Complex division, imag part

Percentage Accurate: 61.4% → 90.3%

Time: 12.3s

Alternatives: 13

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 90.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+33} \lor \neg \left(a \leq 5 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{a} - d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= a -7.4e+33) (not (<= a 5e+46)))
   (* (/ (- (* b (/ c a)) d) (hypot d c)) (/ 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 <= -7.4e+33) || !(a <= 5e+46)) {
		tmp = (((b * (c / a)) - d) / hypot(d, c)) * (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 <= -7.4e+33) || !(a <= 5e+46))
		tmp = Float64(Float64(Float64(Float64(b * Float64(c / a)) - d) / hypot(d, c)) * Float64(a / hypot(d, c)));
	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[a, -7.4e+33], N[Not[LessEqual[a, 5e+46]], $MachinePrecision]], N[(N[(N[(N[(b * N[(c / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $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 < -7.3999999999999997e33 or 5.0000000000000002e46 < a

   1. Initial program 57.4%

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

   3. Taylor expanded in a around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. add-sqr-sqrt 57.4%

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

      3. hypot-undefine 57.4%

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

      4. hypot-undefine 57.4%

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

      5. times-frac 92.1%

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

      6. associate-/l* 96.9%

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

      7. hypot-undefine 60.1%

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

      8. +-commutative 60.1%

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

      9. hypot-define 96.9%

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

      10. hypot-undefine 60.1%

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

      11. +-commutative 60.1%

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

      12. hypot-define 96.9%

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

   5. Applied egg-rr 96.9%

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

   if -7.3999999999999997e33 < a < 5.0000000000000002e46

   1. Initial program 72.2%

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

      1. div-sub 71.3%

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

      2. *-commutative 71.3%

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

      3. fma-define 71.3%

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

      4. add-sqr-sqrt 71.3%

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

      5. times-frac 74.8%

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

      6. fma-neg 74.8%

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

      7. fma-define 74.8%

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

      8. hypot-define 74.9%

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

      9. fma-define 74.9%

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

      10. hypot-define 91.0%

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

      11. associate-/l* 91.9%

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

      12. fma-define 91.9%

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

      13. add-sqr-sqrt 91.9%

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

      14. pow2 91.9%

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

   4. Applied egg-rr 91.9%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+33} \lor \neg \left(a \leq 5 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{a} - d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 10^{+211}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+211)
     (/ (/ t_0 (hypot d c)) (hypot d c))
     (/ (* (/ a (hypot d c)) (fma b (/ c a) (- d))) (hypot d c)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+211) {
		tmp = (t_0 / hypot(d, c)) / hypot(d, c);
	} else {
		tmp = ((a / hypot(d, c)) * fma(b, (c / a), -d)) / hypot(d, c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+211)
		tmp = Float64(Float64(t_0 / hypot(d, c)) / hypot(d, c));
	else
		tmp = Float64(Float64(Float64(a / hypot(d, c)) * fma(b, Float64(c / a), Float64(-d))) / hypot(d, c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+211], N[(N[(t$95$0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(a / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b * N[(c / a), $MachinePrecision] + (-d)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $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))) < 9.9999999999999996e210

   1. Initial program 80.1%

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

      1. fma-neg 80.1%

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

      2. distribute-rgt-neg-out 80.1%

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

      3. +-commutative 80.1%

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

      4. fma-define 80.1%

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

   3. Simplified 80.1%

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

      1. distribute-rgt-neg-out 80.1%

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

      2. fma-neg 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. *-un-lft-identity 80.1%

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

      2. fma-undefine 80.1%

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

      3. +-commutative 80.1%

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

      4. add-sqr-sqrt 80.1%

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

      5. hypot-undefine 80.1%

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

      6. hypot-undefine 80.1%

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

      7. times-frac 96.2%

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

      8. hypot-undefine 80.0%

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

      9. +-commutative 80.0%

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

      10. hypot-define 96.2%

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

      11. *-commutative 96.2%

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

      12. fma-neg 96.2%

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

      13. *-commutative 96.2%

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

      14. hypot-undefine 80.0%

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

      15. +-commutative 80.0%

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

      16. hypot-define 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. associate-*l/ 96.3%

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

      2. *-lft-identity 96.3%

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

      3. fma-undefine 96.3%

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

      4. *-commutative 96.3%

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

      5. unsub-neg 96.3%

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

      6. *-commutative 96.3%

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

   10. Simplified 96.3%

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

   if 9.9999999999999996e210 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 20.9%

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

   3. Taylor expanded in a around inf 20.9%

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

      1. *-commutative 20.9%

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

      2. add-sqr-sqrt 20.9%

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

      3. hypot-undefine 20.9%

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

      4. hypot-undefine 20.9%

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

      5. times-frac 60.1%

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

      6. associate-/l* 69.9%

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

      7. hypot-undefine 22.4%

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

      8. +-commutative 22.4%

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

      9. hypot-define 69.9%

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

      10. hypot-undefine 22.4%

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

      11. +-commutative 22.4%

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

      12. hypot-define 69.9%

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

   5. Applied egg-rr 69.9%

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

      1. associate-*l/ 71.5%

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

      2. fma-neg 71.4%

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

   7. Applied egg-rr 71.4%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+211}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{a}{\mathsf{hypot}\left(d, c\right)} \cdot \mathsf{fma}\left(b, \frac{c}{a}, -d\right)}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{a} - d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 2e+293)
     (/ (/ t_0 (hypot d c)) (hypot d c))
     (* (/ (- (* b (/ c a)) d) (hypot d c)) (/ a (hypot d c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+293) {
		tmp = (t_0 / hypot(d, c)) / hypot(d, c);
	} else {
		tmp = (((b * (c / a)) - d) / hypot(d, c)) * (a / hypot(d, c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+293) {
		tmp = (t_0 / Math.hypot(d, c)) / Math.hypot(d, c);
	} else {
		tmp = (((b * (c / a)) - d) / Math.hypot(d, c)) * (a / Math.hypot(d, c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 2e+293:
		tmp = (t_0 / math.hypot(d, c)) / math.hypot(d, c)
	else:
		tmp = (((b * (c / a)) - d) / math.hypot(d, c)) * (a / math.hypot(d, c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+293)
		tmp = Float64(Float64(t_0 / hypot(d, c)) / hypot(d, c));
	else
		tmp = Float64(Float64(Float64(Float64(b * Float64(c / a)) - d) / hypot(d, c)) * Float64(a / hypot(d, c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 2e+293)
		tmp = (t_0 / hypot(d, c)) / hypot(d, c);
	else
		tmp = (((b * (c / a)) - d) / hypot(d, c)) * (a / hypot(d, c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+293], N[(N[(t$95$0 / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * N[(c / a), $MachinePrecision]), $MachinePrecision] - d), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[d ^ 2 + c ^ 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))) < 1.9999999999999998e293

   1. Initial program 80.6%

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

      1. fma-neg 80.6%

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

      2. distribute-rgt-neg-out 80.6%

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

      3. +-commutative 80.6%

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

      4. fma-define 80.6%

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

   3. Simplified 80.6%

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

      1. distribute-rgt-neg-out 80.6%

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

      2. fma-neg 80.6%

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

   6. Applied egg-rr 80.6%

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

      1. *-un-lft-identity 80.6%

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

      2. fma-undefine 80.6%

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

      3. +-commutative 80.6%

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

      4. add-sqr-sqrt 80.6%

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

      5. hypot-undefine 80.6%

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

      6. hypot-undefine 80.6%

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

      7. times-frac 96.3%

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

      8. hypot-undefine 80.5%

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

      9. +-commutative 80.5%

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

      10. hypot-define 96.3%

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

      11. *-commutative 96.3%

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

      12. fma-neg 96.3%

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

      13. *-commutative 96.3%

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

      14. hypot-undefine 80.5%

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

      15. +-commutative 80.5%

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

      16. hypot-define 96.3%

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

   8. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.4%

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

      2. *-lft-identity 96.4%

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

      3. fma-undefine 96.4%

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

      4. *-commutative 96.4%

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

      5. unsub-neg 96.4%

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

      6. *-commutative 96.4%

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

   10. Simplified 96.4%

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

   if 1.9999999999999998e293 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 13.6%

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

   3. Taylor expanded in a around inf 13.6%

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

      1. *-commutative 13.6%

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

      2. add-sqr-sqrt 13.6%

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

      3. hypot-undefine 13.6%

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

      4. hypot-undefine 13.6%

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

      5. times-frac 58.0%

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

      6. associate-/l* 68.8%

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

      7. hypot-undefine 16.9%

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

      8. +-commutative 16.9%

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

      9. hypot-define 68.8%

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

      10. hypot-undefine 16.9%

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

      11. +-commutative 16.9%

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

      12. hypot-define 68.8%

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

   5. Applied egg-rr 68.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+293}:\\ \;\;\;\;\frac{\frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{a} - d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(d, c)) / hypot(d, 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)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= Inf)
		tmp = (t_0 / hypot(d, c)) / hypot(d, c);
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 79.5%

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

      1. fma-neg 79.5%

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

      2. distribute-rgt-neg-out 79.5%

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

      3. +-commutative 79.5%

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

      4. fma-define 79.5%

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

   3. Simplified 79.5%

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

      1. distribute-rgt-neg-out 79.5%

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

      2. fma-neg 79.5%

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

   6. Applied egg-rr 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. fma-undefine 79.5%

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

      3. +-commutative 79.5%

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

      4. add-sqr-sqrt 79.5%

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

      5. hypot-undefine 79.5%

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

      6. hypot-undefine 79.5%

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

      7. times-frac 95.7%

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

      8. hypot-undefine 79.4%

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

      9. +-commutative 79.4%

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

      10. hypot-define 95.7%

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

      11. *-commutative 95.7%

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

      12. fma-neg 95.7%

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

      13. *-commutative 95.7%

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

      14. hypot-undefine 79.4%

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

      15. +-commutative 79.4%

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

      16. hypot-define 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. associate-*l/ 95.8%

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

      2. *-lft-identity 95.8%

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

      3. fma-undefine 95.8%

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

      4. *-commutative 95.8%

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

      5. unsub-neg 95.8%

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

      6. *-commutative 95.8%

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

   10. Simplified 95.8%

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

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

   1. Initial program 0.0%

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

      1. fma-neg 0.0%

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

      2. distribute-rgt-neg-out 0.0%

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

      3. +-commutative 0.0%

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

      4. fma-define 0.0%

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

   3. Simplified 0.0%

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

      1. distribute-rgt-neg-out 0.0%

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

      2. fma-neg 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. fma-undefine 0.0%

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

      3. +-commutative 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. hypot-undefine 0.0%

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

      6. hypot-undefine 0.0%

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

      7. times-frac 2.7%

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

      8. hypot-undefine 0.0%

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

      9. +-commutative 0.0%

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

      10. hypot-define 2.7%

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

      11. *-commutative 2.7%

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

      12. fma-neg 2.8%

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

      13. *-commutative 2.8%

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

      14. hypot-undefine 0.0%

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

      15. +-commutative 0.0%

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

      16. hypot-define 2.8%

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

   8. Applied egg-rr 2.8%

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

      1. associate-*l/ 2.8%

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

      2. *-lft-identity 2.8%

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

      3. fma-undefine 2.7%

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

      4. *-commutative 2.7%

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

      5. unsub-neg 2.7%

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

      6. *-commutative 2.7%

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

   10. Simplified 2.7%

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

   11. Taylor expanded in c around 0 41.5%

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

       1. +-commutative 41.5%

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

       2. mul-1-neg 41.5%

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

       3. unsub-neg 41.5%

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

       4. *-rgt-identity 41.5%

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

       5. unpow2 41.5%

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

       6. times-frac 44.4%

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

       7. *-commutative 44.4%

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

       8. associate-*r/ 58.6%

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

       9. associate-*r/ 58.6%

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

       10. associate-*r/ 44.4%

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

       11. *-commutative 44.4%

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

       12. *-rgt-identity 44.4%

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

       13. *-commutative 44.4%

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

       14. associate-*r/ 58.6%

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

       15. associate-*r/ 44.4%

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

       16. *-commutative 44.4%

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

       17. div-sub 44.4%

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

       18. associate-/l* 58.6%

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

   13. Simplified 58.6%

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

4. Final simplification 89.7%

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

Alternative 5: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -2.26 \cdot 10^{+86}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= d -2.26e+86)
     (/ (- (* c (/ b d)) a) d)
     (if (<= d -1.25e-159)
       t_0
       (if (<= d 5.8e-47)
         (/ (- b (/ a (/ c d))) c)
         (if (<= d 1.3e+47) t_0 (/ (- (* b (/ c d)) a) (hypot d c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.26e+86) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -1.25e-159) {
		tmp = t_0;
	} else if (d <= 5.8e-47) {
		tmp = (b - (a / (c / d))) / c;
	} else if (d <= 1.3e+47) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / hypot(d, c);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -2.26e+86) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -1.25e-159) {
		tmp = t_0;
	} else if (d <= 5.8e-47) {
		tmp = (b - (a / (c / d))) / c;
	} else if (d <= 1.3e+47) {
		tmp = t_0;
	} else {
		tmp = ((b * (c / d)) - a) / Math.hypot(d, c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -2.26e+86:
		tmp = ((c * (b / d)) - a) / d
	elif d <= -1.25e-159:
		tmp = t_0
	elif d <= 5.8e-47:
		tmp = (b - (a / (c / d))) / c
	elif d <= 1.3e+47:
		tmp = t_0
	else:
		tmp = ((b * (c / d)) - a) / math.hypot(d, c)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -2.26e+86)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -1.25e-159)
		tmp = t_0;
	elseif (d <= 5.8e-47)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	elseif (d <= 1.3e+47)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -2.26e+86)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= -1.25e-159)
		tmp = t_0;
	elseif (d <= 5.8e-47)
		tmp = (b - (a / (c / d))) / c;
	elseif (d <= 1.3e+47)
		tmp = t_0;
	else
		tmp = ((b * (c / d)) - a) / hypot(d, c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.26e+86], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.25e-159], t$95$0, If[LessEqual[d, 5.8e-47], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.3e+47], t$95$0, N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.26e86

   1. Initial program 43.7%

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

   3. Taylor expanded in c around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. mul-1-neg 72.4%

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

      3. unsub-neg 72.4%

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

      4. unpow2 72.4%

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

      5. associate-/r* 78.3%

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

      6. div-sub 78.3%

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

      7. *-commutative 78.3%

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

      8. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -2.26e86 < d < -1.25000000000000008e-159 or 5.8000000000000001e-47 < d < 1.30000000000000002e47

   1. Initial program 87.9%

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

   if -1.25000000000000008e-159 < d < 5.8000000000000001e-47

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf 90.2%

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

      1. remove-double-neg 90.2%

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

      2. mul-1-neg 90.2%

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

      3. neg-mul-1 90.2%

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

      4. distribute-lft-in 90.2%

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

      5. distribute-lft-in 90.2%

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

      6. mul-1-neg 90.2%

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

      7. unsub-neg 90.2%

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

      8. neg-mul-1 90.2%

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

      9. mul-1-neg 90.2%

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

      10. remove-double-neg 90.2%

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

      11. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. un-div-inv 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 1.30000000000000002e47 < d

   1. Initial program 38.3%

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

      1. fma-neg 38.3%

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

      2. distribute-rgt-neg-out 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-define 38.3%

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

   3. Simplified 38.3%

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

      1. distribute-rgt-neg-out 38.3%

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

      2. fma-neg 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-un-lft-identity 38.3%

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

      2. fma-undefine 38.3%

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

      3. +-commutative 38.3%

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

      4. add-sqr-sqrt 38.3%

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

      5. hypot-undefine 38.3%

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

      6. hypot-undefine 38.3%

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

      7. times-frac 68.7%

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

      8. hypot-undefine 38.2%

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

      9. +-commutative 38.2%

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

      10. hypot-define 68.7%

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

      11. *-commutative 68.7%

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

      12. fma-neg 68.8%

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

      13. *-commutative 68.8%

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

      14. hypot-undefine 38.2%

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

      15. +-commutative 38.2%

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

      16. hypot-define 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. associate-*l/ 69.0%

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

      2. *-lft-identity 69.0%

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

      3. fma-undefine 68.9%

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

      4. *-commutative 68.9%

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

      5. unsub-neg 68.9%

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

      6. *-commutative 68.9%

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

   10. Simplified 68.9%

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

   11. Taylor expanded in c around 0 75.4%

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

       1. +-commutative 75.4%

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

       2. mul-1-neg 75.4%

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

       3. sub-neg 75.4%

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

       4. associate-/l* 80.2%

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

   13. Simplified 80.2%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.26 \cdot 10^{+86}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.25 \cdot 10^{-159}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{t\_0}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= d -1.9e+85)
     (/ (- (* c (/ b d)) a) d)
     (if (<= d -1.95e-159)
       (/ t_0 (fma c c (* d d)))
       (if (<= d 3.75e-47)
         (/ (- b (/ a (/ c d))) c)
         (if (<= d 1.7e+47)
           (/ t_0 (+ (* c c) (* d d)))
           (/ (- (* b (/ c d)) a) (hypot d c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if (d <= -1.9e+85) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= -1.95e-159) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (d <= 3.75e-47) {
		tmp = (b - (a / (c / d))) / c;
	} else if (d <= 1.7e+47) {
		tmp = t_0 / ((c * c) + (d * d));
	} else {
		tmp = ((b * (c / d)) - a) / hypot(d, c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (d <= -1.9e+85)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= -1.95e-159)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (d <= 3.75e-47)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	elseif (d <= 1.7e+47)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.9e+85], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.95e-159], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.75e-47], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.7e+47], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.89999999999999996e85

   1. Initial program 43.7%

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

   3. Taylor expanded in c around 0 72.4%

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

      1. +-commutative 72.4%

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

      2. mul-1-neg 72.4%

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

      3. unsub-neg 72.4%

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

      4. unpow2 72.4%

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

      5. associate-/r* 78.3%

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

      6. div-sub 78.3%

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

      7. *-commutative 78.3%

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

      8. associate-/l* 88.3%

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

   5. Simplified 88.3%

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

   if -1.89999999999999996e85 < d < -1.94999999999999988e-159

   1. Initial program 90.2%

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

      1. fma-define 90.2%

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

   3. Simplified 90.2%

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

   if -1.94999999999999988e-159 < d < 3.74999999999999984e-47

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf 90.2%

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

      1. remove-double-neg 90.2%

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

      2. mul-1-neg 90.2%

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

      3. neg-mul-1 90.2%

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

      4. distribute-lft-in 90.2%

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

      5. distribute-lft-in 90.2%

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

      6. mul-1-neg 90.2%

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

      7. unsub-neg 90.2%

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

      8. neg-mul-1 90.2%

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

      9. mul-1-neg 90.2%

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

      10. remove-double-neg 90.2%

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

      11. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. un-div-inv 90.3%

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

   7. Applied egg-rr 90.3%

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

   if 3.74999999999999984e-47 < d < 1.6999999999999999e47

   1. Initial program 82.6%

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

   if 1.6999999999999999e47 < d

   1. Initial program 38.3%

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

      1. fma-neg 38.3%

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

      2. distribute-rgt-neg-out 38.3%

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

      3. +-commutative 38.3%

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

      4. fma-define 38.3%

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

   3. Simplified 38.3%

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

      1. distribute-rgt-neg-out 38.3%

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

      2. fma-neg 38.3%

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

   6. Applied egg-rr 38.3%

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

      1. *-un-lft-identity 38.3%

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

      2. fma-undefine 38.3%

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

      3. +-commutative 38.3%

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

      4. add-sqr-sqrt 38.3%

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

      5. hypot-undefine 38.3%

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

      6. hypot-undefine 38.3%

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

      7. times-frac 68.7%

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

      8. hypot-undefine 38.2%

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

      9. +-commutative 38.2%

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

      10. hypot-define 68.7%

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

      11. *-commutative 68.7%

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

      12. fma-neg 68.8%

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

      13. *-commutative 68.8%

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

      14. hypot-undefine 38.2%

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

      15. +-commutative 38.2%

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

      16. hypot-define 68.8%

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

   8. Applied egg-rr 68.8%

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

      1. associate-*l/ 69.0%

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

      2. *-lft-identity 69.0%

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

      3. fma-undefine 68.9%

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

      4. *-commutative 68.9%

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

      5. unsub-neg 68.9%

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

      6. *-commutative 68.9%

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

   10. Simplified 68.9%

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

   11. Taylor expanded in c around 0 75.4%

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

       1. +-commutative 75.4%

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

       2. mul-1-neg 75.4%

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

       3. sub-neg 75.4%

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

       4. associate-/l* 80.2%

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

   13. Simplified 80.2%

       \[\leadsto \frac{\color{blue}{b \cdot \frac{c}{d} - a}}{\mathsf{hypot}\left(d, c\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.95 \cdot 10^{-159}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{+47}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -2.35e+86)
     t_1
     (if (<= d -1.3e-159)
       t_0
       (if (<= d 3.75e-47)
         (/ (- b (/ a (/ c d))) c)
         (if (<= d 2e+146) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -2.35e+86) {
		tmp = t_1;
	} else if (d <= -1.3e-159) {
		tmp = t_0;
	} else if (d <= 3.75e-47) {
		tmp = (b - (a / (c / d))) / c;
	} else if (d <= 2e+146) {
		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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = ((c * (b / d)) - a) / d
    if (d <= (-2.35d+86)) then
        tmp = t_1
    else if (d <= (-1.3d-159)) then
        tmp = t_0
    else if (d <= 3.75d-47) then
        tmp = (b - (a / (c / d))) / c
    else if (d <= 2d+146) 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 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -2.35e+86) {
		tmp = t_1;
	} else if (d <= -1.3e-159) {
		tmp = t_0;
	} else if (d <= 3.75e-47) {
		tmp = (b - (a / (c / d))) / c;
	} else if (d <= 2e+146) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -2.35e+86:
		tmp = t_1
	elif d <= -1.3e-159:
		tmp = t_0
	elif d <= 3.75e-47:
		tmp = (b - (a / (c / d))) / c
	elif d <= 2e+146:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -2.35e+86)
		tmp = t_1;
	elseif (d <= -1.3e-159)
		tmp = t_0;
	elseif (d <= 3.75e-47)
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	elseif (d <= 2e+146)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -2.35e+86)
		tmp = t_1;
	elseif (d <= -1.3e-159)
		tmp = t_0;
	elseif (d <= 3.75e-47)
		tmp = (b - (a / (c / d))) / c;
	elseif (d <= 2e+146)
		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[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.35e+86], t$95$1, If[LessEqual[d, -1.3e-159], t$95$0, If[LessEqual[d, 3.75e-47], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2e+146], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.3500000000000001e86 or 1.99999999999999987e146 < d

   1. Initial program 37.6%

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

   3. Taylor expanded in c around 0 70.2%

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

      1. +-commutative 70.2%

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

      2. mul-1-neg 70.2%

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

      3. unsub-neg 70.2%

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

      4. unpow2 70.2%

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

      5. associate-/r* 78.6%

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

      6. div-sub 78.6%

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

      7. *-commutative 78.6%

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

      8. associate-/l* 87.3%

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

   5. Simplified 87.3%

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

   if -2.3500000000000001e86 < d < -1.2999999999999999e-159 or 3.74999999999999984e-47 < d < 1.99999999999999987e146

   1. Initial program 84.1%

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

   if -1.2999999999999999e-159 < d < 3.74999999999999984e-47

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf 90.2%

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

      1. remove-double-neg 90.2%

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

      2. mul-1-neg 90.2%

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

      3. neg-mul-1 90.2%

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

      4. distribute-lft-in 90.2%

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

      5. distribute-lft-in 90.2%

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

      6. mul-1-neg 90.2%

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

      7. unsub-neg 90.2%

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

      8. neg-mul-1 90.2%

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

      9. mul-1-neg 90.2%

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

      10. remove-double-neg 90.2%

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

      11. associate-/l* 90.2%

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

   5. Simplified 90.2%

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

      1. clear-num 90.2%

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

      2. un-div-inv 90.3%

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

   7. Applied egg-rr 90.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.35 \cdot 10^{+86}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.75 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+63} \lor \neg \left(c \leq -1.7 \cdot 10^{+19}\right) \land \left(c \leq -2.3 \cdot 10^{-109} \lor \neg \left(c \leq 1.45 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.6e+63)
         (and (not (<= c -1.7e+19))
              (or (<= c -2.3e-109) (not (<= c 1.45e+37)))))
   (/ b c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.6e+63) || (!(c <= -1.7e+19) && ((c <= -2.3e-109) || !(c <= 1.45e+37)))) {
		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 <= (-3.6d+63)) .or. (.not. (c <= (-1.7d+19))) .and. (c <= (-2.3d-109)) .or. (.not. (c <= 1.45d+37))) 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 <= -3.6e+63) || (!(c <= -1.7e+19) && ((c <= -2.3e-109) || !(c <= 1.45e+37)))) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.6e+63) or (not (c <= -1.7e+19) and ((c <= -2.3e-109) or not (c <= 1.45e+37))):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.6e+63) || (!(c <= -1.7e+19) && ((c <= -2.3e-109) || !(c <= 1.45e+37))))
		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 <= -3.6e+63) || (~((c <= -1.7e+19)) && ((c <= -2.3e-109) || ~((c <= 1.45e+37)))))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.6e+63], And[N[Not[LessEqual[c, -1.7e+19]], $MachinePrecision], Or[LessEqual[c, -2.3e-109], N[Not[LessEqual[c, 1.45e+37]], $MachinePrecision]]]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.59999999999999999e63 or -1.7e19 < c < -2.3000000000000001e-109 or 1.44999999999999989e37 < c

   1. Initial program 63.7%

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

   3. Taylor expanded in c around inf 66.7%

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

   if -3.59999999999999999e63 < c < -1.7e19 or -2.3000000000000001e-109 < c < 1.44999999999999989e37

   1. Initial program 69.7%

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

   3. Taylor expanded in c around 0 70.6%

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

      1. associate-*r/ 70.6%

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

      2. neg-mul-1 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+63} \lor \neg \left(c \leq -1.7 \cdot 10^{+19}\right) \land \left(c \leq -2.3 \cdot 10^{-109} \lor \neg \left(c \leq 1.45 \cdot 10^{+37}\right)\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.05 \cdot 10^{-20} \lor \neg \left(d \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.05e-20) (not (<= d 3e+14)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.05e-20) || !(d <= 3e+14)) {
		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 <= (-4.05d-20)) .or. (.not. (d <= 3d+14))) 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 <= -4.05e-20) || !(d <= 3e+14)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.05e-20) or not (d <= 3e+14):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.05e-20) || !(d <= 3e+14))
		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 <= -4.05e-20) || ~((d <= 3e+14)))
		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, -4.05e-20], N[Not[LessEqual[d, 3e+14]], $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 < -4.05000000000000024e-20 or 3e14 < d

   1. Initial program 53.6%

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

   3. Taylor expanded in c around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   5. Simplified 65.8%

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

   if -4.05000000000000024e-20 < d < 3e14

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf 83.1%

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

      1. remove-double-neg 83.1%

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

      2. mul-1-neg 83.1%

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

      3. neg-mul-1 83.1%

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

      4. distribute-lft-in 83.1%

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

      5. distribute-lft-in 83.1%

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

      6. mul-1-neg 83.1%

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

      7. unsub-neg 83.1%

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

      8. neg-mul-1 83.1%

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

      9. mul-1-neg 83.1%

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

      10. remove-double-neg 83.1%

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

      11. associate-/l* 83.1%

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

   5. Simplified 83.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.05 \cdot 10^{-20} \lor \neg \left(d \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{-20} \lor \neg \left(d \leq 3.6 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.2e-20) (not (<= d 3.6e-30)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.2e-20) || !(d <= 3.6e-30)) {
		tmp = ((b * (c / d)) - 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 <= (-4.2d-20)) .or. (.not. (d <= 3.6d-30))) then
        tmp = ((b * (c / d)) - 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 <= -4.2e-20) || !(d <= 3.6e-30)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.2e-20) or not (d <= 3.6e-30):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.2e-20) || !(d <= 3.6e-30))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / 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 <= -4.2e-20) || ~((d <= 3.6e-30)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.2e-20], N[Not[LessEqual[d, 3.6e-30]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / 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 < -4.1999999999999998e-20 or 3.6000000000000003e-30 < d

   1. Initial program 56.5%

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

      1. fma-neg 56.5%

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

      2. distribute-rgt-neg-out 56.5%

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

      3. +-commutative 56.5%

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

      4. fma-define 56.5%

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

   3. Simplified 56.5%

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

      1. distribute-rgt-neg-out 56.5%

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

      2. fma-neg 56.5%

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

   6. Applied egg-rr 56.5%

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

      1. *-un-lft-identity 56.5%

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

      2. fma-undefine 56.5%

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

      3. +-commutative 56.5%

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

      4. add-sqr-sqrt 56.4%

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

      5. hypot-undefine 56.5%

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

      6. hypot-undefine 56.5%

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

      7. times-frac 71.7%

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

      8. hypot-undefine 56.4%

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

      9. +-commutative 56.4%

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

      10. hypot-define 71.7%

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

      11. *-commutative 71.7%

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

      12. fma-neg 71.7%

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

      13. *-commutative 71.7%

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

      14. hypot-undefine 56.4%

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

      15. +-commutative 56.4%

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

      16. hypot-define 71.7%

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

   8. Applied egg-rr 71.7%

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

      1. associate-*l/ 71.9%

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

      2. *-lft-identity 71.9%

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

      3. fma-undefine 71.8%

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

      4. *-commutative 71.8%

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

      5. unsub-neg 71.8%

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

      6. *-commutative 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in c around 0 65.4%

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

       1. +-commutative 65.4%

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

       2. mul-1-neg 65.4%

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

       3. unsub-neg 65.4%

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

       4. *-rgt-identity 65.4%

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

       5. unpow2 65.4%

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

       6. times-frac 70.1%

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

       7. *-commutative 70.1%

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

       8. associate-*r/ 75.0%

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

       9. associate-*r/ 75.0%

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

       10. associate-*r/ 70.1%

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

       11. *-commutative 70.1%

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

       12. *-rgt-identity 70.1%

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

       13. *-commutative 70.1%

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

       14. associate-*r/ 75.0%

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

       15. associate-*r/ 70.1%

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

       16. *-commutative 70.1%

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

       17. div-sub 70.1%

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

       18. associate-/l* 74.4%

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

   13. Simplified 74.4%

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

   if -4.1999999999999998e-20 < d < 3.6000000000000003e-30

   1. Initial program 78.8%

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

   3. Taylor expanded in c around inf 87.0%

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

      1. remove-double-neg 87.0%

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

      2. mul-1-neg 87.0%

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

      3. neg-mul-1 87.0%

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

      4. distribute-lft-in 87.0%

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

      5. distribute-lft-in 87.0%

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

      6. mul-1-neg 87.0%

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

      7. unsub-neg 87.0%

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

      8. neg-mul-1 87.0%

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

      9. mul-1-neg 87.0%

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

      10. remove-double-neg 87.0%

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

      11. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{-20} \lor \neg \left(d \leq 3.6 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.5e-20)
   (/ (- (* c (/ b d)) a) d)
   (if (<= d 2.7e-30) (/ (- b (* a (/ d c))) c) (/ (- (* b (/ c d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.5e-20) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 2.7e-30) {
		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 (d <= (-3.5d-20)) then
        tmp = ((c * (b / d)) - a) / d
    else if (d <= 2.7d-30) 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 (d <= -3.5e-20) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (d <= 2.7e-30) {
		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 d <= -3.5e-20:
		tmp = ((c * (b / d)) - a) / d
	elif d <= 2.7e-30:
		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 (d <= -3.5e-20)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (d <= 2.7e-30)
		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 (d <= -3.5e-20)
		tmp = ((c * (b / d)) - a) / d;
	elseif (d <= 2.7e-30)
		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[LessEqual[d, -3.5e-20], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 2.7e-30], 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 3 regimes

2. if d < -3.50000000000000003e-20

   1. Initial program 60.7%

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

   3. Taylor expanded in c around 0 68.6%

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

      1. +-commutative 68.6%

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

      2. mul-1-neg 68.6%

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

      3. unsub-neg 68.6%

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

      4. unpow2 68.6%

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

      5. associate-/r* 72.3%

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

      6. div-sub 72.3%

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

      7. *-commutative 72.3%

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

      8. associate-/l* 78.6%

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

   5. Simplified 78.6%

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

   if -3.50000000000000003e-20 < d < 2.69999999999999987e-30

   1. Initial program 78.8%

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

   3. Taylor expanded in c around inf 87.0%

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

      1. remove-double-neg 87.0%

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

      2. mul-1-neg 87.0%

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

      3. neg-mul-1 87.0%

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

      4. distribute-lft-in 87.0%

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

      5. distribute-lft-in 87.0%

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

      6. mul-1-neg 87.0%

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

      7. unsub-neg 87.0%

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

      8. neg-mul-1 87.0%

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

      9. mul-1-neg 87.0%

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

      10. remove-double-neg 87.0%

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

      11. associate-/l* 87.0%

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

   5. Simplified 87.0%

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

   if 2.69999999999999987e-30 < d

   1. Initial program 51.3%

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

      1. fma-neg 51.3%

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

      2. distribute-rgt-neg-out 51.3%

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

      3. +-commutative 51.3%

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

      4. fma-define 51.3%

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

   3. Simplified 51.3%

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

      1. distribute-rgt-neg-out 51.3%

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

      2. fma-neg 51.3%

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

   6. Applied egg-rr 51.3%

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

      1. *-un-lft-identity 51.3%

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

      2. fma-undefine 51.3%

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

      3. +-commutative 51.3%

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

      4. add-sqr-sqrt 51.3%

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

      5. hypot-undefine 51.3%

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

      6. hypot-undefine 51.3%

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

      7. times-frac 73.8%

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

      8. hypot-undefine 51.2%

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

      9. +-commutative 51.2%

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

      10. hypot-define 73.8%

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

      11. *-commutative 73.8%

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

      12. fma-neg 73.9%

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

      13. *-commutative 73.9%

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

      14. hypot-undefine 51.2%

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

      15. +-commutative 51.2%

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

      16. hypot-define 73.9%

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

   8. Applied egg-rr 73.9%

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

      1. associate-*l/ 74.0%

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

      2. *-lft-identity 74.0%

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

      3. fma-undefine 74.0%

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

      4. *-commutative 74.0%

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

      5. unsub-neg 74.0%

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

      6. *-commutative 74.0%

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

   10. Simplified 74.0%

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

   11. Taylor expanded in c around 0 61.5%

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

       1. +-commutative 61.5%

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

       2. mul-1-neg 61.5%

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

       3. unsub-neg 61.5%

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

       4. *-rgt-identity 61.5%

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

       5. unpow2 61.5%

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

       6. times-frac 67.4%

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

       7. *-commutative 67.4%

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

       8. associate-*r/ 70.7%

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

       9. associate-*r/ 70.7%

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

       10. associate-*r/ 67.5%

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

       11. *-commutative 67.5%

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

       12. *-rgt-identity 67.5%

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

       13. *-commutative 67.5%

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

       14. associate-*r/ 70.7%

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

       15. associate-*r/ 67.5%

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

       16. *-commutative 67.5%

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

       17. div-sub 67.5%

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

       18. associate-/l* 70.7%

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

   13. Simplified 70.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{-30}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.18 \cdot 10^{+191}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -1.18e+191) (/ b d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.18e+191) {
		tmp = b / 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 <= (-1.18d+191)) then
        tmp = b / 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 <= -1.18e+191) {
		tmp = b / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.18e+191:
		tmp = b / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.18e+191)
		tmp = Float64(b / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.18e+191)
		tmp = b / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.18e+191], N[(b / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.17999999999999994e191

   1. Initial program 48.6%

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

      1. fma-neg 48.6%

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

      2. distribute-rgt-neg-out 48.6%

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

      3. +-commutative 48.6%

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

      4. fma-define 48.6%

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

   3. Simplified 48.6%

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

      1. distribute-rgt-neg-out 48.6%

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

      2. fma-neg 48.6%

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

   6. Applied egg-rr 48.6%

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

      1. *-un-lft-identity 48.6%

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

      2. fma-undefine 48.6%

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

      3. +-commutative 48.6%

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

      4. add-sqr-sqrt 48.6%

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

      5. hypot-undefine 48.6%

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

      6. hypot-undefine 48.6%

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

      7. times-frac 67.5%

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

      8. hypot-undefine 48.6%

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

      9. +-commutative 48.6%

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

      10. hypot-define 67.5%

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

      11. *-commutative 67.5%

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

      12. fma-neg 67.5%

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

      13. *-commutative 67.5%

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

      14. hypot-undefine 48.6%

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

      15. +-commutative 48.6%

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

      16. hypot-define 67.5%

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

   8. Applied egg-rr 67.5%

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

      1. associate-*l/ 67.5%

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

      2. *-lft-identity 67.5%

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

      3. fma-undefine 67.5%

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

      4. *-commutative 67.5%

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

      5. unsub-neg 67.5%

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

      6. *-commutative 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in c around 0 48.8%

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

       1. +-commutative 48.8%

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

       2. mul-1-neg 48.8%

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

       3. sub-neg 48.8%

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

       4. associate-/l* 49.0%

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

   13. Simplified 49.0%

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

   14. Taylor expanded in c around inf 29.1%

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

   if -1.17999999999999994e191 < d

   1. Initial program 68.5%

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

   3. Taylor expanded in c around inf 48.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.18 \cdot 10^{+191}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.9% 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 66.4%

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

3. Taylor expanded in c around inf 44.0%

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

4. Final simplification 44.0%

   \[\leadsto \frac{b}{c} \]
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 2024055 
(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))))