Complex division, imag part

Percentage Accurate: 61.0% → 96.6%

Time: 12.0s

Alternatives: 14

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% 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 64.5%

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

   1. div-sub 63.6%

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

   2. *-commutative 63.6%

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

   3. fma-define 63.6%

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

   4. add-sqr-sqrt 63.6%

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

   5. times-frac 65.5%

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

   6. fmm-def 65.5%

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

   7. fma-define 65.5%

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

   8. hypot-define 65.5%

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

   9. fma-define 65.5%

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

   10. hypot-define 78.1%

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

   11. associate-/l* 79.2%

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

   12. fma-define 79.2%

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

   13. add-sqr-sqrt 79.2%

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

   14. pow2 79.2%

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

4. Applied egg-rr 79.2%

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

   1. *-un-lft-identity 79.2%

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

   2. unpow2 79.2%

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

   3. times-frac 94.4%

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

   4. hypot-undefine 79.3%

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

   5. +-commutative 79.3%

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

   6. hypot-undefine 94.4%

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

   7. hypot-undefine 79.3%

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

   8. +-commutative 79.3%

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

   9. hypot-undefine 94.4%

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

6. Applied egg-rr 94.4%

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

   1. associate-*l/ 94.4%

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

   2. *-lft-identity 94.4%

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

8. Simplified 94.4%

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

9. Final simplification 94.4%

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

Alternative 2: 91.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+24} \lor \neg \left(a \leq 1.5 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{c \cdot \frac{b}{a} - d}} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -3.2e+24) (not (<= a 1.5e-93)))
   (* (/ 1.0 (/ (hypot d c) (- (* c (/ b a)) d))) (/ a (hypot d c)))
   (fma
    (/ c (hypot c d))
    (/ b (hypot c d))
    (* a (/ d (- (pow (hypot c d) 2.0)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -3.2e+24) || !(a <= 1.5e-93)) {
		tmp = (1.0 / (hypot(d, c) / ((c * (b / a)) - d))) * (a / hypot(d, c));
	} else {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a * (d / -pow(hypot(c, d), 2.0))));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((a <= -3.2e+24) || !(a <= 1.5e-93))
		tmp = Float64(Float64(1.0 / Float64(hypot(d, c) / Float64(Float64(c * Float64(b / a)) - d))) * Float64(a / hypot(d, c)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(d / Float64(-(hypot(c, d) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[a, -3.2e+24], N[Not[LessEqual[a, 1.5e-93]], $MachinePrecision]], N[(N[(1.0 / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / N[(N[(c * N[(b / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a * N[(d / (-N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.1999999999999997e24 or 1.5000000000000001e-93 < a

   1. Initial program 56.7%

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

   3. Taylor expanded in a around inf 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-/l* 56.7%

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

   5. Simplified 56.7%

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

      1. *-commutative 56.7%

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

      2. add-sqr-sqrt 56.7%

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

      3. hypot-undefine 56.7%

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

      4. hypot-undefine 56.7%

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

      5. times-frac 95.1%

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

      6. associate-*r/ 87.1%

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

      7. hypot-undefine 55.6%

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

      8. +-commutative 55.6%

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

      9. hypot-undefine 87.1%

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

      10. hypot-undefine 55.5%

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

      11. +-commutative 55.5%

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

      12. hypot-undefine 87.1%

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

   7. Applied egg-rr 87.1%

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

      1. clear-num 86.7%

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

      2. inv-pow 86.7%

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

      3. hypot-undefine 55.1%

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

      4. +-commutative 55.1%

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

      5. hypot-define 86.7%

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

      6. associate-/l* 94.6%

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

      7. fmm-def 94.6%

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

   9. Applied egg-rr 94.6%

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

       1. unpow-1 94.6%

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

       2. hypot-undefine 56.7%

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

       3. unpow2 56.7%

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

       4. unpow2 56.7%

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

       5. +-commutative 56.7%

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

       6. unpow2 56.7%

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

       7. unpow2 56.7%

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

       8. hypot-define 94.6%

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

       9. fmm-undef 94.6%

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

   11. Simplified 94.6%

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

   if -3.1999999999999997e24 < a < 1.5000000000000001e-93

   1. Initial program 73.1%

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

      1. div-sub 72.9%

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

      2. *-commutative 72.9%

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

      3. fma-define 72.9%

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

      4. add-sqr-sqrt 72.9%

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

      5. times-frac 76.6%

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

      6. fmm-def 76.6%

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

      7. fma-define 76.6%

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

      8. hypot-define 76.6%

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

      9. fma-define 76.6%

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

      10. hypot-define 90.7%

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

      11. associate-/l* 90.9%

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

      12. fma-define 90.9%

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

      13. add-sqr-sqrt 90.9%

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

      14. pow2 90.9%

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

   4. Applied egg-rr 90.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.2 \cdot 10^{+24} \lor \neg \left(a \leq 1.5 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{c \cdot \frac{b}{a} - d}} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{d}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -4 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -5.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.6 \cdot 10^{+106}:\\ \;\;\;\;\left(c \cdot \frac{b}{a} - d\right) \cdot \frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -4e+57)
     t_0
     (if (<= c -5.4e-117)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= c 1.25e-28)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 4.6e+106)
           (* (- (* c (/ b a)) d) (/ (/ a (hypot d c)) (hypot d c)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -4e+57) {
		tmp = t_0;
	} else if (c <= -5.4e-117) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 1.25e-28) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 4.6e+106) {
		tmp = ((c * (b / a)) - d) * ((a / hypot(d, c)) / hypot(d, c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -4e+57) {
		tmp = t_0;
	} else if (c <= -5.4e-117) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 1.25e-28) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 4.6e+106) {
		tmp = ((c * (b / a)) - d) * ((a / Math.hypot(d, c)) / Math.hypot(d, c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -4e+57:
		tmp = t_0
	elif c <= -5.4e-117:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif c <= 1.25e-28:
		tmp = (((c * b) / d) - a) / d
	elif c <= 4.6e+106:
		tmp = ((c * (b / a)) - d) * ((a / math.hypot(d, c)) / math.hypot(d, c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -4e+57)
		tmp = t_0;
	elseif (c <= -5.4e-117)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.25e-28)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 4.6e+106)
		tmp = Float64(Float64(Float64(c * Float64(b / a)) - d) * Float64(Float64(a / hypot(d, c)) / hypot(d, c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -4e+57)
		tmp = t_0;
	elseif (c <= -5.4e-117)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (c <= 1.25e-28)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 4.6e+106)
		tmp = ((c * (b / a)) - d) * ((a / hypot(d, c)) / hypot(d, c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4e+57], t$95$0, If[LessEqual[c, -5.4e-117], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.25e-28], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.6e+106], N[(N[(N[(c * N[(b / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision] * N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.00000000000000019e57 or 4.6000000000000004e106 < c

   1. Initial program 42.3%

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

   3. Taylor expanded in c around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. unsub-neg 78.3%

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

      3. unsub-neg 78.3%

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

      4. mul-1-neg 78.3%

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

      5. remove-double-neg 78.3%

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

      6. mul-1-neg 78.3%

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

      7. neg-mul-1 78.3%

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

      8. distribute-lft-in 78.3%

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

      9. distribute-lft-in 78.3%

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

      10. mul-1-neg 78.3%

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

      11. unsub-neg 78.3%

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

      12. neg-mul-1 78.3%

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

      13. mul-1-neg 78.3%

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

      14. remove-double-neg 78.3%

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

      15. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

   if -4.00000000000000019e57 < c < -5.40000000000000005e-117

   1. Initial program 88.9%

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

   if -5.40000000000000005e-117 < c < 1.25e-28

   1. Initial program 73.9%

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

      1. div-sub 71.4%

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

      2. *-commutative 71.4%

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

      3. fma-define 71.4%

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

      4. add-sqr-sqrt 71.4%

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

      5. times-frac 66.4%

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

      6. fmm-def 66.4%

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

      7. fma-define 66.4%

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

      8. hypot-define 66.4%

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

      9. fma-define 66.4%

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

      10. hypot-define 68.9%

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

      11. associate-/l* 69.3%

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

      12. fma-define 69.3%

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

      13. add-sqr-sqrt 69.3%

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

      14. pow2 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in d around inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if 1.25e-28 < c < 4.6000000000000004e106

   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 a around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-/l* 70.1%

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

   5. Simplified 70.1%

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

      1. *-commutative 70.1%

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

      2. add-sqr-sqrt 70.1%

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

      3. hypot-undefine 70.1%

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

      4. hypot-undefine 70.2%

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

      5. times-frac 85.5%

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

      6. associate-*r/ 82.4%

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

      7. hypot-undefine 68.8%

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

      8. +-commutative 68.8%

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

      9. hypot-undefine 82.4%

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

      10. hypot-undefine 68.8%

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

      11. +-commutative 68.8%

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

      12. hypot-undefine 82.4%

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

   7. Applied egg-rr 82.4%

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

      1. associate-*r/ 85.7%

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

      2. associate-/l* 89.0%

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

      3. fmm-def 89.0%

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

      4. hypot-undefine 75.4%

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

      5. +-commutative 75.4%

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

      6. hypot-define 89.0%

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

      7. hypot-undefine 75.3%

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

      8. +-commutative 75.3%

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

      9. hypot-define 89.0%

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

   9. Applied egg-rr 89.0%

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

       1. associate-*l/ 77.6%

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

       2. associate-*r/ 90.2%

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

       3. associate-/l* 86.1%

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

       4. fmm-undef 86.1%

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

       5. hypot-undefine 76.4%

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

       6. unpow2 76.4%

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

       7. unpow2 76.4%

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

       8. +-commutative 76.4%

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

       9. unpow2 76.4%

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

       10. unpow2 76.4%

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

       11. hypot-define 86.1%

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

       12. hypot-undefine 76.4%

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

       13. unpow2 76.4%

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

       14. unpow2 76.4%

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

       15. +-commutative 76.4%

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

       16. unpow2 76.4%

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

       17. unpow2 76.4%

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

       18. hypot-define 86.1%

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

   11. Simplified 86.1%

       \[\leadsto \color{blue}{\left(c \cdot \frac{b}{a} - d\right) \cdot \frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+57}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -5.4 \cdot 10^{-117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.6 \cdot 10^{+106}:\\ \;\;\;\;\left(c \cdot \frac{b}{a} - d\right) \cdot \frac{\frac{a}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -4.6 \cdot 10^{+57}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\left(c \cdot \frac{b}{a} - d\right) \cdot \left(a \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -4.6e+57)
     t_0
     (if (<= c -4.2e-116)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= c 9e-28)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 7.5e+102)
           (* (- (* c (/ b a)) d) (* a (pow (hypot d c) -2.0)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -4.6e+57) {
		tmp = t_0;
	} else if (c <= -4.2e-116) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 9e-28) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 7.5e+102) {
		tmp = ((c * (b / a)) - d) * (a * pow(hypot(d, c), -2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -4.6e+57) {
		tmp = t_0;
	} else if (c <= -4.2e-116) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (c <= 9e-28) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 7.5e+102) {
		tmp = ((c * (b / a)) - d) * (a * Math.pow(Math.hypot(d, c), -2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -4.6e+57:
		tmp = t_0
	elif c <= -4.2e-116:
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d))
	elif c <= 9e-28:
		tmp = (((c * b) / d) - a) / d
	elif c <= 7.5e+102:
		tmp = ((c * (b / a)) - d) * (a * math.pow(math.hypot(d, c), -2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -4.6e+57)
		tmp = t_0;
	elseif (c <= -4.2e-116)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 9e-28)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 7.5e+102)
		tmp = Float64(Float64(Float64(c * Float64(b / a)) - d) * Float64(a * (hypot(d, c) ^ -2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -4.6e+57)
		tmp = t_0;
	elseif (c <= -4.2e-116)
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	elseif (c <= 9e-28)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 7.5e+102)
		tmp = ((c * (b / a)) - d) * (a * (hypot(d, c) ^ -2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.6e+57], t$95$0, If[LessEqual[c, -4.2e-116], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9e-28], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.5e+102], N[(N[(N[(c * N[(b / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision] * N[(a * N[Power[N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.5999999999999998e57 or 7.5e102 < c

   1. Initial program 42.3%

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

   3. Taylor expanded in c around inf 78.3%

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

      1. mul-1-neg 78.3%

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

      2. unsub-neg 78.3%

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

      3. unsub-neg 78.3%

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

      4. mul-1-neg 78.3%

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

      5. remove-double-neg 78.3%

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

      6. mul-1-neg 78.3%

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

      7. neg-mul-1 78.3%

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

      8. distribute-lft-in 78.3%

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

      9. distribute-lft-in 78.3%

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

      10. mul-1-neg 78.3%

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

      11. unsub-neg 78.3%

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

      12. neg-mul-1 78.3%

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

      13. mul-1-neg 78.3%

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

      14. remove-double-neg 78.3%

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

      15. associate-/l* 83.4%

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

   5. Simplified 83.4%

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

   if -4.5999999999999998e57 < c < -4.1999999999999998e-116

   1. Initial program 88.9%

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

   if -4.1999999999999998e-116 < c < 8.9999999999999996e-28

   1. Initial program 73.9%

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

      1. div-sub 71.4%

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

      2. *-commutative 71.4%

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

      3. fma-define 71.4%

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

      4. add-sqr-sqrt 71.4%

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

      5. times-frac 66.4%

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

      6. fmm-def 66.4%

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

      7. fma-define 66.4%

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

      8. hypot-define 66.4%

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

      9. fma-define 66.4%

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

      10. hypot-define 68.9%

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

      11. associate-/l* 69.3%

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

      12. fma-define 69.3%

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

      13. add-sqr-sqrt 69.3%

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

      14. pow2 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in d around inf 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if 8.9999999999999996e-28 < c < 7.5e102

   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 a around inf 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-/l* 70.1%

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

   5. Simplified 70.1%

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

      1. add-sqr-sqrt 70.1%

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

      2. hypot-undefine 70.1%

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

      3. hypot-undefine 70.2%

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

      4. unpow2 70.2%

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

      5. associate-/l* 73.3%

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

      6. div-inv 73.3%

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

      7. associate-*r/ 70.1%

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

      8. pow-flip 72.3%

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

      9. hypot-undefine 70.3%

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

      10. +-commutative 70.3%

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

      11. hypot-undefine 72.3%

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

      12. metadata-eval 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*r* 75.5%

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

      3. associate-/l* 78.6%

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

   9. Simplified 78.6%

      \[\leadsto \color{blue}{\left(a \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}\right) \cdot \left(c \cdot \frac{b}{a} - d\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+57}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+102}:\\ \;\;\;\;\left(c \cdot \frac{b}{a} - d\right) \cdot \left(a \cdot {\left(\mathsf{hypot}\left(d, c\right)\right)}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-76} \lor \neg \left(a \leq 2.65 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{c \cdot \frac{b}{a} - d}} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -3.1e-76) (not (<= a 2.65e-123)))
   (* (/ 1.0 (/ (hypot d c) (- (* c (/ b a)) d))) (/ a (hypot d c)))
   (* (/ c (hypot d c)) (/ b (hypot d c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -3.1e-76) || !(a <= 2.65e-123)) {
		tmp = (1.0 / (hypot(d, c) / ((c * (b / a)) - d))) * (a / hypot(d, c));
	} else {
		tmp = (c / hypot(d, c)) * (b / hypot(d, c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((a <= -3.1e-76) || !(a <= 2.65e-123)) {
		tmp = (1.0 / (Math.hypot(d, c) / ((c * (b / a)) - d))) * (a / Math.hypot(d, c));
	} else {
		tmp = (c / Math.hypot(d, c)) * (b / Math.hypot(d, c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (a <= -3.1e-76) or not (a <= 2.65e-123):
		tmp = (1.0 / (math.hypot(d, c) / ((c * (b / a)) - d))) * (a / math.hypot(d, c))
	else:
		tmp = (c / math.hypot(d, c)) * (b / math.hypot(d, c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((a <= -3.1e-76) || !(a <= 2.65e-123))
		tmp = Float64(Float64(1.0 / Float64(hypot(d, c) / Float64(Float64(c * Float64(b / a)) - d))) * Float64(a / hypot(d, c)));
	else
		tmp = Float64(Float64(c / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((a <= -3.1e-76) || ~((a <= 2.65e-123)))
		tmp = (1.0 / (hypot(d, c) / ((c * (b / a)) - d))) * (a / hypot(d, c));
	else
		tmp = (c / hypot(d, c)) * (b / hypot(d, c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[a, -3.1e-76], N[Not[LessEqual[a, 2.65e-123]], $MachinePrecision]], N[(N[(1.0 / N[(N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision] / N[(N[(c * N[(b / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.0999999999999997e-76 or 2.64999999999999985e-123 < a

   1. Initial program 55.7%

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

   3. Taylor expanded in a around inf 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-/l* 55.7%

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

   5. Simplified 55.7%

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

      1. *-commutative 55.7%

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

      2. add-sqr-sqrt 55.7%

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

      3. hypot-undefine 55.7%

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

      4. hypot-undefine 55.7%

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

      5. times-frac 91.2%

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

      6. associate-*r/ 84.7%

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

      7. hypot-undefine 54.8%

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

      8. +-commutative 54.8%

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

      9. hypot-undefine 84.7%

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

      10. hypot-undefine 54.8%

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

      11. +-commutative 54.8%

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

      12. hypot-undefine 84.7%

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

   7. Applied egg-rr 84.7%

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

      1. clear-num 84.3%

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

      2. inv-pow 84.3%

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

      3. hypot-undefine 54.4%

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

      4. +-commutative 54.4%

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

      5. hypot-define 84.3%

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

      6. associate-/l* 90.8%

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

      7. fmm-def 90.8%

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

   9. Applied egg-rr 90.8%

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

       1. unpow-1 90.8%

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

       2. hypot-undefine 55.8%

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

       3. unpow2 55.8%

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

       4. unpow2 55.8%

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

       5. +-commutative 55.8%

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

       6. unpow2 55.8%

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

       7. unpow2 55.8%

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

       8. hypot-define 90.8%

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

       9. fmm-undef 90.8%

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

   11. Simplified 90.8%

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

   if -3.0999999999999997e-76 < a < 2.64999999999999985e-123

   1. Initial program 80.2%

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

   3. Taylor expanded in b around inf 73.0%

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

      1. *-commutative 73.0%

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

   5. Simplified 73.0%

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

      1. add-sqr-sqrt 54.5%

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

      2. sqrt-div 40.0%

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

      3. +-commutative 40.0%

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

      4. hypot-undefine 40.0%

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

      5. sqrt-div 40.0%

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

      6. +-commutative 40.0%

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

      7. hypot-undefine 44.8%

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

      8. times-frac 40.0%

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

      9. add-sqr-sqrt 73.0%

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

      10. frac-times 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.1 \cdot 10^{-76} \lor \neg \left(a \leq 2.65 \cdot 10^{-123}\right):\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(d, c\right)}{c \cdot \frac{b}{a} - d}} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8200000000:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\frac{c \cdot b}{a} - d}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8200000000.0)
   (* (/ c (hypot d c)) (/ b (hypot d c)))
   (if (<= c 2.6e+77)
     (* (/ a (hypot d c)) (/ (- (/ (* c b) a) d) (hypot d c)))
     (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8200000000.0) {
		tmp = (c / hypot(d, c)) * (b / hypot(d, c));
	} else if (c <= 2.6e+77) {
		tmp = (a / hypot(d, c)) * ((((c * b) / a) - d) / hypot(d, c));
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8200000000.0) {
		tmp = (c / Math.hypot(d, c)) * (b / Math.hypot(d, c));
	} else if (c <= 2.6e+77) {
		tmp = (a / Math.hypot(d, c)) * ((((c * b) / a) - d) / Math.hypot(d, c));
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8200000000.0:
		tmp = (c / math.hypot(d, c)) * (b / math.hypot(d, c))
	elif c <= 2.6e+77:
		tmp = (a / math.hypot(d, c)) * ((((c * b) / a) - d) / math.hypot(d, c))
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8200000000.0)
		tmp = Float64(Float64(c / hypot(d, c)) * Float64(b / hypot(d, c)));
	elseif (c <= 2.6e+77)
		tmp = Float64(Float64(a / hypot(d, c)) * Float64(Float64(Float64(Float64(c * b) / a) - d) / hypot(d, c)));
	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 (c <= -8200000000.0)
		tmp = (c / hypot(d, c)) * (b / hypot(d, c));
	elseif (c <= 2.6e+77)
		tmp = (a / hypot(d, c)) * ((((c * b) / a) - d) / hypot(d, c));
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8200000000.0], N[(N[(c / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+77], N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(c * b), $MachinePrecision] / a), $MachinePrecision] - d), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.2e9

   1. Initial program 49.7%

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

   3. Taylor expanded in b around inf 47.0%

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

      1. *-commutative 47.0%

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

   5. Simplified 47.0%

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

      1. add-sqr-sqrt 34.3%

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

      2. sqrt-div 24.6%

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

      3. +-commutative 24.6%

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

      4. hypot-undefine 24.6%

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

      5. sqrt-div 24.6%

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

      6. +-commutative 24.6%

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

      7. hypot-undefine 30.4%

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

      8. times-frac 24.6%

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

      9. add-sqr-sqrt 47.0%

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

      10. frac-times 83.3%

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

   7. Applied egg-rr 83.3%

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

   if -8.2e9 < c < 2.6000000000000002e77

   1. Initial program 76.3%

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

   3. Taylor expanded in a around inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-/l* 67.1%

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

   5. Simplified 67.1%

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

      1. *-commutative 67.1%

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

      2. add-sqr-sqrt 67.1%

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

      3. hypot-undefine 67.1%

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

      4. hypot-undefine 67.1%

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

      5. times-frac 86.4%

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

      6. associate-*r/ 90.3%

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

      7. hypot-undefine 70.1%

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

      8. +-commutative 70.1%

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

      9. hypot-undefine 90.3%

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

      10. hypot-undefine 70.1%

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

      11. +-commutative 70.1%

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

      12. hypot-undefine 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 2.6000000000000002e77 < c

   1. Initial program 48.5%

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

   3. Taylor expanded in c around inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. unsub-neg 80.2%

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

      3. unsub-neg 80.2%

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

      4. mul-1-neg 80.2%

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

      5. remove-double-neg 80.2%

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

      6. mul-1-neg 80.2%

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

      7. neg-mul-1 80.2%

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

      8. distribute-lft-in 80.2%

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

      9. distribute-lft-in 80.2%

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

      10. mul-1-neg 80.2%

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

      11. unsub-neg 80.2%

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

      12. neg-mul-1 80.2%

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

      13. mul-1-neg 80.2%

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

      14. remove-double-neg 80.2%

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

      15. associate-/l* 85.5%

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

   5. Simplified 85.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8200000000:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{\frac{c \cdot b}{a} - d}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (fma c c (* d d)))))
   (if (<= d -6.2e+119)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -1.8e-159)
       t_0
       (if (<= d 9.4e-150)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 6e+147) t_0 (/ (- (* b (/ c d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / fma(c, c, (d * d));
	double tmp;
	if (d <= -6.2e+119) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -1.8e-159) {
		tmp = t_0;
	} else if (d <= 9.4e-150) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6e+147) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(c, c, Float64(d * d)))
	tmp = 0.0
	if (d <= -6.2e+119)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -1.8e-159)
		tmp = t_0;
	elseif (d <= 9.4e-150)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 6e+147)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e+119], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.8e-159], t$95$0, If[LessEqual[d, 9.4e-150], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6e+147], t$95$0, N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.1999999999999999e119

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 72.4%

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

      6. div-sub 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-/l* 85.3%

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

      9. fmm-def 85.3%

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

   5. Simplified 85.3%

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

   if -6.1999999999999999e119 < d < -1.80000000000000011e-159 or 9.3999999999999998e-150 < d < 5.99999999999999987e147

   1. Initial program 83.9%

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

      1. fma-define 83.9%

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

   3. Simplified 83.9%

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

   if -1.80000000000000011e-159 < d < 9.3999999999999998e-150

   1. Initial program 57.5%

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

   3. Taylor expanded in c around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

      3. unsub-neg 96.4%

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

      4. remove-double-neg 96.4%

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

      5. mul-1-neg 96.4%

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

      6. neg-mul-1 96.4%

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

      7. mul-1-neg 96.4%

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

      8. distribute-lft-in 96.4%

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

      9. distribute-lft-in 96.4%

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

      10. neg-mul-1 96.4%

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

      11. mul-1-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      13. associate-*r/ 96.4%

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

      14. associate-*r* 96.4%

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

      15. neg-mul-1 96.4%

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

   5. Simplified 96.4%

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

   if 5.99999999999999987e147 < d

   1. Initial program 34.4%

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

   3. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. unpow2 79.8%

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

      5. associate-/r* 88.1%

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

      6. div-sub 88.1%

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

      7. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 9.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+147}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -5.8e+119)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -1.3e-158)
       t_0
       (if (<= d 1.05e-147)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 6e+147) t_0 (/ (- (* b (/ c d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -5.8e+119) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -1.3e-158) {
		tmp = t_0;
	} else if (d <= 1.05e-147) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6e+147) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -5.8e+119)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -1.3e-158)
		tmp = t_0;
	elseif (d <= 1.05e-147)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 6e+147)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.8e+119], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.3e-158], t$95$0, If[LessEqual[d, 1.05e-147], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6e+147], t$95$0, N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.80000000000000014e119

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. mul-1-neg 68.5%

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

      3. unsub-neg 68.5%

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

      4. unpow2 68.5%

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

      5. associate-/r* 72.4%

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

      6. div-sub 72.4%

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

      7. *-commutative 72.4%

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

      8. associate-/l* 85.3%

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

      9. fmm-def 85.3%

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

   5. Simplified 85.3%

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

   if -5.80000000000000014e119 < d < -1.3e-158 or 1.05e-147 < d < 5.99999999999999987e147

   1. Initial program 83.9%

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

   if -1.3e-158 < d < 1.05e-147

   1. Initial program 57.5%

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

   3. Taylor expanded in c around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

      3. unsub-neg 96.4%

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

      4. remove-double-neg 96.4%

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

      5. mul-1-neg 96.4%

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

      6. neg-mul-1 96.4%

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

      7. mul-1-neg 96.4%

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

      8. distribute-lft-in 96.4%

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

      9. distribute-lft-in 96.4%

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

      10. neg-mul-1 96.4%

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

      11. mul-1-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      13. associate-*r/ 96.4%

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

      14. associate-*r* 96.4%

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

      15. neg-mul-1 96.4%

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

   5. Simplified 96.4%

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

   if 5.99999999999999987e147 < d

   1. Initial program 34.4%

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

   3. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. unpow2 79.8%

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

      5. associate-/r* 88.1%

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

      6. div-sub 88.1%

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

      7. associate-/l* 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+119}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-158}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.05 \cdot 10^{-147}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6 \cdot 10^{+147}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-149}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* b (/ c d)) a) d)))
   (if (<= d -6.2e+119)
     t_1
     (if (<= d -1.45e-159)
       t_0
       (if (<= d 1.75e-149)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 6.8e+149) 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 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -6.2e+119) {
		tmp = t_1;
	} else if (d <= -1.45e-159) {
		tmp = t_0;
	} else if (d <= 1.75e-149) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6.8e+149) {
		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 = ((b * (c / d)) - a) / d
    if (d <= (-6.2d+119)) then
        tmp = t_1
    else if (d <= (-1.45d-159)) then
        tmp = t_0
    else if (d <= 1.75d-149) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 6.8d+149) 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 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -6.2e+119) {
		tmp = t_1;
	} else if (d <= -1.45e-159) {
		tmp = t_0;
	} else if (d <= 1.75e-149) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 6.8e+149) {
		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 = ((b * (c / d)) - a) / d
	tmp = 0
	if d <= -6.2e+119:
		tmp = t_1
	elif d <= -1.45e-159:
		tmp = t_0
	elif d <= 1.75e-149:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 6.8e+149:
		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(b * Float64(c / d)) - a) / d)
	tmp = 0.0
	if (d <= -6.2e+119)
		tmp = t_1;
	elseif (d <= -1.45e-159)
		tmp = t_0;
	elseif (d <= 1.75e-149)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 6.8e+149)
		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 = ((b * (c / d)) - a) / d;
	tmp = 0.0;
	if (d <= -6.2e+119)
		tmp = t_1;
	elseif (d <= -1.45e-159)
		tmp = t_0;
	elseif (d <= 1.75e-149)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 6.8e+149)
		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[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -6.2e+119], t$95$1, If[LessEqual[d, -1.45e-159], t$95$0, If[LessEqual[d, 1.75e-149], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6.8e+149], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.1999999999999999e119 or 6.7999999999999997e149 < d

   1. Initial program 34.0%

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

   3. Taylor expanded in c around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unsub-neg 73.7%

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

      4. unpow2 73.7%

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

      5. associate-/r* 79.7%

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

      6. div-sub 79.7%

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

      7. associate-/l* 85.1%

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

   5. Simplified 85.1%

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

   if -6.1999999999999999e119 < d < -1.44999999999999995e-159 or 1.75e-149 < d < 6.7999999999999997e149

   1. Initial program 83.9%

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

   if -1.44999999999999995e-159 < d < 1.75e-149

   1. Initial program 57.5%

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

   3. Taylor expanded in c around inf 96.4%

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

      1. mul-1-neg 96.4%

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

      2. unsub-neg 96.4%

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

      3. unsub-neg 96.4%

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

      4. remove-double-neg 96.4%

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

      5. mul-1-neg 96.4%

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

      6. neg-mul-1 96.4%

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

      7. mul-1-neg 96.4%

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

      8. distribute-lft-in 96.4%

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

      9. distribute-lft-in 96.4%

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

      10. neg-mul-1 96.4%

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

      11. mul-1-neg 96.4%

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

      12. remove-double-neg 96.4%

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

      13. associate-*r/ 96.4%

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

      14. associate-*r* 96.4%

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

      15. neg-mul-1 96.4%

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

   5. Simplified 96.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+119}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-159}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-149}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{+149}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.1 \cdot 10^{+132} \lor \neg \left(d \leq -5.5 \cdot 10^{+107}\right) \land \left(d \leq -8.6 \cdot 10^{+18} \lor \neg \left(d \leq 9.8 \cdot 10^{-17}\right)\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.1e+132)
         (and (not (<= d -5.5e+107))
              (or (<= d -8.6e+18) (not (<= d 9.8e-17)))))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.1e+132) || (!(d <= -5.5e+107) && ((d <= -8.6e+18) || !(d <= 9.8e-17)))) {
		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 <= (-8.1d+132)) .or. (.not. (d <= (-5.5d+107))) .and. (d <= (-8.6d+18)) .or. (.not. (d <= 9.8d-17))) 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 <= -8.1e+132) || (!(d <= -5.5e+107) && ((d <= -8.6e+18) || !(d <= 9.8e-17)))) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.1e+132) or (not (d <= -5.5e+107) and ((d <= -8.6e+18) or not (d <= 9.8e-17))):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.1e+132) || (!(d <= -5.5e+107) && ((d <= -8.6e+18) || !(d <= 9.8e-17))))
		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 <= -8.1e+132) || (~((d <= -5.5e+107)) && ((d <= -8.6e+18) || ~((d <= 9.8e-17)))))
		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, -8.1e+132], And[N[Not[LessEqual[d, -5.5e+107]], $MachinePrecision], Or[LessEqual[d, -8.6e+18], N[Not[LessEqual[d, 9.8e-17]], $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 < -8.09999999999999988e132 or -5.5000000000000003e107 < d < -8.6e18 or 9.80000000000000024e-17 < d

   1. Initial program 56.8%

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

   3. Taylor expanded in c around 0 69.8%

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

      1. associate-*r/ 69.8%

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

      2. neg-mul-1 69.8%

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

   5. Simplified 69.8%

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

   if -8.09999999999999988e132 < d < -5.5000000000000003e107 or -8.6e18 < d < 9.80000000000000024e-17

   1. Initial program 71.1%

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

   3. Taylor expanded in c around inf 80.0%

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

      1. mul-1-neg 80.0%

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

      2. unsub-neg 80.0%

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

      3. unsub-neg 80.0%

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

      4. mul-1-neg 80.0%

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

      5. remove-double-neg 80.0%

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

      6. mul-1-neg 80.0%

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

      7. neg-mul-1 80.0%

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

      8. distribute-lft-in 80.0%

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

      9. distribute-lft-in 80.0%

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

      10. mul-1-neg 80.0%

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

      11. unsub-neg 80.0%

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

      12. neg-mul-1 80.0%

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

      13. mul-1-neg 80.0%

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

      14. remove-double-neg 80.0%

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

      15. associate-/l* 79.7%

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

   5. Simplified 79.7%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.1 \cdot 10^{+132} \lor \neg \left(d \leq -5.5 \cdot 10^{+107}\right) \land \left(d \leq -8.6 \cdot 10^{+18} \lor \neg \left(d \leq 9.8 \cdot 10^{-17}\right)\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-84} \lor \neg \left(c \leq 28000000\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8.5e-84) (not (<= c 28000000.0)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e-84) || !(c <= 28000000.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-8.5d-84)) .or. (.not. (c <= 28000000.0d0))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e-84) || !(c <= 28000000.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -8.5e-84) or not (c <= 28000000.0):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8.5e-84) || !(c <= 28000000.0))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -8.5e-84) || ~((c <= 28000000.0)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8.5e-84], N[Not[LessEqual[c, 28000000.0]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.4999999999999994e-84 or 2.8e7 < c

   1. Initial program 56.7%

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

   3. Taylor expanded in c around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. unsub-neg 73.2%

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

      3. unsub-neg 73.2%

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

      4. mul-1-neg 73.2%

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

      5. remove-double-neg 73.2%

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

      6. mul-1-neg 73.2%

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

      7. neg-mul-1 73.2%

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

      8. distribute-lft-in 73.2%

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

      9. distribute-lft-in 73.2%

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

      10. mul-1-neg 73.2%

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

      11. unsub-neg 73.2%

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

      12. neg-mul-1 73.2%

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

      13. mul-1-neg 73.2%

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

      14. remove-double-neg 73.2%

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

      15. associate-/l* 75.6%

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

   5. Simplified 75.6%

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

   if -8.4999999999999994e-84 < c < 2.8e7

   1. Initial program 75.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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* 88.9%

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

      6. div-sub 89.9%

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

      7. associate-/l* 88.7%

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

   5. Simplified 88.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-84} \lor \neg \left(c \leq 28000000\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-84} \lor \neg \left(c \leq 48000000000000\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8.5e-84) (not (<= c 48000000000000.0)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ (* c b) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e-84) || !(c <= 48000000000000.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-8.5d-84)) .or. (.not. (c <= 48000000000000.0d0))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = (((c * b) / d) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e-84) || !(c <= 48000000000000.0)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -8.5e-84) or not (c <= 48000000000000.0):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8.5e-84) || !(c <= 48000000000000.0))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -8.5e-84) || ~((c <= 48000000000000.0)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8.5e-84], N[Not[LessEqual[c, 48000000000000.0]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.4999999999999994e-84 or 4.8e13 < c

   1. Initial program 56.7%

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

   3. Taylor expanded in c around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. unsub-neg 73.2%

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

      3. unsub-neg 73.2%

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

      4. mul-1-neg 73.2%

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

      5. remove-double-neg 73.2%

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

      6. mul-1-neg 73.2%

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

      7. neg-mul-1 73.2%

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

      8. distribute-lft-in 73.2%

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

      9. distribute-lft-in 73.2%

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

      10. mul-1-neg 73.2%

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

      11. unsub-neg 73.2%

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

      12. neg-mul-1 73.2%

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

      13. mul-1-neg 73.2%

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

      14. remove-double-neg 73.2%

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

      15. associate-/l* 75.6%

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

   5. Simplified 75.6%

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

   if -8.4999999999999994e-84 < c < 4.8e13

   1. Initial program 75.4%

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

      1. div-sub 73.2%

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

      2. *-commutative 73.2%

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

      3. fma-define 73.2%

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

      4. add-sqr-sqrt 73.2%

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

      5. times-frac 68.9%

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

      6. fmm-def 68.9%

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

      7. fma-define 68.9%

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

      8. hypot-define 68.9%

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

      9. fma-define 68.9%

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

      10. hypot-define 71.0%

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

      11. associate-/l* 71.4%

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

      12. fma-define 71.4%

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

      13. add-sqr-sqrt 71.4%

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

      14. pow2 71.4%

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

   4. Applied egg-rr 71.4%

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

   5. Taylor expanded in d around inf 89.9%

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

      1. *-commutative 89.9%

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

   7. Simplified 89.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-84} \lor \neg \left(c \leq 48000000000000\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -0.000225 \lor \neg \left(c \leq 6.6 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -0.000225) (not (<= c 6.6e+47))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -0.000225) || !(c <= 6.6e+47)) {
		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 <= (-0.000225d0)) .or. (.not. (c <= 6.6d+47))) 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 <= -0.000225) || !(c <= 6.6e+47)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -0.000225) or not (c <= 6.6e+47):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -0.000225) || !(c <= 6.6e+47))
		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 <= -0.000225) || ~((c <= 6.6e+47)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -0.000225], N[Not[LessEqual[c, 6.6e+47]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.2499999999999999e-4 or 6.5999999999999998e47 < c

   1. Initial program 51.1%

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

   3. Taylor expanded in c around inf 68.6%

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

   if -2.2499999999999999e-4 < c < 6.5999999999999998e47

   1. Initial program 76.4%

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

   3. Taylor expanded in c around 0 59.0%

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

      1. associate-*r/ 59.0%

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

      2. neg-mul-1 59.0%

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

   5. Simplified 59.0%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.000225 \lor \neg \left(c \leq 6.6 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.3% 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 64.5%

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

3. Taylor expanded in c around inf 40.5%

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

4. Final simplification 40.5%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024089 
(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))))