Complex division, imag part

Percentage Accurate: 62.0% → 87.8%

Time: 13.3s

Alternatives: 12

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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 87.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = fma(t_0, Float64(b / Float64(hypot(c, d) / c)), Float64(Float64(-a) / d));
	end
	return 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[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $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))) < +inf.0

   1. Initial program 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. add-sqr-sqrt 77.1%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 94.7%

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

   3. Applied egg-rr 94.7%

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

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

   1. Initial program 0.0%

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

      1. div-sub 0.0%

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

      2. sub-neg 0.0%

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

      3. *-un-lft-identity 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. times-frac 0.0%

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

      6. fma-def 0.0%

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

      7. hypot-def 0.0%

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

      8. hypot-def 1.3%

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

      9. add-sqr-sqrt 1.3%

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

      10. pow2 1.3%

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

      11. hypot-def 1.3%

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

   3. Applied egg-rr 1.3%

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

      1. associate-/l* 49.6%

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

      2. distribute-neg-frac 49.6%

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

      3. distribute-rgt-neg-out 49.6%

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

   5. Simplified 49.6%

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

   6. Taylor expanded in d around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. neg-mul-1 74.2%

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

   8. Simplified 74.2%

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

4. Final simplification 90.7%

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

Alternative 2: 85.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], 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[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 77.1%

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

      1. *-un-lft-identity 77.1%

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

      2. add-sqr-sqrt 77.1%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 94.7%

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

   3. Applied egg-rr 94.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in c around inf 41.6%

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

      1. +-commutative 41.6%

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

      2. mul-1-neg 41.6%

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

      3. unsub-neg 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-/l* 44.3%

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

   4. Simplified 44.3%

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

      1. div-inv 44.1%

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

      2. fma-neg 44.1%

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

      3. div-inv 44.1%

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

      4. clear-num 44.1%

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

   6. Applied egg-rr 44.1%

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

      1. fma-udef 44.1%

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

      2. unsub-neg 44.1%

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

      3. associate-*r/ 41.3%

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

      4. *-commutative 41.3%

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

   8. Simplified 41.3%

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

      1. *-commutative 41.3%

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

      2. unpow2 41.3%

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

      3. times-frac 49.7%

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

   10. Applied egg-rr 49.7%

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

4. Final simplification 86.0%

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

Alternative 3: 79.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := a \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;t_1 \cdot \left(t_2 - b\right)\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-135}:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.4 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;t_0 \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b - t_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d)))
        (t_1 (/ 1.0 (hypot c d)))
        (t_2 (* a (/ d c))))
   (if (<= c -2.6e+41)
     (* t_1 (- t_2 b))
     (if (<= c -1.25e-135)
       (/ t_0 (+ (* c c) (* d d)))
       (if (<= c 7.4e-247)
         (- (/ b (/ (pow d 2.0) c)) (/ a d))
         (if (<= c 3.1e+86)
           (* t_0 (/ 1.0 (pow (hypot c d) 2.0)))
           (* t_1 (- b t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = 1.0 / hypot(c, d);
	double t_2 = a * (d / c);
	double tmp;
	if (c <= -2.6e+41) {
		tmp = t_1 * (t_2 - b);
	} else if (c <= -1.25e-135) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 7.4e-247) {
		tmp = (b / (pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 3.1e+86) {
		tmp = t_0 * (1.0 / pow(hypot(c, d), 2.0));
	} else {
		tmp = t_1 * (b - t_2);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = 1.0 / Math.hypot(c, d);
	double t_2 = a * (d / c);
	double tmp;
	if (c <= -2.6e+41) {
		tmp = t_1 * (t_2 - b);
	} else if (c <= -1.25e-135) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 7.4e-247) {
		tmp = (b / (Math.pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 3.1e+86) {
		tmp = t_0 * (1.0 / Math.pow(Math.hypot(c, d), 2.0));
	} else {
		tmp = t_1 * (b - t_2);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	t_1 = 1.0 / math.hypot(c, d)
	t_2 = a * (d / c)
	tmp = 0
	if c <= -2.6e+41:
		tmp = t_1 * (t_2 - b)
	elif c <= -1.25e-135:
		tmp = t_0 / ((c * c) + (d * d))
	elif c <= 7.4e-247:
		tmp = (b / (math.pow(d, 2.0) / c)) - (a / d)
	elif c <= 3.1e+86:
		tmp = t_0 * (1.0 / math.pow(math.hypot(c, d), 2.0))
	else:
		tmp = t_1 * (b - t_2)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(1.0 / hypot(c, d))
	t_2 = Float64(a * Float64(d / c))
	tmp = 0.0
	if (c <= -2.6e+41)
		tmp = Float64(t_1 * Float64(t_2 - b));
	elseif (c <= -1.25e-135)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 7.4e-247)
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	elseif (c <= 3.1e+86)
		tmp = Float64(t_0 * Float64(1.0 / (hypot(c, d) ^ 2.0)));
	else
		tmp = Float64(t_1 * Float64(b - t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	t_1 = 1.0 / hypot(c, d);
	t_2 = a * (d / c);
	tmp = 0.0;
	if (c <= -2.6e+41)
		tmp = t_1 * (t_2 - b);
	elseif (c <= -1.25e-135)
		tmp = t_0 / ((c * c) + (d * d));
	elseif (c <= 7.4e-247)
		tmp = (b / ((d ^ 2.0) / c)) - (a / d);
	elseif (c <= 3.1e+86)
		tmp = t_0 * (1.0 / (hypot(c, d) ^ 2.0));
	else
		tmp = t_1 * (b - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.6e+41], N[(t$95$1 * N[(t$95$2 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.25e-135], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.4e-247], N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.1e+86], N[(t$95$0 * N[(1.0 / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(b - t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.6000000000000001e41

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 67.4%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in c around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. neg-mul-1 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 82.5%

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

   6. Simplified 82.5%

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

   if -2.6000000000000001e41 < c < -1.25000000000000005e-135

   1. Initial program 90.4%

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

   if -1.25000000000000005e-135 < c < 7.40000000000000021e-247

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 87.4%

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

   4. Simplified 87.4%

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

   if 7.40000000000000021e-247 < c < 3.1000000000000002e86

   1. Initial program 82.1%

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

      1. clear-num 81.3%

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

      2. associate-/r/ 82.2%

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

      3. add-sqr-sqrt 82.1%

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

      4. pow2 82.1%

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

      5. hypot-def 82.1%

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

   3. Applied egg-rr 82.1%

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

   if 3.1000000000000002e86 < c

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. hypot-def 61.1%

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

   3. Applied egg-rr 61.1%

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

   4. Taylor expanded in c around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. associate-*r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b - a \cdot \frac{d}{c}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a \cdot \frac{d}{c} - b\right)\\ \mathbf{elif}\;c \leq -1.25 \cdot 10^{-135}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.4 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{c}\right)\\ \end{array} \]

Alternative 4: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := a \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;t_1 \cdot \left(t_2 - b\right)\\ \mathbf{elif}\;c \leq -7.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b - t_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d)))
        (t_1 (/ 1.0 (hypot c d)))
        (t_2 (* a (/ d c))))
   (if (<= c -1.6e+41)
     (* t_1 (- t_2 b))
     (if (<= c -7.8e-138)
       (/ t_0 (+ (* c c) (* d d)))
       (if (<= c 1.8e-252)
         (- (/ b (/ (pow d 2.0) c)) (/ a d))
         (if (<= c 6.5e+86) (/ t_0 (fma c c (* d d))) (* t_1 (- b t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = 1.0 / hypot(c, d);
	double t_2 = a * (d / c);
	double tmp;
	if (c <= -1.6e+41) {
		tmp = t_1 * (t_2 - b);
	} else if (c <= -7.8e-138) {
		tmp = t_0 / ((c * c) + (d * d));
	} else if (c <= 1.8e-252) {
		tmp = (b / (pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 6.5e+86) {
		tmp = t_0 / fma(c, c, (d * d));
	} else {
		tmp = t_1 * (b - t_2);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(1.0 / hypot(c, d))
	t_2 = Float64(a * Float64(d / c))
	tmp = 0.0
	if (c <= -1.6e+41)
		tmp = Float64(t_1 * Float64(t_2 - b));
	elseif (c <= -7.8e-138)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.8e-252)
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	elseif (c <= 6.5e+86)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(t_1 * Float64(b - t_2));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.6e+41], N[(t$95$1 * N[(t$95$2 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -7.8e-138], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.8e-252], N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.5e+86], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(b - t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.60000000000000005e41

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 67.4%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in c around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. neg-mul-1 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 82.5%

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

   6. Simplified 82.5%

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

   if -1.60000000000000005e41 < c < -7.7999999999999999e-138

   1. Initial program 90.4%

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

   if -7.7999999999999999e-138 < c < 1.80000000000000011e-252

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 87.4%

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

   4. Simplified 87.4%

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

   if 1.80000000000000011e-252 < c < 6.49999999999999996e86

   1. Initial program 82.1%

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

      1. fma-def 82.1%

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

   3. Simplified 82.1%

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

   if 6.49999999999999996e86 < c

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. hypot-def 61.1%

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

   3. Applied egg-rr 61.1%

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

   4. Taylor expanded in c around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. associate-*r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b - a \cdot \frac{d}{c}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a \cdot \frac{d}{c} - b\right)\\ \mathbf{elif}\;c \leq -7.8 \cdot 10^{-138}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{c}\right)\\ \end{array} \]

Alternative 5: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -7.5 \cdot 10^{+125}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -7.5e+125)
     (- (* b (/ 1.0 c)) (* (/ d c) (/ a c)))
     (if (<= c -5e-139)
       t_0
       (if (<= c 7.5e-247)
         (- (/ b (/ (pow d 2.0) c)) (/ a d))
         (if (<= c 5.4e+86)
           t_0
           (* (/ 1.0 (hypot c d)) (- b (* a (/ d c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7.5e+125) {
		tmp = (b * (1.0 / c)) - ((d / c) * (a / c));
	} else if (c <= -5e-139) {
		tmp = t_0;
	} else if (c <= 7.5e-247) {
		tmp = (b / (pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 5.4e+86) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (a * (d / c)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7.5e+125) {
		tmp = (b * (1.0 / c)) - ((d / c) * (a / c));
	} else if (c <= -5e-139) {
		tmp = t_0;
	} else if (c <= 7.5e-247) {
		tmp = (b / (Math.pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 5.4e+86) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b - (a * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -7.5e+125:
		tmp = (b * (1.0 / c)) - ((d / c) * (a / c))
	elif c <= -5e-139:
		tmp = t_0
	elif c <= 7.5e-247:
		tmp = (b / (math.pow(d, 2.0) / c)) - (a / d)
	elif c <= 5.4e+86:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b - (a * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -7.5e+125)
		tmp = Float64(Float64(b * Float64(1.0 / c)) - Float64(Float64(d / c) * Float64(a / c)));
	elseif (c <= -5e-139)
		tmp = t_0;
	elseif (c <= 7.5e-247)
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	elseif (c <= 5.4e+86)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(a * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -7.5e+125)
		tmp = (b * (1.0 / c)) - ((d / c) * (a / c));
	elseif (c <= -5e-139)
		tmp = t_0;
	elseif (c <= 7.5e-247)
		tmp = (b / ((d ^ 2.0) / c)) - (a / d);
	elseif (c <= 5.4e+86)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b - (a * (d / c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.5e+125], N[(N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5e-139], t$95$0, If[LessEqual[c, 7.5e-247], N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.4e+86], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.5000000000000006e125

   1. Initial program 31.2%

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

   2. Taylor expanded in c around inf 73.7%

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

      1. +-commutative 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unsub-neg 73.7%

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

      4. *-commutative 73.7%

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

      5. associate-/l* 71.9%

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

   4. Simplified 71.9%

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

      1. div-inv 71.7%

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

      2. fma-neg 71.7%

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

      3. div-inv 71.7%

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

      4. clear-num 71.7%

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

   6. Applied egg-rr 71.7%

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

      1. fma-udef 71.7%

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

      2. unsub-neg 71.7%

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

      3. associate-*r/ 73.5%

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

      4. *-commutative 73.5%

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

   8. Simplified 73.5%

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

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. times-frac 84.9%

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

   10. Applied egg-rr 84.9%

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

   if -7.5000000000000006e125 < c < -5.00000000000000034e-139 or 7.5e-247 < c < 5.40000000000000036e86

   1. Initial program 83.4%

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

   if -5.00000000000000034e-139 < c < 7.5e-247

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 87.4%

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

   4. Simplified 87.4%

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

   if 5.40000000000000036e86 < c

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. hypot-def 61.1%

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

   3. Applied egg-rr 61.1%

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

   4. Taylor expanded in c around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. associate-*r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b - a \cdot \frac{d}{c}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+125}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -5 \cdot 10^{-139}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{c}\right)\\ \end{array} \]

Alternative 6: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_2 := a \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -3 \cdot 10^{+41}:\\ \;\;\;\;t_1 \cdot \left(t_2 - b\right)\\ \mathbf{elif}\;c \leq -1.2 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(b - t_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (/ 1.0 (hypot c d)))
        (t_2 (* a (/ d c))))
   (if (<= c -3e+41)
     (* t_1 (- t_2 b))
     (if (<= c -1.2e-135)
       t_0
       (if (<= c 7.5e-247)
         (- (/ b (/ (pow d 2.0) c)) (/ a d))
         (if (<= c 7e+86) t_0 (* t_1 (- b t_2))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = 1.0 / hypot(c, d);
	double t_2 = a * (d / c);
	double tmp;
	if (c <= -3e+41) {
		tmp = t_1 * (t_2 - b);
	} else if (c <= -1.2e-135) {
		tmp = t_0;
	} else if (c <= 7.5e-247) {
		tmp = (b / (pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 7e+86) {
		tmp = t_0;
	} else {
		tmp = t_1 * (b - t_2);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = 1.0 / Math.hypot(c, d);
	double t_2 = a * (d / c);
	double tmp;
	if (c <= -3e+41) {
		tmp = t_1 * (t_2 - b);
	} else if (c <= -1.2e-135) {
		tmp = t_0;
	} else if (c <= 7.5e-247) {
		tmp = (b / (Math.pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 7e+86) {
		tmp = t_0;
	} else {
		tmp = t_1 * (b - t_2);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = 1.0 / math.hypot(c, d)
	t_2 = a * (d / c)
	tmp = 0
	if c <= -3e+41:
		tmp = t_1 * (t_2 - b)
	elif c <= -1.2e-135:
		tmp = t_0
	elif c <= 7.5e-247:
		tmp = (b / (math.pow(d, 2.0) / c)) - (a / d)
	elif c <= 7e+86:
		tmp = t_0
	else:
		tmp = t_1 * (b - t_2)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(1.0 / hypot(c, d))
	t_2 = Float64(a * Float64(d / c))
	tmp = 0.0
	if (c <= -3e+41)
		tmp = Float64(t_1 * Float64(t_2 - b));
	elseif (c <= -1.2e-135)
		tmp = t_0;
	elseif (c <= 7.5e-247)
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	elseif (c <= 7e+86)
		tmp = t_0;
	else
		tmp = Float64(t_1 * Float64(b - t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = 1.0 / hypot(c, d);
	t_2 = a * (d / c);
	tmp = 0.0;
	if (c <= -3e+41)
		tmp = t_1 * (t_2 - b);
	elseif (c <= -1.2e-135)
		tmp = t_0;
	elseif (c <= 7.5e-247)
		tmp = (b / ((d ^ 2.0) / c)) - (a / d);
	elseif (c <= 7e+86)
		tmp = t_0;
	else
		tmp = t_1 * (b - t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3e+41], N[(t$95$1 * N[(t$95$2 - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.2e-135], t$95$0, If[LessEqual[c, 7.5e-247], N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7e+86], t$95$0, N[(t$95$1 * N[(b - t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.9999999999999998e41

   1. Initial program 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. add-sqr-sqrt 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. hypot-def 67.4%

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

   3. Applied egg-rr 67.4%

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

   4. Taylor expanded in c around -inf 80.7%

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

      1. +-commutative 80.7%

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

      2. neg-mul-1 80.7%

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

      3. unsub-neg 80.7%

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

      4. associate-*r/ 82.5%

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

   6. Simplified 82.5%

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

   if -2.9999999999999998e41 < c < -1.1999999999999999e-135 or 7.5e-247 < c < 7.00000000000000038e86

   1. Initial program 84.7%

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

   if -1.1999999999999999e-135 < c < 7.5e-247

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 87.4%

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

   4. Simplified 87.4%

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

   if 7.00000000000000038e86 < c

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.5%

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

      4. hypot-def 35.5%

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

      5. hypot-def 61.1%

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

   3. Applied egg-rr 61.1%

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

   4. Taylor expanded in c around inf 78.2%

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

      1. mul-1-neg 78.2%

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

      2. unsub-neg 78.2%

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

      3. associate-*r/ 84.1%

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

   6. Simplified 84.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b - a \cdot \frac{d}{c}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(a \cdot \frac{d}{c} - b\right)\\ \mathbf{elif}\;c \leq -1.2 \cdot 10^{-135}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-247}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - a \cdot \frac{d}{c}\right)\\ \end{array} \]

Alternative 7: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -7 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (* b (/ 1.0 c)) (* (/ d c) (/ a c)))))
   (if (<= c -7e+125)
     t_1
     (if (<= c -1.9e-137)
       t_0
       (if (<= c 3.1e-248)
         (- (/ b (/ (pow d 2.0) c)) (/ a d))
         (if (<= c 4.8e+86) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -7e+125) {
		tmp = t_1;
	} else if (c <= -1.9e-137) {
		tmp = t_0;
	} else if (c <= 3.1e-248) {
		tmp = (b / (pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 4.8e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b * (1.0d0 / c)) - ((d / c) * (a / c))
    if (c <= (-7d+125)) then
        tmp = t_1
    else if (c <= (-1.9d-137)) then
        tmp = t_0
    else if (c <= 3.1d-248) then
        tmp = (b / ((d ** 2.0d0) / c)) - (a / d)
    else if (c <= 4.8d+86) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -7e+125) {
		tmp = t_1;
	} else if (c <= -1.9e-137) {
		tmp = t_0;
	} else if (c <= 3.1e-248) {
		tmp = (b / (Math.pow(d, 2.0) / c)) - (a / d);
	} else if (c <= 4.8e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c))
	tmp = 0
	if c <= -7e+125:
		tmp = t_1
	elif c <= -1.9e-137:
		tmp = t_0
	elif c <= 3.1e-248:
		tmp = (b / (math.pow(d, 2.0) / c)) - (a / d)
	elif c <= 4.8e+86:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b * Float64(1.0 / c)) - Float64(Float64(d / c) * Float64(a / c)))
	tmp = 0.0
	if (c <= -7e+125)
		tmp = t_1;
	elseif (c <= -1.9e-137)
		tmp = t_0;
	elseif (c <= 3.1e-248)
		tmp = Float64(Float64(b / Float64((d ^ 2.0) / c)) - Float64(a / d));
	elseif (c <= 4.8e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	tmp = 0.0;
	if (c <= -7e+125)
		tmp = t_1;
	elseif (c <= -1.9e-137)
		tmp = t_0;
	elseif (c <= 3.1e-248)
		tmp = (b / ((d ^ 2.0) / c)) - (a / d);
	elseif (c <= 4.8e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7e+125], t$95$1, If[LessEqual[c, -1.9e-137], t$95$0, If[LessEqual[c, 3.1e-248], N[(N[(b / N[(N[Power[d, 2.0], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.00000000000000023e125 or 4.8000000000000001e86 < c

   1. Initial program 33.6%

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

   2. Taylor expanded in c around inf 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. *-commutative 74.2%

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

      5. associate-/l* 73.7%

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

   4. Simplified 73.7%

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

      1. div-inv 73.5%

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

      2. fma-neg 73.5%

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

      3. div-inv 73.5%

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

      4. clear-num 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. fma-udef 73.5%

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

      2. unsub-neg 73.5%

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

      3. associate-*r/ 73.9%

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

      4. *-commutative 73.9%

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

   8. Simplified 73.9%

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

      1. *-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. times-frac 84.3%

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

   10. Applied egg-rr 84.3%

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

   if -7.00000000000000023e125 < c < -1.89999999999999999e-137 or 3.1000000000000002e-248 < c < 4.8000000000000001e86

   1. Initial program 83.4%

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

   if -1.89999999999999999e-137 < c < 3.1000000000000002e-248

   1. Initial program 67.3%

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

   2. Taylor expanded in c around 0 89.5%

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

      1. +-commutative 89.5%

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

      2. mul-1-neg 89.5%

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

      3. unsub-neg 89.5%

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

      4. associate-/l* 87.4%

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

   4. Simplified 87.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+125}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -1.9 \cdot 10^{-137}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{-248}:\\ \;\;\;\;\frac{b}{\frac{{d}^{2}}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 8: 78.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -7.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (* b (/ 1.0 c)) (* (/ d c) (/ a c)))))
   (if (<= c -1.3e+127)
     t_1
     (if (<= c -3e-139)
       t_0
       (if (<= c -7.4e-304) (/ (- a) d) (if (<= c 7e+86) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -1.3e+127) {
		tmp = t_1;
	} else if (c <= -3e-139) {
		tmp = t_0;
	} else if (c <= -7.4e-304) {
		tmp = -a / d;
	} else if (c <= 7e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b * (1.0d0 / c)) - ((d / c) * (a / c))
    if (c <= (-1.3d+127)) then
        tmp = t_1
    else if (c <= (-3d-139)) then
        tmp = t_0
    else if (c <= (-7.4d-304)) then
        tmp = -a / d
    else if (c <= 7d+86) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -1.3e+127) {
		tmp = t_1;
	} else if (c <= -3e-139) {
		tmp = t_0;
	} else if (c <= -7.4e-304) {
		tmp = -a / d;
	} else if (c <= 7e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c))
	tmp = 0
	if c <= -1.3e+127:
		tmp = t_1
	elif c <= -3e-139:
		tmp = t_0
	elif c <= -7.4e-304:
		tmp = -a / d
	elif c <= 7e+86:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b * Float64(1.0 / c)) - Float64(Float64(d / c) * Float64(a / c)))
	tmp = 0.0
	if (c <= -1.3e+127)
		tmp = t_1;
	elseif (c <= -3e-139)
		tmp = t_0;
	elseif (c <= -7.4e-304)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 7e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	tmp = 0.0;
	if (c <= -1.3e+127)
		tmp = t_1;
	elseif (c <= -3e-139)
		tmp = t_0;
	elseif (c <= -7.4e-304)
		tmp = -a / d;
	elseif (c <= 7e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.3e+127], t$95$1, If[LessEqual[c, -3e-139], t$95$0, If[LessEqual[c, -7.4e-304], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 7e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.3000000000000001e127 or 7.00000000000000038e86 < c

   1. Initial program 33.6%

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

   2. Taylor expanded in c around inf 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. *-commutative 74.2%

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

      5. associate-/l* 73.7%

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

   4. Simplified 73.7%

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

      1. div-inv 73.5%

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

      2. fma-neg 73.5%

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

      3. div-inv 73.5%

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

      4. clear-num 73.5%

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

   6. Applied egg-rr 73.5%

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

      1. fma-udef 73.5%

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

      2. unsub-neg 73.5%

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

      3. associate-*r/ 73.9%

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

      4. *-commutative 73.9%

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

   8. Simplified 73.9%

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

      1. *-commutative 73.9%

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

      2. unpow2 73.9%

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

      3. times-frac 84.3%

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

   10. Applied egg-rr 84.3%

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

   if -1.3000000000000001e127 < c < -2.9999999999999999e-139 or -7.4000000000000006e-304 < c < 7.00000000000000038e86

   1. Initial program 83.9%

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

   if -2.9999999999999999e-139 < c < -7.4000000000000006e-304

   1. Initial program 60.8%

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

   2. Taylor expanded in c around 0 86.4%

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

      1. associate-*r/ 86.4%

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

      2. neg-mul-1 86.4%

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

   4. Simplified 86.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+127}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -3 \cdot 10^{-139}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq -7.4 \cdot 10^{-304}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 7 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 9: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{c \cdot c + d \cdot d}\\ t_1 := b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{if}\;c \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.45 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (+ (* c c) (* d d))))
        (t_1 (- (* b (/ 1.0 c)) (* (/ d c) (/ a c)))))
   (if (<= c -1.2e+37)
     t_1
     (if (<= c -1.45e-136)
       t_0
       (if (<= c 6.3e-55) (/ (- a) d) (if (<= c 1.9e+86) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -1.2e+37) {
		tmp = t_1;
	} else if (c <= -1.45e-136) {
		tmp = t_0;
	} else if (c <= 6.3e-55) {
		tmp = -a / d;
	} else if (c <= 1.9e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * c) / ((c * c) + (d * d))
    t_1 = (b * (1.0d0 / c)) - ((d / c) * (a / c))
    if (c <= (-1.2d+37)) then
        tmp = t_1
    else if (c <= (-1.45d-136)) then
        tmp = t_0
    else if (c <= 6.3d-55) then
        tmp = -a / d
    else if (c <= 1.9d+86) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	double tmp;
	if (c <= -1.2e+37) {
		tmp = t_1;
	} else if (c <= -1.45e-136) {
		tmp = t_0;
	} else if (c <= 6.3e-55) {
		tmp = -a / d;
	} else if (c <= 1.9e+86) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) / ((c * c) + (d * d))
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c))
	tmp = 0
	if c <= -1.2e+37:
		tmp = t_1
	elif c <= -1.45e-136:
		tmp = t_0
	elif c <= 6.3e-55:
		tmp = -a / d
	elif c <= 1.9e+86:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b * Float64(1.0 / c)) - Float64(Float64(d / c) * Float64(a / c)))
	tmp = 0.0
	if (c <= -1.2e+37)
		tmp = t_1;
	elseif (c <= -1.45e-136)
		tmp = t_0;
	elseif (c <= 6.3e-55)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 1.9e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) / ((c * c) + (d * d));
	t_1 = (b * (1.0 / c)) - ((d / c) * (a / c));
	tmp = 0.0;
	if (c <= -1.2e+37)
		tmp = t_1;
	elseif (c <= -1.45e-136)
		tmp = t_0;
	elseif (c <= 6.3e-55)
		tmp = -a / d;
	elseif (c <= 1.9e+86)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.2e+37], t$95$1, If[LessEqual[c, -1.45e-136], t$95$0, If[LessEqual[c, 6.3e-55], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 1.9e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.2e37 or 1.89999999999999989e86 < c

   1. Initial program 39.2%

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

   2. Taylor expanded in c around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-/l* 73.9%

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

   4. Simplified 73.9%

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

      1. div-inv 73.6%

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

      2. fma-neg 73.6%

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

      3. div-inv 73.7%

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

      4. clear-num 73.6%

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

   6. Applied egg-rr 73.6%

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

      1. fma-udef 73.6%

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

      2. unsub-neg 73.6%

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

      3. associate-*r/ 74.0%

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

      4. *-commutative 74.0%

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

   8. Simplified 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. times-frac 82.9%

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

   10. Applied egg-rr 82.9%

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

   if -1.2e37 < c < -1.44999999999999997e-136 or 6.2999999999999997e-55 < c < 1.89999999999999989e86

   1. Initial program 87.7%

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

   2. Taylor expanded in b around inf 64.9%

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

      1. *-commutative 64.9%

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

   4. Simplified 64.9%

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

   if -1.44999999999999997e-136 < c < 6.2999999999999997e-55

   1. Initial program 72.6%

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

   2. Taylor expanded in c around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. neg-mul-1 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+37}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -1.45 \cdot 10^{-136}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.3 \cdot 10^{-55}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{1}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 10: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -1.26 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (+ (* c c) (* d d)))))
   (if (<= c -3e+41)
     (/ b c)
     (if (<= c -1.26e-135)
       t_0
       (if (<= c 1.5e-56) (/ (- a) d) (if (<= c 5.4e+86) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double tmp;
	if (c <= -3e+41) {
		tmp = b / c;
	} else if (c <= -1.26e-135) {
		tmp = t_0;
	} else if (c <= 1.5e-56) {
		tmp = -a / d;
	} else if (c <= 5.4e+86) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (b * c) / ((c * c) + (d * d))
    if (c <= (-3d+41)) then
        tmp = b / c
    else if (c <= (-1.26d-135)) then
        tmp = t_0
    else if (c <= 1.5d-56) then
        tmp = -a / d
    else if (c <= 5.4d+86) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double tmp;
	if (c <= -3e+41) {
		tmp = b / c;
	} else if (c <= -1.26e-135) {
		tmp = t_0;
	} else if (c <= 1.5e-56) {
		tmp = -a / d;
	} else if (c <= 5.4e+86) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) / ((c * c) + (d * d))
	tmp = 0
	if c <= -3e+41:
		tmp = b / c
	elif c <= -1.26e-135:
		tmp = t_0
	elif c <= 1.5e-56:
		tmp = -a / d
	elif c <= 5.4e+86:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -3e+41)
		tmp = Float64(b / c);
	elseif (c <= -1.26e-135)
		tmp = t_0;
	elseif (c <= 1.5e-56)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 5.4e+86)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -3e+41)
		tmp = b / c;
	elseif (c <= -1.26e-135)
		tmp = t_0;
	elseif (c <= 1.5e-56)
		tmp = -a / d;
	elseif (c <= 5.4e+86)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3e+41], N[(b / c), $MachinePrecision], If[LessEqual[c, -1.26e-135], t$95$0, If[LessEqual[c, 1.5e-56], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 5.4e+86], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.9999999999999998e41 or 5.40000000000000036e86 < c

   1. Initial program 39.2%

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

   2. Taylor expanded in c around inf 72.2%

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

   if -2.9999999999999998e41 < c < -1.2600000000000001e-135 or 1.49999999999999995e-56 < c < 5.40000000000000036e86

   1. Initial program 87.7%

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

   2. Taylor expanded in b around inf 64.9%

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

      1. *-commutative 64.9%

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

   4. Simplified 64.9%

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

   if -1.2600000000000001e-135 < c < 1.49999999999999995e-56

   1. Initial program 72.6%

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

   2. Taylor expanded in c around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. neg-mul-1 76.5%

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

   4. Simplified 76.5%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{+41}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -1.26 \cdot 10^{-135}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 11: 63.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-17} \lor \neg \left(c \leq 1.22 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.5e-17) (not (<= c 1.22e-9))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e-17) || !(c <= 1.22e-9)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.5d-17)) .or. (.not. (c <= 1.22d-9))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e-17) || !(c <= 1.22e-9)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.5e-17) or not (c <= 1.22e-9):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.5e-17) || !(c <= 1.22e-9))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.5e-17) || ~((c <= 1.22e-9)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.5e-17], N[Not[LessEqual[c, 1.22e-9]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.49999999999999978e-17 or 1.2199999999999999e-9 < c

   1. Initial program 49.1%

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

   2. Taylor expanded in c around inf 67.5%

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

   if -4.49999999999999978e-17 < c < 1.2199999999999999e-9

   1. Initial program 78.0%

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

   2. Taylor expanded in c around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-17} \lor \neg \left(c \leq 1.22 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 12: 42.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

2. Taylor expanded in c around inf 46.2%

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

3. Final simplification 46.2%

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

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 2023314 
(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))))