Complex division, imag part

Percentage Accurate: 61.8% → 92.5%

Time: 12.5s

Alternatives: 15

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 3.55 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (/ c (hypot c d))
          (/ b (hypot c d))
          (/ (- a) (/ (pow (hypot c d) 2.0) d))))
        (t_1 (- (* (/ 1.0 (hypot c d)) (/ c (/ (hypot c d) b))) (/ a d))))
   (if (<= d -1.75e+155)
     t_1
     (if (<= d -1.15e-135)
       t_0
       (if (<= d 5.8e-180)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 3.55e+90) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / (pow(hypot(c, d), 2.0) / d)));
	double t_1 = ((1.0 / hypot(c, d)) * (c / (hypot(c, d) / b))) - (a / d);
	double tmp;
	if (d <= -1.75e+155) {
		tmp = t_1;
	} else if (d <= -1.15e-135) {
		tmp = t_0;
	} else if (d <= 5.8e-180) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 3.55e+90) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / Float64((hypot(c, d) ^ 2.0) / d)))
	t_1 = Float64(Float64(Float64(1.0 / hypot(c, d)) * Float64(c / Float64(hypot(c, d) / b))) - Float64(a / d))
	tmp = 0.0
	if (d <= -1.75e+155)
		tmp = t_1;
	elseif (d <= -1.15e-135)
		tmp = t_0;
	elseif (d <= 5.8e-180)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 3.55e+90)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / N[(N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.75e+155], t$95$1, If[LessEqual[d, -1.15e-135], t$95$0, If[LessEqual[d, 5.8e-180], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 3.55e+90], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.74999999999999992e155 or 3.54999999999999988e90 < d

   1. Initial program 36.6%

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

      1. div-sub 36.6%

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

      2. sub-neg 36.6%

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

      3. *-un-lft-identity 36.6%

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

      4. add-sqr-sqrt 36.6%

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

      5. times-frac 36.6%

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

      6. fma-def 36.6%

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

      7. hypot-def 36.6%

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

      8. hypot-def 49.2%

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

      9. associate-/l* 50.4%

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

      10. add-sqr-sqrt 50.4%

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

      11. pow2 50.4%

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

      12. hypot-def 50.4%

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

   3. Applied egg-rr 50.4%

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

      1. fma-neg 50.4%

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

      2. *-commutative 50.4%

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

      3. associate-/l* 52.2%

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

      4. associate-/r/ 51.0%

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

      5. *-commutative 51.0%

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

   5. Simplified 51.0%

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

   6. Taylor expanded in d around inf 96.5%

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

   if -1.74999999999999992e155 < d < -1.15e-135 or 5.79999999999999961e-180 < d < 3.54999999999999988e90

   1. Initial program 77.5%

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

      1. div-sub 76.7%

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

      2. sub-neg 76.7%

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

      3. *-commutative 76.7%

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

      4. add-sqr-sqrt 76.7%

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

      5. times-frac 78.8%

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

      6. fma-def 78.8%

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

      7. hypot-def 78.8%

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

      8. hypot-def 90.9%

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

      9. associate-/l* 95.8%

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

      10. add-sqr-sqrt 95.8%

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

      11. pow2 95.8%

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

      12. hypot-def 95.8%

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

   3. Applied egg-rr 95.8%

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

   if -1.15e-135 < d < 5.79999999999999961e-180

   1. Initial program 71.7%

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

      1. *-un-lft-identity 71.7%

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

      2. add-sqr-sqrt 71.7%

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

      3. times-frac 71.8%

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

      4. hypot-def 71.9%

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

      5. hypot-def 84.7%

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

   3. Applied egg-rr 84.7%

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

   4. Taylor expanded in c around -inf 50.4%

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

      1. +-commutative 50.4%

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

      2. neg-mul-1 50.4%

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

      3. unsub-neg 50.4%

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

      4. associate-/l* 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in c around -inf 94.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{+155}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-180}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 3.55 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{\frac{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}{d}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \]

Alternative 2: 90.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}}\\ t_1 := t_0 - d \cdot \frac{a}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ t_2 := t_0 - \frac{a}{d}\\ \mathbf{if}\;d \leq -4.1 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 2.95 \cdot 10^{-170}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ c (/ (hypot c d) b))))
        (t_1 (- t_0 (* d (/ a (pow (hypot c d) 2.0)))))
        (t_2 (- t_0 (/ a d))))
   (if (<= d -4.1e+89)
     t_2
     (if (<= d -1.55e-69)
       t_1
       (if (<= d 2.95e-170)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 1.85e+90) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (c / (hypot(c, d) / b));
	double t_1 = t_0 - (d * (a / pow(hypot(c, d), 2.0)));
	double t_2 = t_0 - (a / d);
	double tmp;
	if (d <= -4.1e+89) {
		tmp = t_2;
	} else if (d <= -1.55e-69) {
		tmp = t_1;
	} else if (d <= 2.95e-170) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.85e+90) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / Math.hypot(c, d)) * (c / (Math.hypot(c, d) / b));
	double t_1 = t_0 - (d * (a / Math.pow(Math.hypot(c, d), 2.0)));
	double t_2 = t_0 - (a / d);
	double tmp;
	if (d <= -4.1e+89) {
		tmp = t_2;
	} else if (d <= -1.55e-69) {
		tmp = t_1;
	} else if (d <= 2.95e-170) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.85e+90) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / math.hypot(c, d)) * (c / (math.hypot(c, d) / b))
	t_1 = t_0 - (d * (a / math.pow(math.hypot(c, d), 2.0)))
	t_2 = t_0 - (a / d)
	tmp = 0
	if d <= -4.1e+89:
		tmp = t_2
	elif d <= -1.55e-69:
		tmp = t_1
	elif d <= 2.95e-170:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 1.85e+90:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(c / Float64(hypot(c, d) / b)))
	t_1 = Float64(t_0 - Float64(d * Float64(a / (hypot(c, d) ^ 2.0))))
	t_2 = Float64(t_0 - Float64(a / d))
	tmp = 0.0
	if (d <= -4.1e+89)
		tmp = t_2;
	elseif (d <= -1.55e-69)
		tmp = t_1;
	elseif (d <= 2.95e-170)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 1.85e+90)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / hypot(c, d)) * (c / (hypot(c, d) / b));
	t_1 = t_0 - (d * (a / (hypot(c, d) ^ 2.0)));
	t_2 = t_0 - (a / d);
	tmp = 0.0;
	if (d <= -4.1e+89)
		tmp = t_2;
	elseif (d <= -1.55e-69)
		tmp = t_1;
	elseif (d <= 2.95e-170)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 1.85e+90)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(d * N[(a / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.1e+89], t$95$2, If[LessEqual[d, -1.55e-69], t$95$1, If[LessEqual[d, 2.95e-170], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e+90], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.09999999999999985e89 or 1.85e90 < d

   1. Initial program 45.0%

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

      1. div-sub 45.0%

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

      2. sub-neg 45.0%

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

      3. *-un-lft-identity 45.0%

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

      4. add-sqr-sqrt 45.0%

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

      5. times-frac 45.0%

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

      6. fma-def 45.0%

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

      7. hypot-def 45.0%

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

      8. hypot-def 55.7%

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

      9. associate-/l* 57.9%

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

      10. add-sqr-sqrt 57.9%

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

      11. pow2 57.9%

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

      12. hypot-def 57.9%

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

   3. Applied egg-rr 57.9%

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

      1. fma-neg 57.9%

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

      2. *-commutative 57.9%

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

      3. associate-/l* 59.4%

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

      4. associate-/r/ 57.2%

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

      5. *-commutative 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in d around inf 97.0%

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

   if -4.09999999999999985e89 < d < -1.55e-69 or 2.9499999999999999e-170 < d < 1.85e90

   1. Initial program 75.0%

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

      1. div-sub 74.0%

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

      2. sub-neg 74.0%

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

      3. *-un-lft-identity 74.0%

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

      4. add-sqr-sqrt 74.0%

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

      5. times-frac 73.9%

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

      6. fma-def 73.9%

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

      7. hypot-def 73.9%

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

      8. hypot-def 80.2%

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

      9. associate-/l* 85.2%

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

      10. add-sqr-sqrt 85.2%

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

      11. pow2 85.2%

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

      12. hypot-def 85.2%

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

   3. Applied egg-rr 85.2%

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

      1. fma-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-/l* 95.4%

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

      4. associate-/r/ 93.2%

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

      5. *-commutative 93.2%

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

   5. Simplified 93.2%

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

   if -1.55e-69 < d < 2.9499999999999999e-170

   1. Initial program 73.2%

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

      1. *-un-lft-identity 73.2%

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

      2. add-sqr-sqrt 73.2%

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

      3. times-frac 73.2%

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

      4. hypot-def 73.2%

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

      5. hypot-def 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in c around -inf 51.4%

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

      1. +-commutative 51.4%

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

      2. neg-mul-1 51.4%

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

      3. unsub-neg 51.4%

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

      4. associate-/l* 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in c around -inf 92.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -1.55 \cdot 10^{-69}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - d \cdot \frac{a}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{elif}\;d \leq 2.95 \cdot 10^{-170}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+90}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - d \cdot \frac{a}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \]

Alternative 3: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 5e+280)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (* (/ -1.0 c) (- (/ a (/ c d)) b)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+280) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+280) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 5e+280:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 5e+280)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 5e+280)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $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))) < 5.0000000000000002e280

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 80.0%

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

      4. hypot-def 80.0%

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

      5. hypot-def 96.7%

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

   3. Applied egg-rr 96.7%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.6%

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

      1. *-un-lft-identity 16.6%

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

      2. add-sqr-sqrt 16.6%

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

      3. times-frac 16.6%

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

      4. hypot-def 16.6%

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

      5. hypot-def 24.3%

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

   3. Applied egg-rr 24.3%

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

   4. Taylor expanded in c around -inf 32.9%

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

      1. +-commutative 32.9%

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

      2. neg-mul-1 32.9%

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

      3. unsub-neg 32.9%

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

      4. associate-/l* 36.1%

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

   6. Simplified 36.1%

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

   7. Taylor expanded in c around -inf 63.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \end{array} \]

Alternative 4: 88.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_1}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))) (t_1 (- (* c b) (* d a))))
   (if (<= (/ t_1 (+ (* c c) (* d d))) 5e+280)
     (* t_0 (/ t_1 (hypot c d)))
     (- (* t_0 (/ c (/ (hypot c d) b))) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 5e+280) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / Math.hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 5e+280) {
		tmp = t_0 * (t_1 / Math.hypot(c, d));
	} else {
		tmp = (t_0 * (c / (Math.hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 1.0 / math.hypot(c, d)
	t_1 = (c * b) - (d * a)
	tmp = 0
	if (t_1 / ((c * c) + (d * d))) <= 5e+280:
		tmp = t_0 * (t_1 / math.hypot(c, d))
	else:
		tmp = (t_0 * (c / (math.hypot(c, d) / b))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(c * c) + Float64(d * d))) <= 5e+280)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = Float64(Float64(t_0 * Float64(c / Float64(hypot(c, d) / b))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / hypot(c, d);
	t_1 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_1 / ((c * c) + (d * d))) <= 5e+280)
		tmp = t_0 * (t_1 / hypot(c, d));
	else
		tmp = (t_0 * (c / (hypot(c, d) / b))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $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))) < 5.0000000000000002e280

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 80.0%

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

      4. hypot-def 80.0%

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

      5. hypot-def 96.7%

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

   3. Applied egg-rr 96.7%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.6%

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

      1. div-sub 6.7%

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

      2. sub-neg 6.7%

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

      3. *-un-lft-identity 6.7%

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

      4. add-sqr-sqrt 6.7%

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

      5. times-frac 6.7%

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

      6. fma-def 6.7%

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

      7. hypot-def 6.7%

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

      8. hypot-def 12.4%

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

      9. associate-/l* 15.5%

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

      10. add-sqr-sqrt 15.5%

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

      11. pow2 15.5%

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

      12. hypot-def 15.5%

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

   3. Applied egg-rr 15.5%

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

      1. fma-neg 15.5%

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

      2. *-commutative 15.5%

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

      3. associate-/l* 50.7%

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

      4. associate-/r/ 50.7%

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

      5. *-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in d around inf 67.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \]

Alternative 5: 89.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{\frac{d \cdot a - c \cdot b}{\mathsf{hypot}\left(c, d\right)}}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* c b) (* d a)) (+ (* c c) (* d d))) 5e+280)
   (/ (/ (- (* d a) (* c b)) (hypot c d)) (- (hypot c d)))
   (- (* (/ 1.0 (hypot c d)) (/ c (/ (hypot c d) b))) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((c * b) - (d * a)) / ((c * c) + (d * d))) <= 5e+280) {
		tmp = (((d * a) - (c * b)) / hypot(c, d)) / -hypot(c, d);
	} else {
		tmp = ((1.0 / hypot(c, d)) * (c / (hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((((c * b) - (d * a)) / ((c * c) + (d * d))) <= 5e+280) {
		tmp = (((d * a) - (c * b)) / Math.hypot(c, d)) / -Math.hypot(c, d);
	} else {
		tmp = ((1.0 / Math.hypot(c, d)) * (c / (Math.hypot(c, d) / b))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (((c * b) - (d * a)) / ((c * c) + (d * d))) <= 5e+280:
		tmp = (((d * a) - (c * b)) / math.hypot(c, d)) / -math.hypot(c, d)
	else:
		tmp = ((1.0 / math.hypot(c, d)) * (c / (math.hypot(c, d) / b))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+280)
		tmp = Float64(Float64(Float64(Float64(d * a) - Float64(c * b)) / hypot(c, d)) / Float64(-hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(1.0 / hypot(c, d)) * Float64(c / Float64(hypot(c, d) / b))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((((c * b) - (d * a)) / ((c * c) + (d * d))) <= 5e+280)
		tmp = (((d * a) - (c * b)) / hypot(c, d)) / -hypot(c, d);
	else
		tmp = ((1.0 / hypot(c, d)) * (c / (hypot(c, d) / b))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+280], N[(N[(N[(N[(d * a), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])), $MachinePrecision], N[(N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(c / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $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))) < 5.0000000000000002e280

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 80.0%

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

      4. hypot-def 80.0%

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

      5. hypot-def 96.7%

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

   3. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.8%

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

      2. *-un-lft-identity 96.8%

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

      3. frac-2neg 96.8%

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

      4. fma-neg 96.8%

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

      5. distribute-rgt-neg-in 96.8%

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

   5. Applied egg-rr 96.8%

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

      1. distribute-neg-frac 96.8%

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

      2. fma-def 96.8%

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

      3. +-commutative 96.8%

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

      4. distribute-neg-in 96.8%

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

      5. distribute-rgt-neg-out 96.8%

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

      6. remove-double-neg 96.8%

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

      7. *-commutative 96.8%

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

      8. distribute-rgt-neg-in 96.8%

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

   7. Simplified 96.8%

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

   if 5.0000000000000002e280 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.6%

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

      1. div-sub 6.7%

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

      2. sub-neg 6.7%

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

      3. *-un-lft-identity 6.7%

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

      4. add-sqr-sqrt 6.7%

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

      5. times-frac 6.7%

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

      6. fma-def 6.7%

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

      7. hypot-def 6.7%

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

      8. hypot-def 12.4%

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

      9. associate-/l* 15.5%

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

      10. add-sqr-sqrt 15.5%

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

      11. pow2 15.5%

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

      12. hypot-def 15.5%

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

   3. Applied egg-rr 15.5%

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

      1. fma-neg 15.5%

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

      2. *-commutative 15.5%

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

      3. associate-/l* 50.7%

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

      4. associate-/r/ 50.7%

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

      5. *-commutative 50.7%

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

   5. Simplified 50.7%

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

   6. Taylor expanded in d around inf 67.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+280}:\\ \;\;\;\;\frac{\frac{d \cdot a - c \cdot b}{\mathsf{hypot}\left(c, d\right)}}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c}{\frac{\mathsf{hypot}\left(c, d\right)}{b}} - \frac{a}{d}\\ \end{array} \]

Alternative 6: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ t_1 := \frac{b}{\frac{d}{c}}\\ t_2 := -\mathsf{hypot}\left(c, d\right)\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1 - a}{t_2}\\ \mathbf{elif}\;d \leq -6.9 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t_1}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (- (* c b) (* d a)) (pow (hypot c d) -2.0)))
        (t_1 (/ b (/ d c)))
        (t_2 (- (hypot c d))))
   (if (<= d -3.4e+143)
     (/ (- t_1 a) t_2)
     (if (<= d -6.9e-74)
       t_0
       (if (<= d 6.8e-105)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 7.5e+28) t_0 (/ (- a t_1) t_2)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) * pow(hypot(c, d), -2.0);
	double t_1 = b / (d / c);
	double t_2 = -hypot(c, d);
	double tmp;
	if (d <= -3.4e+143) {
		tmp = (t_1 - a) / t_2;
	} else if (d <= -6.9e-74) {
		tmp = t_0;
	} else if (d <= 6.8e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - t_1) / t_2;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) * Math.pow(Math.hypot(c, d), -2.0);
	double t_1 = b / (d / c);
	double t_2 = -Math.hypot(c, d);
	double tmp;
	if (d <= -3.4e+143) {
		tmp = (t_1 - a) / t_2;
	} else if (d <= -6.9e-74) {
		tmp = t_0;
	} else if (d <= 6.8e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - t_1) / t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) * math.pow(math.hypot(c, d), -2.0)
	t_1 = b / (d / c)
	t_2 = -math.hypot(c, d)
	tmp = 0
	if d <= -3.4e+143:
		tmp = (t_1 - a) / t_2
	elif d <= -6.9e-74:
		tmp = t_0
	elif d <= 6.8e-105:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 7.5e+28:
		tmp = t_0
	else:
		tmp = (a - t_1) / t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) * (hypot(c, d) ^ -2.0))
	t_1 = Float64(b / Float64(d / c))
	t_2 = Float64(-hypot(c, d))
	tmp = 0.0
	if (d <= -3.4e+143)
		tmp = Float64(Float64(t_1 - a) / t_2);
	elseif (d <= -6.9e-74)
		tmp = t_0;
	elseif (d <= 6.8e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = Float64(Float64(a - t_1) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) * (hypot(c, d) ^ -2.0);
	t_1 = b / (d / c);
	t_2 = -hypot(c, d);
	tmp = 0.0;
	if (d <= -3.4e+143)
		tmp = (t_1 - a) / t_2;
	elseif (d <= -6.9e-74)
		tmp = t_0;
	elseif (d <= 6.8e-105)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = (a - t_1) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])}, If[LessEqual[d, -3.4e+143], N[(N[(t$95$1 - a), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[d, -6.9e-74], t$95$0, If[LessEqual[d, 6.8e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+28], t$95$0, N[(N[(a - t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.39999999999999982e143

   1. Initial program 39.9%

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

      1. *-un-lft-identity 39.9%

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

      2. add-sqr-sqrt 39.9%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. hypot-def 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. associate-*l/ 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. frac-2neg 72.0%

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

      4. fma-neg 72.1%

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

      5. distribute-rgt-neg-in 72.1%

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

   5. Applied egg-rr 72.1%

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

      1. distribute-neg-frac 72.1%

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

      2. fma-def 72.0%

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

      3. +-commutative 72.0%

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

      4. distribute-neg-in 72.0%

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

      5. distribute-rgt-neg-out 72.0%

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

      6. remove-double-neg 72.0%

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

      7. *-commutative 72.0%

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

      8. distribute-rgt-neg-in 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in d around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-/l* 92.0%

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

   10. Simplified 92.0%

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

   if -3.39999999999999982e143 < d < -6.89999999999999981e-74 or 6.79999999999999984e-105 < d < 7.4999999999999998e28

   1. Initial program 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. times-frac 82.7%

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

      4. hypot-def 82.7%

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

      5. hypot-def 87.8%

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

   3. Applied egg-rr 87.8%

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

      1. frac-times 82.9%

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

      2. *-un-lft-identity 82.9%

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

      3. *-commutative 82.9%

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

      4. unpow2 82.9%

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

      5. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{c \cdot b - a \cdot d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\right)} \]

      6. expm1-udef 37.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{c \cdot b - a \cdot d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} - 1} \]

      7. div-inv 37.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(c \cdot b - a \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}}\right)} - 1 \]

      8. *-commutative 37.1%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\color{blue}{b \cdot c} - a \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} - 1 \]

      9. fma-neg 37.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(b, c, -a \cdot d\right)} \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} - 1 \]

      10. distribute-rgt-neg-in 37.1%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(b, c, \color{blue}{a \cdot \left(-d\right)}\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)} - 1 \]

      11. pow-flip 37.1%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot \color{blue}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{\left(-2\right)}}\right)} - 1 \]

      12. metadata-eval 37.1%

          \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{\color{blue}{-2}}\right)} - 1 \]

   5. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\right)\right)} \]

      2. expm1-log1p 83.0%

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

      3. fma-def 83.0%

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

      4. distribute-rgt-neg-out 83.0%

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

      5. unsub-neg 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   if -6.89999999999999981e-74 < d < 6.79999999999999984e-105

   1. Initial program 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. times-frac 70.4%

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

      4. hypot-def 70.4%

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

      5. hypot-def 81.1%

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

   3. Applied egg-rr 81.1%

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

   4. Taylor expanded in c around -inf 46.4%

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

      1. +-commutative 46.4%

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

      2. neg-mul-1 46.4%

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

      3. unsub-neg 46.4%

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

      4. associate-/l* 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in c around -inf 88.5%

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

   if 7.4999999999999998e28 < d

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.4%

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

      4. hypot-def 46.4%

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

      5. hypot-def 68.0%

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

   3. Applied egg-rr 68.0%

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

      1. associate-*l/ 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. frac-2neg 68.1%

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

      4. fma-neg 68.2%

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

      5. distribute-rgt-neg-in 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. distribute-neg-frac 68.2%

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

      2. fma-def 68.1%

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

      3. +-commutative 68.1%

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

      4. distribute-neg-in 68.1%

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

      5. distribute-rgt-neg-out 68.1%

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

      6. remove-double-neg 68.1%

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

      7. *-commutative 68.1%

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

      8. distribute-rgt-neg-in 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in d around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. associate-/l* 90.1%

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

   10. Simplified 90.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -6.9 \cdot 10^{-74}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{elif}\;d \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\left(c \cdot b - d \cdot a\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - \frac{b}{\frac{d}{c}}}{-\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 7: 81.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a - \frac{b}{\frac{d}{c}}}{-\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -6.2e+143)
     (/ (- a) d)
     (if (<= d -1.45e-69)
       t_0
       (if (<= d 4.8e-106)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 7.5e+28) t_0 (/ (- a (/ b (/ d c))) (- (hypot c d)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+143) {
		tmp = -a / d;
	} else if (d <= -1.45e-69) {
		tmp = t_0;
	} else if (d <= 4.8e-106) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - (b / (d / c))) / -hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+143) {
		tmp = -a / d;
	} else if (d <= -1.45e-69) {
		tmp = t_0;
	} else if (d <= 4.8e-106) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - (b / (d / c))) / -Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -6.2e+143:
		tmp = -a / d
	elif d <= -1.45e-69:
		tmp = t_0
	elif d <= 4.8e-106:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 7.5e+28:
		tmp = t_0
	else:
		tmp = (a - (b / (d / c))) / -math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -6.2e+143)
		tmp = Float64(Float64(-a) / d);
	elseif (d <= -1.45e-69)
		tmp = t_0;
	elseif (d <= 4.8e-106)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = Float64(Float64(a - Float64(b / Float64(d / c))) / Float64(-hypot(c, d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -6.2e+143)
		tmp = -a / d;
	elseif (d <= -1.45e-69)
		tmp = t_0;
	elseif (d <= 4.8e-106)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = (a - (b / (d / c))) / -hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e+143], N[((-a) / d), $MachinePrecision], If[LessEqual[d, -1.45e-69], t$95$0, If[LessEqual[d, 4.8e-106], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+28], t$95$0, N[(N[(a - N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.1999999999999998e143

   1. Initial program 39.9%

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

   2. Taylor expanded in c around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. neg-mul-1 79.1%

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

   4. Simplified 79.1%

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

   if -6.1999999999999998e143 < d < -1.4499999999999999e-69 or 4.7999999999999995e-106 < d < 7.4999999999999998e28

   1. Initial program 82.4%

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

   if -1.4499999999999999e-69 < d < 4.7999999999999995e-106

   1. Initial program 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. add-sqr-sqrt 70.9%

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

      3. times-frac 71.0%

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

      4. hypot-def 71.0%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in c around -inf 47.6%

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

      1. +-commutative 47.6%

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

      2. neg-mul-1 47.6%

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

      3. unsub-neg 47.6%

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

      4. associate-/l* 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c around -inf 88.8%

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

   if 7.4999999999999998e28 < d

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.4%

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

      4. hypot-def 46.4%

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

      5. hypot-def 68.0%

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

   3. Applied egg-rr 68.0%

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

      1. associate-*l/ 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. frac-2neg 68.1%

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

      4. fma-neg 68.2%

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

      5. distribute-rgt-neg-in 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. distribute-neg-frac 68.2%

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

      2. fma-def 68.1%

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

      3. +-commutative 68.1%

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

      4. distribute-neg-in 68.1%

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

      5. distribute-rgt-neg-out 68.1%

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

      6. remove-double-neg 68.1%

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

      7. *-commutative 68.1%

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

      8. distribute-rgt-neg-in 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in d around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. associate-/l* 90.1%

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

   10. Simplified 90.1%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - \frac{b}{\frac{d}{c}}}{-\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 8: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{\frac{d}{c}}\\ t_2 := -\mathsf{hypot}\left(c, d\right)\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1 - a}{t_2}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t_1}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ b (/ d c)))
        (t_2 (- (hypot c d))))
   (if (<= d -3.4e+143)
     (/ (- t_1 a) t_2)
     (if (<= d -1.45e-69)
       t_0
       (if (<= d 1.1e-104)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 7.5e+28) t_0 (/ (- a t_1) t_2)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = b / (d / c);
	double t_2 = -hypot(c, d);
	double tmp;
	if (d <= -3.4e+143) {
		tmp = (t_1 - a) / t_2;
	} else if (d <= -1.45e-69) {
		tmp = t_0;
	} else if (d <= 1.1e-104) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - t_1) / t_2;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = b / (d / c);
	double t_2 = -Math.hypot(c, d);
	double tmp;
	if (d <= -3.4e+143) {
		tmp = (t_1 - a) / t_2;
	} else if (d <= -1.45e-69) {
		tmp = t_0;
	} else if (d <= 1.1e-104) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 7.5e+28) {
		tmp = t_0;
	} else {
		tmp = (a - t_1) / t_2;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = b / (d / c)
	t_2 = -math.hypot(c, d)
	tmp = 0
	if d <= -3.4e+143:
		tmp = (t_1 - a) / t_2
	elif d <= -1.45e-69:
		tmp = t_0
	elif d <= 1.1e-104:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 7.5e+28:
		tmp = t_0
	else:
		tmp = (a - t_1) / t_2
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(b / Float64(d / c))
	t_2 = Float64(-hypot(c, d))
	tmp = 0.0
	if (d <= -3.4e+143)
		tmp = Float64(Float64(t_1 - a) / t_2);
	elseif (d <= -1.45e-69)
		tmp = t_0;
	elseif (d <= 1.1e-104)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = Float64(Float64(a - t_1) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = b / (d / c);
	t_2 = -hypot(c, d);
	tmp = 0.0;
	if (d <= -3.4e+143)
		tmp = (t_1 - a) / t_2;
	elseif (d <= -1.45e-69)
		tmp = t_0;
	elseif (d <= 1.1e-104)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 7.5e+28)
		tmp = t_0;
	else
		tmp = (a - t_1) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])}, If[LessEqual[d, -3.4e+143], N[(N[(t$95$1 - a), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[d, -1.45e-69], t$95$0, If[LessEqual[d, 1.1e-104], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 7.5e+28], t$95$0, N[(N[(a - t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.39999999999999982e143

   1. Initial program 39.9%

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

      1. *-un-lft-identity 39.9%

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

      2. add-sqr-sqrt 39.9%

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

      3. times-frac 40.0%

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

      4. hypot-def 40.0%

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

      5. hypot-def 72.0%

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

   3. Applied egg-rr 72.0%

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

      1. associate-*l/ 72.0%

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

      2. *-un-lft-identity 72.0%

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

      3. frac-2neg 72.0%

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

      4. fma-neg 72.1%

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

      5. distribute-rgt-neg-in 72.1%

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

   5. Applied egg-rr 72.1%

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

      1. distribute-neg-frac 72.1%

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

      2. fma-def 72.0%

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

      3. +-commutative 72.0%

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

      4. distribute-neg-in 72.0%

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

      5. distribute-rgt-neg-out 72.0%

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

      6. remove-double-neg 72.0%

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

      7. *-commutative 72.0%

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

      8. distribute-rgt-neg-in 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in d around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-/l* 92.0%

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

   10. Simplified 92.0%

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

   if -3.39999999999999982e143 < d < -1.4499999999999999e-69 or 1.10000000000000006e-104 < d < 7.4999999999999998e28

   1. Initial program 82.4%

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

   if -1.4499999999999999e-69 < d < 1.10000000000000006e-104

   1. Initial program 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. add-sqr-sqrt 70.9%

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

      3. times-frac 71.0%

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

      4. hypot-def 71.0%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in c around -inf 47.6%

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

      1. +-commutative 47.6%

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

      2. neg-mul-1 47.6%

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

      3. unsub-neg 47.6%

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

      4. associate-/l* 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c around -inf 88.8%

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

   if 7.4999999999999998e28 < d

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.4%

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

      4. hypot-def 46.4%

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

      5. hypot-def 68.0%

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

   3. Applied egg-rr 68.0%

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

      1. associate-*l/ 68.1%

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

      2. *-un-lft-identity 68.1%

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

      3. frac-2neg 68.1%

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

      4. fma-neg 68.2%

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

      5. distribute-rgt-neg-in 68.2%

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

   5. Applied egg-rr 68.2%

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

      1. distribute-neg-frac 68.2%

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

      2. fma-def 68.1%

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

      3. +-commutative 68.1%

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

      4. distribute-neg-in 68.1%

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

      5. distribute-rgt-neg-out 68.1%

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

      6. remove-double-neg 68.1%

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

      7. *-commutative 68.1%

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

      8. distribute-rgt-neg-in 68.1%

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

   7. Simplified 68.1%

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

   8. Taylor expanded in d around inf 88.3%

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

      1. mul-1-neg 88.3%

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

      2. unsub-neg 88.3%

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

      3. associate-/l* 90.1%

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

   10. Simplified 90.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.1 \cdot 10^{-104}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - \frac{b}{\frac{d}{c}}}{-\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 9: 79.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -6.2e+143)
     (/ (- a) d)
     (if (<= d -1.65e-69)
       t_0
       (if (<= d 1.18e-105)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 1.8e+28) t_0 (- (/ b (/ (pow d 2.0) c)) (/ a d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+143) {
		tmp = -a / d;
	} else if (d <= -1.65e-69) {
		tmp = t_0;
	} else if (d <= 1.18e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.8e+28) {
		tmp = t_0;
	} else {
		tmp = (b / (pow(d, 2.0) / c)) - (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) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-6.2d+143)) then
        tmp = -a / d
    else if (d <= (-1.65d-69)) then
        tmp = t_0
    else if (d <= 1.18d-105) then
        tmp = ((-1.0d0) / c) * ((a / (c / d)) - b)
    else if (d <= 1.8d+28) then
        tmp = t_0
    else
        tmp = (b / ((d ** 2.0d0) / c)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -6.2e+143) {
		tmp = -a / d;
	} else if (d <= -1.65e-69) {
		tmp = t_0;
	} else if (d <= 1.18e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.8e+28) {
		tmp = t_0;
	} else {
		tmp = (b / (Math.pow(d, 2.0) / c)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -6.2e+143:
		tmp = -a / d
	elif d <= -1.65e-69:
		tmp = t_0
	elif d <= 1.18e-105:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 1.8e+28:
		tmp = t_0
	else:
		tmp = (b / (math.pow(d, 2.0) / c)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -6.2e+143)
		tmp = Float64(Float64(-a) / d);
	elseif (d <= -1.65e-69)
		tmp = t_0;
	elseif (d <= 1.18e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 1.8e+28)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -6.2e+143)
		tmp = -a / d;
	elseif (d <= -1.65e-69)
		tmp = t_0;
	elseif (d <= 1.18e-105)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 1.8e+28)
		tmp = t_0;
	else
		tmp = (b / ((d ^ 2.0) / c)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.2e+143], N[((-a) / d), $MachinePrecision], If[LessEqual[d, -1.65e-69], t$95$0, If[LessEqual[d, 1.18e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.8e+28], t$95$0, N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.1999999999999998e143

   1. Initial program 39.9%

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

   2. Taylor expanded in c around 0 79.1%

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

      1. associate-*r/ 79.1%

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

      2. neg-mul-1 79.1%

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

   4. Simplified 79.1%

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

   if -6.1999999999999998e143 < d < -1.65e-69 or 1.1799999999999999e-105 < d < 1.8e28

   1. Initial program 82.4%

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

   if -1.65e-69 < d < 1.1799999999999999e-105

   1. Initial program 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. add-sqr-sqrt 70.9%

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

      3. times-frac 71.0%

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

      4. hypot-def 71.0%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in c around -inf 47.6%

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

      1. +-commutative 47.6%

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

      2. neg-mul-1 47.6%

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

      3. unsub-neg 47.6%

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

      4. associate-/l* 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c around -inf 88.8%

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

   if 1.8e28 < d

   1. Initial program 46.5%

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

   2. Taylor expanded in c around 0 83.4%

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

      1. +-commutative 83.4%

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

      2. mul-1-neg 83.4%

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

      3. unsub-neg 83.4%

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

      4. associate-/l* 86.8%

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

   4. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.65 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \end{array} \]

Alternative 10: 79.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -1.05 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.72 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -1.05e+144)
     t_0
     (if (<= d -1.45e-69)
       t_1
       (if (<= d 1.45e-105)
         (* (/ -1.0 c) (- (/ a (/ c d)) b))
         (if (<= d 1.72e+130) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.05e+144) {
		tmp = t_0;
	} else if (d <= -1.45e-69) {
		tmp = t_1;
	} else if (d <= 1.45e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.72e+130) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = -a / d
    t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (d <= (-1.05d+144)) then
        tmp = t_0
    else if (d <= (-1.45d-69)) then
        tmp = t_1
    else if (d <= 1.45d-105) then
        tmp = ((-1.0d0) / c) * ((a / (c / d)) - b)
    else if (d <= 1.72d+130) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -1.05e+144) {
		tmp = t_0;
	} else if (d <= -1.45e-69) {
		tmp = t_1;
	} else if (d <= 1.45e-105) {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	} else if (d <= 1.72e+130) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if d <= -1.05e+144:
		tmp = t_0
	elif d <= -1.45e-69:
		tmp = t_1
	elif d <= 1.45e-105:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	elif d <= 1.72e+130:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -1.05e+144)
		tmp = t_0;
	elseif (d <= -1.45e-69)
		tmp = t_1;
	elseif (d <= 1.45e-105)
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	elseif (d <= 1.72e+130)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (d <= -1.05e+144)
		tmp = t_0;
	elseif (d <= -1.45e-69)
		tmp = t_1;
	elseif (d <= 1.45e-105)
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	elseif (d <= 1.72e+130)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.05e+144], t$95$0, If[LessEqual[d, -1.45e-69], t$95$1, If[LessEqual[d, 1.45e-105], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.72e+130], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.04999999999999998e144 or 1.72000000000000008e130 < d

   1. Initial program 36.1%

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

   2. Taylor expanded in c around 0 82.5%

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

      1. associate-*r/ 82.5%

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

      2. neg-mul-1 82.5%

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

   4. Simplified 82.5%

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

   if -1.04999999999999998e144 < d < -1.4499999999999999e-69 or 1.45000000000000002e-105 < d < 1.72000000000000008e130

   1. Initial program 80.8%

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

   if -1.4499999999999999e-69 < d < 1.45000000000000002e-105

   1. Initial program 70.9%

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

      1. *-un-lft-identity 70.9%

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

      2. add-sqr-sqrt 70.9%

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

      3. times-frac 71.0%

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

      4. hypot-def 71.0%

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

      5. hypot-def 81.5%

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

   3. Applied egg-rr 81.5%

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

   4. Taylor expanded in c around -inf 47.6%

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

      1. +-commutative 47.6%

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

      2. neg-mul-1 47.6%

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

      3. unsub-neg 47.6%

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

      4. associate-/l* 48.6%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in c around -inf 88.8%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+144}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-69}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.45 \cdot 10^{-105}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \mathbf{elif}\;d \leq 1.72 \cdot 10^{+130}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 11: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-31} \lor \neg \left(d \leq 3.25 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.4e-31) (not (<= d 3.25e-68)))
   (/ (- a) d)
   (* (/ -1.0 c) (- (/ a (/ c d)) b))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e-31) || !(d <= 3.25e-68)) {
		tmp = -a / d;
	} else {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	}
	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 <= (-1.4d-31)) .or. (.not. (d <= 3.25d-68))) then
        tmp = -a / d
    else
        tmp = ((-1.0d0) / c) * ((a / (c / d)) - b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e-31) || !(d <= 3.25e-68)) {
		tmp = -a / d;
	} else {
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.4e-31) or not (d <= 3.25e-68):
		tmp = -a / d
	else:
		tmp = (-1.0 / c) * ((a / (c / d)) - b)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.4e-31) || !(d <= 3.25e-68))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(-1.0 / c) * Float64(Float64(a / Float64(c / d)) - b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.4e-31) || ~((d <= 3.25e-68)))
		tmp = -a / d;
	else
		tmp = (-1.0 / c) * ((a / (c / d)) - b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.4e-31], N[Not[LessEqual[d, 3.25e-68]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(-1.0 / c), $MachinePrecision] * N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.3999999999999999e-31 or 3.2499999999999999e-68 < d

   1. Initial program 59.3%

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

   2. Taylor expanded in c around 0 68.8%

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

      1. associate-*r/ 68.8%

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

      2. neg-mul-1 68.8%

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

   4. Simplified 68.8%

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

   if -1.3999999999999999e-31 < d < 3.2499999999999999e-68

   1. Initial program 71.0%

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

      1. *-un-lft-identity 71.0%

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

      2. add-sqr-sqrt 71.0%

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

      3. times-frac 71.0%

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

      4. hypot-def 71.1%

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

      5. hypot-def 82.2%

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

   3. Applied egg-rr 82.2%

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

   4. Taylor expanded in c around -inf 45.8%

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

      1. +-commutative 45.8%

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

      2. neg-mul-1 45.8%

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

      3. unsub-neg 45.8%

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

      4. associate-/l* 46.7%

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

   6. Simplified 46.7%

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

   7. Taylor expanded in c around -inf 82.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{-31} \lor \neg \left(d \leq 3.25 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{c} \cdot \left(\frac{a}{\frac{c}{d}} - b\right)\\ \end{array} \]

Alternative 12: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.92 \cdot 10^{-60} \lor \neg \left(d \leq 1.35 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.92e-60) (not (<= d 1.35e-69))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.92e-60) || !(d <= 1.35e-69)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.92d-60)) .or. (.not. (d <= 1.35d-69))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.92e-60) || !(d <= 1.35e-69)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.92e-60) or not (d <= 1.35e-69):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.92e-60) || !(d <= 1.35e-69))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.92e-60) || ~((d <= 1.35e-69)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.92e-60], N[Not[LessEqual[d, 1.35e-69]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.9200000000000001e-60 or 1.3499999999999999e-69 < d

   1. Initial program 58.9%

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

   2. Taylor expanded in c around 0 66.8%

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

      1. associate-*r/ 66.8%

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

      2. neg-mul-1 66.8%

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

   4. Simplified 66.8%

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

   if -1.9200000000000001e-60 < d < 1.3499999999999999e-69

   1. Initial program 72.8%

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

   2. Taylor expanded in c around inf 69.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.92 \cdot 10^{-60} \lor \neg \left(d \leq 1.35 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13: 46.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+129} \lor \neg \left(d \leq 2.8 \cdot 10^{+183}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.8e+129) (not (<= d 2.8e+183))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.8e+129) || !(d <= 2.8e+183)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.8d+129)) .or. (.not. (d <= 2.8d+183))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.8e+129) || !(d <= 2.8e+183)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.8e+129) or not (d <= 2.8e+183):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.8e+129) || !(d <= 2.8e+183))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.8e+129) || ~((d <= 2.8e+183)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.8e+129], N[Not[LessEqual[d, 2.8e+183]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.80000000000000005e129 or 2.80000000000000018e183 < d

   1. Initial program 39.3%

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

      1. *-un-lft-identity 39.3%

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

      2. add-sqr-sqrt 39.3%

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

      3. times-frac 39.3%

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

      4. hypot-def 39.3%

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

      5. hypot-def 68.4%

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

   3. Applied egg-rr 68.4%

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

   4. Taylor expanded in c around 0 60.5%

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

      1. neg-mul-1 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in d around -inf 35.4%

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

   if -5.80000000000000005e129 < d < 2.80000000000000018e183

   1. Initial program 72.9%

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

   2. Taylor expanded in c around inf 48.7%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+129} \lor \neg \left(d \leq 2.8 \cdot 10^{+183}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 14: 9.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-un-lft-identity 64.4%

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

   2. add-sqr-sqrt 64.4%

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

   3. times-frac 64.4%

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

   4. hypot-def 64.4%

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

   5. hypot-def 78.8%

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

3. Applied egg-rr 78.8%

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

4. Taylor expanded in c around 0 34.6%

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

   1. neg-mul-1 34.6%

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

6. Simplified 34.6%

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

7. Taylor expanded in c around -inf 7.2%

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

8. Final simplification 7.2%

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

Alternative 15: 11.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.4%

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

   1. *-un-lft-identity 64.4%

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

   2. add-sqr-sqrt 64.4%

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

   3. times-frac 64.4%

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

   4. hypot-def 64.4%

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

   5. hypot-def 78.8%

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

3. Applied egg-rr 78.8%

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

4. Taylor expanded in c around 0 34.6%

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

   1. neg-mul-1 34.6%

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

6. Simplified 34.6%

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

7. Taylor expanded in d around -inf 11.8%

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

8. Final simplification 11.8%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

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