-def 67.0%

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

   3. Applied egg-rr 67.0%

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

   4. Taylor expanded in d around -inf 52.3%

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

      1. mul-1-neg 52.3%

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

      2. unsub-neg 52.3%

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

      3. associate-/l* 52.3%

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

   6. Simplified 52.3%

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

   7. Taylor expanded in c around 0 33.1%

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

   if -9.49999999999999913e158 < d < 9e89

   1. Initial program 71.7%

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

   2. Taylor expanded in c around inf 53.2%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+158}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 9: 11.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.1%

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

   1. *-un-lft-identity 63.1%

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

   2. add-sqr-sqrt 63.1%

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

   3. times-frac 63.1%

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

   4. hypot-def 63.2%

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

   5. hypot-def 78.5%

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

3. Applied egg-rr 78.5%

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

4. Taylor expanded in d around -inf 32.5%

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

   1. mul-1-neg 32.5%

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

   2. unsub-neg 32.5%

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

   3. associate-/l* 32.9%

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

6. Simplified 32.9%

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

7. Taylor expanded in c around 0 12.2%

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

8. Final simplification 12.2%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|d\right| < \left|c\right|:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-a\right) + b \cdot \frac{c}{d}}{d + c \cdot \frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (fabs(d) < fabs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

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

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (abs(d) < abs(c))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * Float64(c / d))) / Float64(d + Float64(c * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (abs(d) < abs(c))
		tmp = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (c / d))) / (d + (c * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2023271 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (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))))