Complex division, imag part

Percentage Accurate: 61.8% → 91.0%

Time: 9.7s

Alternatives: 9

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+129} \lor \neg \left(d \leq 4.2 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(d, c\right)}, c \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}, \frac{a}{-d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.4e+129) (not (<= d 4.2e+136)))
   (fma (/ 1.0 (hypot d c)) (* c (/ b (hypot d c))) (/ a (- d)))
   (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 ((d <= -2.4e+129) || !(d <= 4.2e+136)) {
		tmp = fma((1.0 / hypot(d, c)), (c * (b / hypot(d, c))), (a / -d));
	} 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 ((d <= -2.4e+129) || !(d <= 4.2e+136))
		tmp = fma(Float64(1.0 / hypot(d, c)), Float64(c * Float64(b / hypot(d, c))), Float64(a / Float64(-d)));
	else
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(-d) / (hypot(c, d) ^ 2.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.4e+129], N[Not[LessEqual[d, 4.2e+136]], $MachinePrecision]], N[(N[(1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / (-d)), $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 d < -2.3999999999999999e129 or 4.1999999999999998e136 < d

   1. Initial program 28.1%

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

      1. div-sub 28.1%

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

      2. *-un-lft-identity 28.1%

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

      3. add-sqr-sqrt 28.1%

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

      4. times-frac 28.1%

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

      5. fma-neg 28.1%

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

      6. hypot-define 28.1%

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

      7. hypot-define 31.4%

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

      8. associate-/l* 37.8%

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

      9. add-sqr-sqrt 37.8%

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

      10. pow2 37.8%

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

      11. hypot-define 37.8%

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

   4. Applied egg-rr 37.8%

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

      1. hypot-undefine 34.4%

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

      2. unpow2 34.4%

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

      3. unpow2 34.4%

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

      4. +-commutative 34.4%

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

      5. unpow2 34.4%

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

      6. unpow2 34.4%

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

      7. hypot-define 37.8%

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

      8. *-commutative 37.8%

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

      9. associate-/l* 43.5%

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

      10. hypot-undefine 34.8%

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

      11. unpow2 34.8%

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

      12. unpow2 34.8%

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

      13. +-commutative 34.8%

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

      14. unpow2 34.8%

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

      15. unpow2 34.8%

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

      16. hypot-define 43.5%

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

      17. distribute-rgt-neg-in 43.5%

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

      18. distribute-frac-neg 43.5%

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

   6. Simplified 43.5%

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

   7. Taylor expanded in d around inf 94.0%

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

      1. associate-*r/ 94.0%

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

      2. neg-mul-1 94.0%

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

   9. Simplified 94.0%

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

   if -2.3999999999999999e129 < d < 4.1999999999999998e136

   1. Initial program 69.2%

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

      1. div-sub 68.0%

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

      2. *-commutative 68.0%

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

      3. add-sqr-sqrt 68.0%

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

      4. times-frac 72.9%

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

      5. fma-neg 72.9%

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

      6. hypot-define 72.9%

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

      7. hypot-define 89.3%

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

      8. associate-/l* 91.6%

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

      9. add-sqr-sqrt 91.6%

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

      10. pow2 91.6%

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

      11. hypot-define 91.6%

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

   4. Applied egg-rr 91.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.4 \cdot 10^{+129} \lor \neg \left(d \leq 4.2 \cdot 10^{+136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(d, c\right)}, c \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}, \frac{a}{-d}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+215}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(d, c\right)}, c \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+215)))
     (fma (/ 1.0 (hypot d c)) (* c (/ b (hypot d c))) (/ (- a) d))
     (* (/ 1.0 (hypot c d)) (/ (fma b c (* d (- a))) (hypot c 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 ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+215)) {
		tmp = fma((1.0 / hypot(d, c)), (c * (b / hypot(d, c))), (-a / d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (fma(b, c, (d * -a)) / hypot(c, 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 ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+215))
		tmp = fma(Float64(1.0 / hypot(d, c)), Float64(c * Float64(b / hypot(d, c))), Float64(Float64(-a) / d));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(b, c, Float64(d * Float64(-a))) / hypot(c, 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[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+215]], $MachinePrecision]], N[(N[(1.0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -inf.0 or 1.99999999999999981e215 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.5%

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

      1. div-sub 14.2%

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

      2. *-un-lft-identity 14.2%

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

      3. add-sqr-sqrt 14.2%

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

      4. times-frac 14.2%

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

      5. fma-neg 14.2%

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

      6. hypot-define 14.2%

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

      7. hypot-define 15.9%

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

      8. associate-/l* 25.9%

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

      9. add-sqr-sqrt 25.9%

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

      10. pow2 25.9%

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

      11. hypot-define 25.9%

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

   4. Applied egg-rr 25.9%

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

      1. hypot-undefine 24.0%

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

      2. unpow2 24.0%

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

      3. unpow2 24.0%

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

      4. +-commutative 24.0%

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

      5. unpow2 24.0%

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

      6. unpow2 24.0%

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

      7. hypot-define 25.9%

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

      8. *-commutative 25.9%

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

      9. associate-/l* 57.2%

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

      10. hypot-undefine 34.6%

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

      11. unpow2 34.6%

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

      12. unpow2 34.6%

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

      13. +-commutative 34.6%

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

      14. unpow2 34.6%

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

      15. unpow2 34.6%

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

      16. hypot-define 57.2%

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

      17. distribute-rgt-neg-in 57.2%

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

      18. distribute-frac-neg 57.2%

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

   6. Simplified 57.2%

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

   7. Taylor expanded in d around inf 74.0%

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

      1. associate-*r/ 74.0%

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

      2. neg-mul-1 74.0%

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

   9. Simplified 74.0%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1.99999999999999981e215

   1. Initial program 78.6%

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

      1. *-un-lft-identity 78.6%

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

      2. add-sqr-sqrt 78.6%

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

      3. times-frac 78.6%

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

      4. hypot-define 78.6%

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

      5. fma-neg 78.6%

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

      6. distribute-rgt-neg-in 78.6%

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

      7. hypot-define 97.8%

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

   4. Applied egg-rr 97.8%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+215}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(d, c\right)}, c \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -2.05e+136)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d -7.3e-78)
     (* (/ 1.0 (hypot c d)) (/ (fma b c (* d (- a))) (hypot c d)))
     (if (<= d 1.9e+49) (/ (- b (* (/ d c) a)) c) (/ (- (* c (/ b d)) a) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.05e+136) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -7.3e-78) {
		tmp = (1.0 / hypot(c, d)) * (fma(b, c, (d * -a)) / hypot(c, d));
	} else if (d <= 1.9e+49) {
		tmp = (b - ((d / c) * a)) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.05e+136)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= -7.3e-78)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(b, c, Float64(d * Float64(-a))) / hypot(c, d)));
	elseif (d <= 1.9e+49)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.05e+136], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -7.3e-78], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e+49], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.0499999999999999e136

   1. Initial program 25.2%

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

      1. div-sub 25.2%

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

      2. *-un-lft-identity 25.2%

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

      3. add-sqr-sqrt 25.2%

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

      4. times-frac 25.2%

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

      5. fma-neg 25.2%

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

      6. hypot-define 25.2%

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

      7. hypot-define 27.4%

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

      8. associate-/l* 35.9%

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

      9. add-sqr-sqrt 35.9%

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

      10. pow2 35.9%

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

      11. hypot-define 35.9%

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

   4. Applied egg-rr 35.9%

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

      1. hypot-undefine 33.5%

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

      2. unpow2 33.5%

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

      3. unpow2 33.5%

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

      4. +-commutative 33.5%

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

      5. unpow2 33.5%

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

      6. unpow2 33.5%

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

      7. hypot-define 35.9%

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

      8. *-commutative 35.9%

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

      9. associate-/l* 41.7%

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

      10. hypot-undefine 34.0%

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

      11. unpow2 34.0%

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

      12. unpow2 34.0%

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

      13. +-commutative 34.0%

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

      14. unpow2 34.0%

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

      15. unpow2 34.0%

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

      16. hypot-define 41.7%

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

      17. distribute-rgt-neg-in 41.7%

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

      18. distribute-frac-neg 41.7%

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

   6. Simplified 41.7%

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

   7. Taylor expanded in d around inf 86.6%

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

      1. associate-*r/ 97.7%

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

      2. +-commutative 97.7%

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

      3. associate-*r/ 86.6%

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

      4. neg-mul-1 86.6%

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

      5. sub-neg 86.6%

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

      6. associate-*r/ 97.7%

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

   9. Simplified 97.7%

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

   if -2.0499999999999999e136 < d < -7.29999999999999981e-78

   1. Initial program 67.0%

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

      1. *-un-lft-identity 67.0%

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

      2. add-sqr-sqrt 67.0%

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

      3. times-frac 67.2%

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

      4. hypot-define 67.2%

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

      5. fma-neg 67.2%

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

      6. distribute-rgt-neg-in 67.2%

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

      7. hypot-define 85.6%

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

   4. Applied egg-rr 85.6%

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

   if -7.29999999999999981e-78 < d < 1.8999999999999999e49

   1. Initial program 68.9%

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

   3. Taylor expanded in c around inf 87.9%

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

      1. mul-1-neg 87.9%

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

      2. unsub-neg 87.9%

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

      3. associate-/l* 88.0%

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

   5. Simplified 88.0%

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

   if 1.8999999999999999e49 < d

   1. Initial program 45.5%

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

   3. Taylor expanded in d around -inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. +-commutative 77.8%

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

      3. remove-double-neg 77.8%

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

      4. sub-neg 77.8%

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

      5. neg-mul-1 77.8%

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

      6. distribute-neg-frac2 77.8%

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

      7. cancel-sign-sub-inv 77.8%

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

      8. metadata-eval 77.8%

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

      9. *-lft-identity 77.8%

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

      10. +-commutative 77.8%

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

      11. mul-1-neg 77.8%

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

      12. unsub-neg 77.8%

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

      13. *-commutative 77.8%

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

      14. associate-/l* 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.05 \cdot 10^{+136}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+20} \lor \neg \left(d \leq 3 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.8e+20) (not (<= d 3e+49)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (* (/ d c) a)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.8e+20) || !(d <= 3e+49)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.8d+20)) .or. (.not. (d <= 3d+49))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - ((d / c) * a)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.8e+20) || !(d <= 3e+49)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.8e+20) or not (d <= 3e+49):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((d / c) * a)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.8e+20) || !(d <= 3e+49))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.8e+20) || ~((d <= 3e+49)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((d / c) * a)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.8e+20], N[Not[LessEqual[d, 3e+49]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.8e20 or 3.0000000000000002e49 < d

   1. Initial program 40.5%

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

      1. div-sub 40.5%

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

      2. *-un-lft-identity 40.5%

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

      3. add-sqr-sqrt 40.5%

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

      4. times-frac 40.4%

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

      5. fma-neg 40.5%

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

      6. hypot-define 40.5%

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

      7. hypot-define 45.5%

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

      8. associate-/l* 52.4%

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

      9. add-sqr-sqrt 52.4%

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

      10. pow2 52.4%

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

      11. hypot-define 52.4%

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

   4. Applied egg-rr 52.4%

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

      1. hypot-undefine 47.3%

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

      2. unpow2 47.3%

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

      3. unpow2 47.3%

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

      4. +-commutative 47.3%

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

      5. unpow2 47.3%

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

      6. unpow2 47.3%

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

      7. hypot-define 52.4%

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

      8. *-commutative 52.4%

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

      9. associate-/l* 59.5%

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

      10. hypot-undefine 49.2%

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

      11. unpow2 49.2%

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

      12. unpow2 49.2%

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

      13. +-commutative 49.2%

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

      14. unpow2 49.2%

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

      15. unpow2 49.2%

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

      16. hypot-define 59.5%

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

      17. distribute-rgt-neg-in 59.5%

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

      18. distribute-frac-neg 59.5%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in d around inf 80.0%

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

      1. associate-*r/ 84.4%

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

      2. +-commutative 84.4%

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

      3. associate-*r/ 80.0%

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

      4. neg-mul-1 80.0%

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

      5. sub-neg 80.0%

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

      6. associate-*r/ 84.4%

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

   9. Simplified 84.4%

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

   if -1.8e20 < d < 3.0000000000000002e49

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.8 \cdot 10^{+20} \lor \neg \left(d \leq 3 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{+23} \lor \neg \left(d \leq 5.8 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.15e+23) (not (<= d 5.8e+57)))
   (/ (- a) d)
   (/ (- b (* (/ d c) a)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.15e+23) || !(d <= 5.8e+57)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.15d+23)) .or. (.not. (d <= 5.8d+57))) then
        tmp = -a / d
    else
        tmp = (b - ((d / c) * a)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.15e+23) || !(d <= 5.8e+57)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.15e+23) or not (d <= 5.8e+57):
		tmp = -a / d
	else:
		tmp = (b - ((d / c) * a)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.15e+23) || !(d <= 5.8e+57))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.15e+23) || ~((d <= 5.8e+57)))
		tmp = -a / d;
	else
		tmp = (b - ((d / c) * a)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.15e+23], N[Not[LessEqual[d, 5.8e+57]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.1499999999999999e23 or 5.8000000000000003e57 < d

   1. Initial program 40.3%

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

   3. Taylor expanded in c around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. neg-mul-1 74.1%

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

   5. Simplified 74.1%

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

   if -2.1499999999999999e23 < d < 5.8000000000000003e57

   1. Initial program 69.6%

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

   3. Taylor expanded in c around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. unsub-neg 85.8%

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

      3. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.15 \cdot 10^{+23} \lor \neg \left(d \leq 5.8 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -6.8e+16)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d 1.9e+49) (/ (- b (* (/ d c) a)) c) (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -6.8e+16) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 1.9e+49) {
		tmp = (b - ((d / c) * a)) / 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 (d <= (-6.8d+16)) then
        tmp = ((b * (c / d)) - a) / d
    else if (d <= 1.9d+49) then
        tmp = (b - ((d / c) * a)) / 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 (d <= -6.8e+16) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 1.9e+49) {
		tmp = (b - ((d / c) * a)) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -6.8e+16:
		tmp = ((b * (c / d)) - a) / d
	elif d <= 1.9e+49:
		tmp = (b - ((d / c) * a)) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -6.8e+16)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= 1.9e+49)
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -6.8e+16)
		tmp = ((b * (c / d)) - a) / d;
	elseif (d <= 1.9e+49)
		tmp = (b - ((d / c) * a)) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -6.8e+16], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.9e+49], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.8e16

   1. Initial program 36.5%

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

      1. div-sub 36.5%

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

      2. *-un-lft-identity 36.5%

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

      3. add-sqr-sqrt 36.5%

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

      4. times-frac 36.5%

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

      5. fma-neg 36.5%

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

      6. hypot-define 36.5%

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

      7. hypot-define 41.9%

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

      8. associate-/l* 52.0%

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

      9. add-sqr-sqrt 52.0%

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

      10. pow2 52.0%

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

      11. hypot-define 52.0%

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

   4. Applied egg-rr 52.0%

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

      1. hypot-undefine 46.5%

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

      2. unpow2 46.5%

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

      3. unpow2 46.5%

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

      4. +-commutative 46.5%

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

      5. unpow2 46.5%

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

      6. unpow2 46.5%

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

      7. hypot-define 52.0%

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

      8. *-commutative 52.0%

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

      9. associate-/l* 58.9%

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

      10. hypot-undefine 46.9%

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

      11. unpow2 46.9%

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

      12. unpow2 46.9%

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

      13. +-commutative 46.9%

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

      14. unpow2 46.9%

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

      15. unpow2 46.9%

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

      16. hypot-define 58.9%

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

      17. distribute-rgt-neg-in 58.9%

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

      18. distribute-frac-neg 58.9%

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

   6. Simplified 58.9%

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

   7. Taylor expanded in d around inf 81.6%

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

      1. associate-*r/ 89.1%

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

      2. +-commutative 89.1%

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

      3. associate-*r/ 81.6%

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

      4. neg-mul-1 81.6%

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

      5. sub-neg 81.6%

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

      6. associate-*r/ 89.1%

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

   9. Simplified 89.1%

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

   if -6.8e16 < d < 1.8999999999999999e49

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 86.3%

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

      1. mul-1-neg 86.3%

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

      2. unsub-neg 86.3%

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

      3. associate-/l* 86.4%

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

   5. Simplified 86.4%

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

   if 1.8999999999999999e49 < d

   1. Initial program 45.5%

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

   3. Taylor expanded in d around -inf 77.8%

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

      1. mul-1-neg 77.8%

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

      2. +-commutative 77.8%

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

      3. remove-double-neg 77.8%

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

      4. sub-neg 77.8%

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

      5. neg-mul-1 77.8%

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

      6. distribute-neg-frac2 77.8%

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

      7. cancel-sign-sub-inv 77.8%

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

      8. metadata-eval 77.8%

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

      9. *-lft-identity 77.8%

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

      10. +-commutative 77.8%

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

      11. mul-1-neg 77.8%

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

      12. unsub-neg 77.8%

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

      13. *-commutative 77.8%

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

      14. associate-/l* 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+14} \lor \neg \left(d \leq 2.5 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.35e+14) (not (<= d 2.5e+49))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.35e+14) || !(d <= 2.5e+49)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.35d+14)) .or. (.not. (d <= 2.5d+49))) then
        tmp = a / -d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.35e+14) || !(d <= 2.5e+49)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.35e+14) or not (d <= 2.5e+49):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.35e+14) || !(d <= 2.5e+49))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.35e+14) || ~((d <= 2.5e+49)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.35e+14], N[Not[LessEqual[d, 2.5e+49]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.35e14 or 2.5000000000000002e49 < d

   1. Initial program 40.5%

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

   3. Taylor expanded in c around 0 73.7%

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

      1. associate-*r/ 73.7%

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

      2. neg-mul-1 73.7%

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

   5. Simplified 73.7%

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

   if -2.35e14 < d < 2.5000000000000002e49

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 67.8%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+14} \lor \neg \left(d \leq 2.5 \cdot 10^{+49}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

3. Taylor expanded in c around inf 42.8%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Add Preprocessing

Alternative 9: 10.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return a / 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 = a / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.5%

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

   1. *-un-lft-identity 56.5%

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

   2. add-sqr-sqrt 56.5%

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

   3. times-frac 56.5%

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

   4. hypot-define 56.5%

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

   5. fma-neg 56.5%

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

   6. distribute-rgt-neg-in 56.5%

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

   7. hypot-define 71.3%

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

4. Applied egg-rr 71.3%

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

5. Taylor expanded in c around -inf 31.4%

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

   1. +-commutative 31.4%

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

   2. neg-mul-1 31.4%

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

   3. unsub-neg 31.4%

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

   4. associate-/l* 33.0%

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

7. Simplified 33.0%

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

8. Taylor expanded in c around 0 7.9%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
9. Add Preprocessing

Developer Target 1: 99.3% 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 2024114 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))