Complex division, imag part

Percentage Accurate: 61.8% → 85.5%

Time: 12.6s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.1× speedup

Math

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

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 c) (/ (* d (/ a c)) 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 / c) - ((d * (a / c)) / 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 / c) - ((d * (a / c)) / 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 / c) - ((d * (a / c)) / 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 / c) - Float64(Float64(d * Float64(a / c)) / 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 / c) - ((d * (a / c)) / 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 / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $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 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. add-sqr-sqrt 76.8%

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

      3. times-frac 76.8%

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

      4. hypot-def 76.8%

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

      5. hypot-def 93.3%

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

   4. Applied egg-rr 93.3%

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

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. hypot-def 0.0%

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

      5. hypot-def 2.7%

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

   4. Applied egg-rr 2.7%

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

   5. Taylor expanded in c around inf 39.2%

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

      1. +-commutative 39.2%

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

      2. mul-1-neg 39.2%

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

      3. unsub-neg 39.2%

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

      4. *-lft-identity 39.2%

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

      5. times-frac 46.5%

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

      6. /-rgt-identity 46.5%

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

   7. Simplified 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. unpow2 46.5%

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

      3. times-frac 58.0%

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

   9. Applied egg-rr 58.0%

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

       1. associate-*r* 59.8%

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

       2. associate-*r/ 59.8%

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

       3. un-div-inv 59.9%

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

   11. Applied egg-rr 59.9%

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

4. Final simplification 89.5%

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

Alternative 2: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4.6 \cdot 10^{-234}:\\ \;\;\;\;\frac{t\_0 + 2 \cdot \mathsf{fma}\left(a, -d, a \cdot d\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+114}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))) (t_1 (- (/ b c) (/ (* d (/ a c)) c))))
   (if (<= c -2.5e+44)
     t_1
     (if (<= c -4.6e-234)
       (/ (+ t_0 (* 2.0 (fma a (- d) (* a d)))) (+ (* c c) (* d d)))
       (if (<= c 5.8e-140)
         (- (* b (/ (/ c d) d)) (/ a d))
         (if (<= c 7.6e+114) (* t_0 (pow (hypot c d) -2.0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = (b / c) - ((d * (a / c)) / c);
	double tmp;
	if (c <= -2.5e+44) {
		tmp = t_1;
	} else if (c <= -4.6e-234) {
		tmp = (t_0 + (2.0 * fma(a, -d, (a * d)))) / ((c * c) + (d * d));
	} else if (c <= 5.8e-140) {
		tmp = (b * ((c / d) / d)) - (a / d);
	} else if (c <= 7.6e+114) {
		tmp = t_0 * pow(hypot(c, d), -2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c))
	tmp = 0.0
	if (c <= -2.5e+44)
		tmp = t_1;
	elseif (c <= -4.6e-234)
		tmp = Float64(Float64(t_0 + Float64(2.0 * fma(a, Float64(-d), Float64(a * d)))) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 5.8e-140)
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	elseif (c <= 7.6e+114)
		tmp = Float64(t_0 * (hypot(c, d) ^ -2.0));
	else
		tmp = t_1;
	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[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.5e+44], t$95$1, If[LessEqual[c, -4.6e-234], N[(N[(t$95$0 + N[(2.0 * N[(a * (-d) + N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.8e-140], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 7.6e+114], N[(t$95$0 * N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.4999999999999998e44 or 7.6000000000000001e114 < c

   1. Initial program 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. add-sqr-sqrt 48.7%

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

      3. times-frac 48.7%

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

      4. hypot-def 48.7%

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

      5. hypot-def 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-lft-identity 74.6%

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

      5. times-frac 77.0%

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

      6. /-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*r* 82.9%

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

       2. associate-*r/ 84.5%

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

       3. un-div-inv 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -2.4999999999999998e44 < c < -4.59999999999999981e-234

   1. Initial program 87.9%

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

      1. prod-diff 87.9%

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

      2. *-commutative 87.9%

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

      3. fma-neg 87.9%

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

      4. prod-diff 87.9%

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

      5. *-commutative 87.9%

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

      6. fma-neg 87.9%

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

      7. associate-+l+ 87.9%

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

      8. *-commutative 87.9%

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

      9. fma-udef 87.9%

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

      10. distribute-lft-neg-in 87.9%

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

      11. *-commutative 87.9%

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

      12. distribute-rgt-neg-in 87.9%

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

      13. fma-def 87.9%

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

      14. *-commutative 87.9%

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

      15. fma-udef 87.9%

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

      16. distribute-lft-neg-in 87.9%

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

      17. *-commutative 87.9%

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

      18. distribute-rgt-neg-in 87.9%

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

      19. fma-def 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. count-2 87.9%

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

   6. Simplified 87.9%

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

   if -4.59999999999999981e-234 < c < 5.79999999999999995e-140

   1. Initial program 71.2%

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

   3. Taylor expanded in c around 0 81.1%

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

      1. +-commutative 81.1%

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

      2. mul-1-neg 81.1%

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

      3. unsub-neg 81.1%

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

      4. associate-/l* 81.3%

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

   5. Simplified 81.3%

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

      1. pow2 81.3%

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

      2. *-un-lft-identity 81.3%

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

      3. times-frac 95.6%

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

   7. Applied egg-rr 95.6%

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

      1. clear-num 95.7%

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

      2. associate-/r/ 95.6%

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

      3. /-rgt-identity 95.6%

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

      4. *-commutative 95.6%

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

      5. associate-/r* 95.6%

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

      6. clear-num 95.6%

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

   9. Applied egg-rr 95.6%

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

   if 5.79999999999999995e-140 < c < 7.6000000000000001e114

   1. Initial program 77.9%

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

      1. clear-num 77.9%

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

      2. associate-/r/ 77.8%

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

      3. add-sqr-sqrt 77.8%

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

      4. pow2 77.8%

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

      5. hypot-def 77.8%

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

   4. Applied egg-rr 77.8%

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

      1. expm1-log1p-u 75.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\right)} \cdot \left(b \cdot c - a \cdot d\right) \]

      2. expm1-udef 38.2%

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

      3. pow-flip 38.2%

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

      4. metadata-eval 38.2%

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

   6. Applied egg-rr 38.2%

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

      1. expm1-def 76.5%

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

      2. expm1-log1p 78.6%

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

   8. Simplified 78.6%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -4.6 \cdot 10^{-234}:\\ \;\;\;\;\frac{\left(b \cdot c - a \cdot d\right) + 2 \cdot \mathsf{fma}\left(a, -d, a \cdot d\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{+114}:\\ \;\;\;\;\left(b \cdot c - a \cdot d\right) \cdot {\left(\mathsf{hypot}\left(c, d\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ t_2 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-234}:\\ \;\;\;\;\frac{t\_0 + 2 \cdot \mathsf{fma}\left(a, -d, a \cdot d\right)}{t\_2}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{t\_0}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d)))
        (t_1 (- (/ b c) (/ (* d (/ a c)) c)))
        (t_2 (+ (* c c) (* d d))))
   (if (<= c -2.1e+44)
     t_1
     (if (<= c -8.2e-234)
       (/ (+ t_0 (* 2.0 (fma a (- d) (* a d)))) t_2)
       (if (<= c 2.4e-77)
         (- (* (/ 1.0 d) (/ b (/ d c))) (/ a d))
         (if (<= c 3.3e+115) (/ t_0 t_2) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = (b / c) - ((d * (a / c)) / c);
	double t_2 = (c * c) + (d * d);
	double tmp;
	if (c <= -2.1e+44) {
		tmp = t_1;
	} else if (c <= -8.2e-234) {
		tmp = (t_0 + (2.0 * fma(a, -d, (a * d)))) / t_2;
	} else if (c <= 2.4e-77) {
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	} else if (c <= 3.3e+115) {
		tmp = t_0 / t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c))
	t_2 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (c <= -2.1e+44)
		tmp = t_1;
	elseif (c <= -8.2e-234)
		tmp = Float64(Float64(t_0 + Float64(2.0 * fma(a, Float64(-d), Float64(a * d)))) / t_2);
	elseif (c <= 2.4e-77)
		tmp = Float64(Float64(Float64(1.0 / d) * Float64(b / Float64(d / c))) - Float64(a / d));
	elseif (c <= 3.3e+115)
		tmp = Float64(t_0 / t_2);
	else
		tmp = t_1;
	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[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.1e+44], t$95$1, If[LessEqual[c, -8.2e-234], N[(N[(t$95$0 + N[(2.0 * N[(a * (-d) + N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[c, 2.4e-77], N[(N[(N[(1.0 / d), $MachinePrecision] * N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.3e+115], N[(t$95$0 / t$95$2), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.09999999999999987e44 or 3.30000000000000005e115 < c

   1. Initial program 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. add-sqr-sqrt 48.7%

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

      3. times-frac 48.7%

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

      4. hypot-def 48.7%

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

      5. hypot-def 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-lft-identity 74.6%

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

      5. times-frac 77.0%

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

      6. /-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*r* 82.9%

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

       2. associate-*r/ 84.5%

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

       3. un-div-inv 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -2.09999999999999987e44 < c < -8.20000000000000021e-234

   1. Initial program 87.9%

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

      1. prod-diff 87.9%

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

      2. *-commutative 87.9%

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

      3. fma-neg 87.9%

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

      4. prod-diff 87.9%

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

      5. *-commutative 87.9%

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

      6. fma-neg 87.9%

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

      7. associate-+l+ 87.9%

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

      8. *-commutative 87.9%

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

      9. fma-udef 87.9%

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

      10. distribute-lft-neg-in 87.9%

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

      11. *-commutative 87.9%

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

      12. distribute-rgt-neg-in 87.9%

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

      13. fma-def 87.9%

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

      14. *-commutative 87.9%

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

      15. fma-udef 87.9%

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

      16. distribute-lft-neg-in 87.9%

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

      17. *-commutative 87.9%

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

      18. distribute-rgt-neg-in 87.9%

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

      19. fma-def 87.9%

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

   4. Applied egg-rr 87.9%

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

      1. *-commutative 87.9%

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

      2. count-2 87.9%

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

   6. Simplified 87.9%

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

   if -8.20000000000000021e-234 < c < 2.3999999999999999e-77

   1. Initial program 72.6%

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

   3. Taylor expanded in c around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. mul-1-neg 79.5%

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

      3. unsub-neg 79.5%

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

      4. associate-/l* 79.7%

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

   5. Simplified 79.7%

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

      1. pow2 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 90.9%

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

   7. Applied egg-rr 90.9%

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

      1. /-rgt-identity 90.9%

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

      2. *-un-lft-identity 90.9%

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

      3. times-frac 92.1%

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

   9. Applied egg-rr 92.1%

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

   if 2.3999999999999999e-77 < c < 3.30000000000000005e115

   1. Initial program 77.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing
3. Recombined 4 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-234}:\\ \;\;\;\;\frac{\left(b \cdot c - a \cdot d\right) + 2 \cdot \mathsf{fma}\left(a, -d, a \cdot d\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.3 \cdot 10^{+115}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-234}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 8.7 \cdot 10^{+114}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (* d (/ a c)) c))))
   (if (<= c -2.1e+44)
     t_1
     (if (<= c -8.2e-234)
       t_0
       (if (<= c 2.5e-77)
         (- (* (/ 1.0 d) (/ b (/ d c))) (/ a d))
         (if (<= c 8.7e+114) 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 / c) - ((d * (a / c)) / c);
	double tmp;
	if (c <= -2.1e+44) {
		tmp = t_1;
	} else if (c <= -8.2e-234) {
		tmp = t_0;
	} else if (c <= 2.5e-77) {
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	} else if (c <= 8.7e+114) {
		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 / c) - ((d * (a / c)) / c)
    if (c <= (-2.1d+44)) then
        tmp = t_1
    else if (c <= (-8.2d-234)) then
        tmp = t_0
    else if (c <= 2.5d-77) then
        tmp = ((1.0d0 / d) * (b / (d / c))) - (a / d)
    else if (c <= 8.7d+114) 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 / c) - ((d * (a / c)) / c);
	double tmp;
	if (c <= -2.1e+44) {
		tmp = t_1;
	} else if (c <= -8.2e-234) {
		tmp = t_0;
	} else if (c <= 2.5e-77) {
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	} else if (c <= 8.7e+114) {
		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 / c) - ((d * (a / c)) / c)
	tmp = 0
	if c <= -2.1e+44:
		tmp = t_1
	elif c <= -8.2e-234:
		tmp = t_0
	elif c <= 2.5e-77:
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d)
	elif c <= 8.7e+114:
		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 / c) - Float64(Float64(d * Float64(a / c)) / c))
	tmp = 0.0
	if (c <= -2.1e+44)
		tmp = t_1;
	elseif (c <= -8.2e-234)
		tmp = t_0;
	elseif (c <= 2.5e-77)
		tmp = Float64(Float64(Float64(1.0 / d) * Float64(b / Float64(d / c))) - Float64(a / d));
	elseif (c <= 8.7e+114)
		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 / c) - ((d * (a / c)) / c);
	tmp = 0.0;
	if (c <= -2.1e+44)
		tmp = t_1;
	elseif (c <= -8.2e-234)
		tmp = t_0;
	elseif (c <= 2.5e-77)
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	elseif (c <= 8.7e+114)
		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 / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.1e+44], t$95$1, If[LessEqual[c, -8.2e-234], t$95$0, If[LessEqual[c, 2.5e-77], N[(N[(N[(1.0 / d), $MachinePrecision] * N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.7e+114], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.09999999999999987e44 or 8.7000000000000001e114 < c

   1. Initial program 48.7%

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

      1. *-un-lft-identity 48.7%

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

      2. add-sqr-sqrt 48.7%

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

      3. times-frac 48.7%

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

      4. hypot-def 48.7%

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

      5. hypot-def 70.0%

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

   4. Applied egg-rr 70.0%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. +-commutative 74.6%

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

      2. mul-1-neg 74.6%

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

      3. unsub-neg 74.6%

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

      4. *-lft-identity 74.6%

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

      5. times-frac 77.0%

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

      6. /-rgt-identity 77.0%

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

   7. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. unpow2 77.0%

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

      3. times-frac 77.4%

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

   9. Applied egg-rr 77.4%

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

       1. associate-*r* 82.9%

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

       2. associate-*r/ 84.5%

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

       3. un-div-inv 84.6%

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

   11. Applied egg-rr 84.6%

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

   if -2.09999999999999987e44 < c < -8.20000000000000021e-234 or 2.49999999999999982e-77 < c < 8.7000000000000001e114

   1. Initial program 83.7%

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

   if -8.20000000000000021e-234 < c < 2.49999999999999982e-77

   1. Initial program 72.6%

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

   3. Taylor expanded in c around 0 79.5%

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

      1. +-commutative 79.5%

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

      2. mul-1-neg 79.5%

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

      3. unsub-neg 79.5%

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

      4. associate-/l* 79.7%

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

   5. Simplified 79.7%

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

      1. pow2 79.7%

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

      2. *-un-lft-identity 79.7%

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

      3. times-frac 90.9%

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

   7. Applied egg-rr 90.9%

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

      1. /-rgt-identity 90.9%

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

      2. *-un-lft-identity 90.9%

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

      3. times-frac 92.1%

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

   9. Applied egg-rr 92.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-234}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \mathbf{elif}\;c \leq 8.7 \cdot 10^{+114}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+48} \lor \neg \left(c \leq 7.2 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6.5e+48) (not (<= c 7.2e-72)))
   (- (/ b c) (/ (* d (/ a c)) c))
   (- (* (/ 1.0 d) (/ b (/ d c))) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.5e+48) || !(c <= 7.2e-72)) {
		tmp = (b / c) - ((d * (a / c)) / c);
	} else {
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-6.5d+48)) .or. (.not. (c <= 7.2d-72))) then
        tmp = (b / c) - ((d * (a / c)) / c)
    else
        tmp = ((1.0d0 / d) * (b / (d / c))) - (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 <= -6.5e+48) || !(c <= 7.2e-72)) {
		tmp = (b / c) - ((d * (a / c)) / c);
	} else {
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6.5e+48) or not (c <= 7.2e-72):
		tmp = (b / c) - ((d * (a / c)) / c)
	else:
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6.5e+48) || !(c <= 7.2e-72))
		tmp = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c));
	else
		tmp = Float64(Float64(Float64(1.0 / d) * Float64(b / Float64(d / c))) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -6.5e+48) || ~((c <= 7.2e-72)))
		tmp = (b / c) - ((d * (a / c)) / c);
	else
		tmp = ((1.0 / d) * (b / (d / c))) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6.5e+48], N[Not[LessEqual[c, 7.2e-72]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / d), $MachinePrecision] * N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.49999999999999972e48 or 7.2e-72 < c

   1. Initial program 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. add-sqr-sqrt 57.9%

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

      3. times-frac 57.8%

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

      4. hypot-def 57.8%

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

      5. hypot-def 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in c around inf 71.3%

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

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. *-lft-identity 71.3%

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

      5. times-frac 72.3%

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

      6. /-rgt-identity 72.3%

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

   7. Simplified 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. unpow2 72.3%

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

      3. times-frac 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. associate-*r* 77.4%

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

       2. associate-*r/ 78.5%

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

       3. un-div-inv 78.5%

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

   11. Applied egg-rr 78.5%

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

   if -6.49999999999999972e48 < c < 7.2e-72

   1. Initial program 77.2%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. /-rgt-identity 81.8%

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

      2. *-un-lft-identity 81.8%

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

      3. times-frac 83.7%

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+48} \lor \neg \left(c \leq 7.2 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \frac{b}{\frac{d}{c}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+51} \lor \neg \left(c \leq 1.1 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.05e+51) (not (<= c 1.1e+62)))
   (/ b c)
   (- (* (/ c d) (/ b d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.05e+51) || !(c <= 1.1e+62)) {
		tmp = b / c;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.05d+51)) .or. (.not. (c <= 1.1d+62))) then
        tmp = b / c
    else
        tmp = ((c / d) * (b / d)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.05e+51) || !(c <= 1.1e+62)) {
		tmp = b / c;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.05e+51) or not (c <= 1.1e+62):
		tmp = b / c
	else:
		tmp = ((c / d) * (b / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.05e+51) || !(c <= 1.1e+62))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.05e+51) || ~((c <= 1.1e+62)))
		tmp = b / c;
	else
		tmp = ((c / d) * (b / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.05e+51], N[Not[LessEqual[c, 1.1e+62]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.0500000000000001e51 or 1.10000000000000007e62 < c

   1. Initial program 52.0%

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

   3. Taylor expanded in c around inf 67.3%

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

   if -1.0500000000000001e51 < c < 1.10000000000000007e62

   1. Initial program 77.1%

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

   3. Taylor expanded in c around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. associate-/l* 70.4%

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

   5. Simplified 70.4%

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

      1. pow2 70.4%

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

      2. *-un-lft-identity 70.4%

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

      3. times-frac 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. /-rgt-identity 76.0%

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

      2. *-un-lft-identity 76.0%

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

      3. *-commutative 76.0%

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

      4. times-frac 74.1%

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

      5. clear-num 74.1%

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

   9. Applied egg-rr 74.1%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{+51} \lor \neg \left(c \leq 1.1 \cdot 10^{+62}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+51} \lor \neg \left(c \leq 2.3 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.5e+51) (not (<= c 2.3e+63)))
   (/ b c)
   (- (* b (/ (/ c d) d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e+51) || !(c <= 2.3e+63)) {
		tmp = b / c;
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.5d+51)) .or. (.not. (c <= 2.3d+63))) then
        tmp = b / c
    else
        tmp = (b * ((c / d) / d)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e+51) || !(c <= 2.3e+63)) {
		tmp = b / c;
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.5e+51) or not (c <= 2.3e+63):
		tmp = b / c
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.5e+51) || !(c <= 2.3e+63))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.5e+51) || ~((c <= 2.3e+63)))
		tmp = b / c;
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.5e+51], N[Not[LessEqual[c, 2.3e+63]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.5e51 or 2.29999999999999993e63 < c

   1. Initial program 52.0%

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

   3. Taylor expanded in c around inf 67.3%

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

   if -4.5e51 < c < 2.29999999999999993e63

   1. Initial program 77.1%

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

   3. Taylor expanded in c around 0 71.7%

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

      1. +-commutative 71.7%

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

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. associate-/l* 70.4%

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

   5. Simplified 70.4%

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

      1. pow2 70.4%

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

      2. *-un-lft-identity 70.4%

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

      3. times-frac 76.0%

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

   7. Applied egg-rr 76.0%

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

      1. clear-num 76.1%

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

      2. associate-/r/ 76.0%

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

      3. /-rgt-identity 76.0%

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

      4. *-commutative 76.0%

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

      5. associate-/r* 76.6%

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

      6. clear-num 76.6%

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

   9. Applied egg-rr 76.6%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{+51} \lor \neg \left(c \leq 2.3 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+48} \lor \neg \left(c \leq 5.4 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - a \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8e+48) (not (<= c 5.4e-72)))
   (- (/ b c) (* a (/ (/ d c) c)))
   (- (* b (/ (/ c d) d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8e+48) || !(c <= 5.4e-72)) {
		tmp = (b / c) - (a * ((d / c) / c));
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-8d+48)) .or. (.not. (c <= 5.4d-72))) then
        tmp = (b / c) - (a * ((d / c) / c))
    else
        tmp = (b * ((c / d) / d)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8e+48) || !(c <= 5.4e-72)) {
		tmp = (b / c) - (a * ((d / c) / c));
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -8e+48) or not (c <= 5.4e-72):
		tmp = (b / c) - (a * ((d / c) / c))
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8e+48) || !(c <= 5.4e-72))
		tmp = Float64(Float64(b / c) - Float64(a * Float64(Float64(d / c) / c)));
	else
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -8e+48) || ~((c <= 5.4e-72)))
		tmp = (b / c) - (a * ((d / c) / c));
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8e+48], N[Not[LessEqual[c, 5.4e-72]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(a * N[(N[(d / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.00000000000000035e48 or 5.4e-72 < c

   1. Initial program 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. add-sqr-sqrt 57.9%

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

      3. times-frac 57.8%

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

      4. hypot-def 57.8%

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

      5. hypot-def 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in c around inf 71.3%

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

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. *-lft-identity 71.3%

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

      5. times-frac 72.3%

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

      6. /-rgt-identity 72.3%

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

   7. Simplified 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. unpow2 72.3%

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

      3. times-frac 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. associate-*l/ 72.6%

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

       2. *-lft-identity 72.6%

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

   11. Simplified 72.6%

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

   if -8.00000000000000035e48 < c < 5.4e-72

   1. Initial program 77.2%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. clear-num 81.9%

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

      2. associate-/r/ 81.8%

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

      3. /-rgt-identity 81.8%

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

      4. *-commutative 81.8%

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

      5. associate-/r* 82.5%

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

      6. clear-num 82.5%

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

   9. Applied egg-rr 82.5%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+48} \lor \neg \left(c \leq 5.4 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - a \cdot \frac{\frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+48} \lor \neg \left(c \leq 1.2 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6.5e+48) (not (<= c 1.2e-72)))
   (- (/ b c) (/ (* d (/ a c)) c))
   (- (* b (/ (/ c d) d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.5e+48) || !(c <= 1.2e-72)) {
		tmp = (b / c) - ((d * (a / c)) / c);
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-6.5d+48)) .or. (.not. (c <= 1.2d-72))) then
        tmp = (b / c) - ((d * (a / c)) / c)
    else
        tmp = (b * ((c / d) / d)) - (a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.5e+48) || !(c <= 1.2e-72)) {
		tmp = (b / c) - ((d * (a / c)) / c);
	} else {
		tmp = (b * ((c / d) / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6.5e+48) or not (c <= 1.2e-72):
		tmp = (b / c) - ((d * (a / c)) / c)
	else:
		tmp = (b * ((c / d) / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6.5e+48) || !(c <= 1.2e-72))
		tmp = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c));
	else
		tmp = Float64(Float64(b * Float64(Float64(c / d) / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -6.5e+48) || ~((c <= 1.2e-72)))
		tmp = (b / c) - ((d * (a / c)) / c);
	else
		tmp = (b * ((c / d) / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6.5e+48], N[Not[LessEqual[c, 1.2e-72]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(c / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.49999999999999972e48 or 1.2e-72 < c

   1. Initial program 57.9%

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

      1. *-un-lft-identity 57.9%

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

      2. add-sqr-sqrt 57.9%

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

      3. times-frac 57.8%

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

      4. hypot-def 57.8%

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

      5. hypot-def 74.9%

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

   4. Applied egg-rr 74.9%

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

   5. Taylor expanded in c around inf 71.3%

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

      1. +-commutative 71.3%

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

      2. mul-1-neg 71.3%

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

      3. unsub-neg 71.3%

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

      4. *-lft-identity 71.3%

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

      5. times-frac 72.3%

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

      6. /-rgt-identity 72.3%

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

   7. Simplified 72.3%

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

      1. *-un-lft-identity 72.3%

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

      2. unpow2 72.3%

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

      3. times-frac 72.6%

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

   9. Applied egg-rr 72.6%

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

       1. associate-*r* 77.4%

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

       2. associate-*r/ 78.5%

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

       3. un-div-inv 78.5%

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

   11. Applied egg-rr 78.5%

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

   if -6.49999999999999972e48 < c < 1.2e-72

   1. Initial program 77.2%

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

   3. Taylor expanded in c around 0 76.6%

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

      1. +-commutative 76.6%

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

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 81.8%

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

   7. Applied egg-rr 81.8%

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

      1. clear-num 81.9%

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

      2. associate-/r/ 81.8%

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

      3. /-rgt-identity 81.8%

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

      4. *-commutative 81.8%

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

      5. associate-/r* 82.5%

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

      6. clear-num 82.5%

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

   9. Applied egg-rr 82.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+48} \lor \neg \left(c \leq 1.2 \cdot 10^{-72}\right):\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{\frac{c}{d}}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -0.0132 \lor \neg \left(c \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -0.0132) (not (<= c 2e+46))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -0.0132) || !(c <= 2e+46)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-0.0132d0)) .or. (.not. (c <= 2d+46))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -0.0132) || !(c <= 2e+46)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -0.0132) or not (c <= 2e+46):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -0.0132) || !(c <= 2e+46))
		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 <= -0.0132) || ~((c <= 2e+46)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -0.0132], N[Not[LessEqual[c, 2e+46]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -0.0132 or 2e46 < c

   1. Initial program 54.2%

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

   3. Taylor expanded in c around inf 63.8%

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

   if -0.0132 < c < 2e46

   1. Initial program 78.0%

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

   3. Taylor expanded in c around 0 63.2%

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

      1. associate-*r/ 63.2%

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

      2. neg-mul-1 63.2%

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

   5. Simplified 63.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.0132 \lor \neg \left(c \leq 2 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 15.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+129} \lor \neg \left(d \leq 7.8 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.8e+129) (not (<= d 7.8e+105))) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.8e+129) || !(d <= 7.8e+105)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-3.8d+129)) .or. (.not. (d <= 7.8d+105))) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.8e+129) || !(d <= 7.8e+105)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.8e+129) or not (d <= 7.8e+105):
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.8e+129) || !(d <= 7.8e+105))
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.8e+129) || ~((d <= 7.8e+105)))
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.8e+129], N[Not[LessEqual[d, 7.8e+105]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.80000000000000005e129 or 7.79999999999999957e105 < d

   1. Initial program 43.6%

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

      1. *-un-lft-identity 43.6%

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

      2. add-sqr-sqrt 43.6%

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

      3. times-frac 43.5%

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

      4. hypot-def 43.5%

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

      5. hypot-def 68.8%

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

   4. Applied egg-rr 68.8%

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

   5. Taylor expanded in c around 0 52.0%

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

      1. mul-1-neg 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in d around -inf 33.3%

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

   if -3.80000000000000005e129 < d < 7.79999999999999957e105

   1. Initial program 77.7%

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

      1. *-un-lft-identity 77.7%

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

      2. add-sqr-sqrt 77.7%

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

      3. times-frac 77.7%

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

      4. hypot-def 77.7%

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

      5. hypot-def 88.6%

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

   4. Applied egg-rr 88.6%

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

   5. Taylor expanded in c around 0 19.2%

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

      1. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

   8. Taylor expanded in c around -inf 10.3%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+129} \lor \neg \left(d \leq 7.8 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+166} \lor \neg \left(d \leq 5.2 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.7e+166) (not (<= d 5.2e+144))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e+166) || !(d <= 5.2e+144)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-5.7d+166)) .or. (.not. (d <= 5.2d+144))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e+166) || !(d <= 5.2e+144)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.7e+166) or not (d <= 5.2e+144):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.7e+166) || !(d <= 5.2e+144))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -5.7e+166) || ~((d <= 5.2e+144)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.7e+166], N[Not[LessEqual[d, 5.2e+144]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.69999999999999977e166 or 5.1999999999999998e144 < d

   1. Initial program 40.5%

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

      1. *-un-lft-identity 40.5%

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

      2. add-sqr-sqrt 40.5%

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

      3. times-frac 40.5%

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

      4. hypot-def 40.5%

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

      5. hypot-def 68.1%

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

   4. Applied egg-rr 68.1%

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

   5. Taylor expanded in c around 0 59.2%

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

      1. mul-1-neg 59.2%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in d around -inf 40.8%

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

   if -5.69999999999999977e166 < d < 5.1999999999999998e144

   1. Initial program 75.1%

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

   3. Taylor expanded in c around inf 45.4%

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

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+166} \lor \neg \left(d \leq 5.2 \cdot 10^{+144}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 9.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

   1. *-un-lft-identity 68.1%

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

   2. add-sqr-sqrt 68.1%

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

   3. times-frac 68.1%

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

   4. hypot-def 68.1%

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

   5. hypot-def 83.0%

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

4. Applied egg-rr 83.0%

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

5. Taylor expanded in c around 0 28.4%

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

   1. mul-1-neg 28.4%

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

7. Simplified 28.4%

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

8. Taylor expanded in c around -inf 10.1%

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

9. Final simplification 10.1%

   \[\leadsto \frac{a}{c} \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024040 
(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))))