Complex division, imag part

Percentage Accurate: 61.5% → 86.8%

Time: 10.6s

Alternatives: 10

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: 61.5% 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: 86.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.32 \cdot 10^{+174}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-153} \lor \neg \left(c \leq 3.4 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.32e+174)
   (/ (- b (* a (/ d c))) c)
   (if (or (<= c -3.1e-153) (not (<= c 3.4e-136)))
     (fma
      (/ c (hypot c d))
      (/ b (hypot c d))
      (* a (/ (- d) (pow (hypot c d) 2.0))))
     (/ (- (* b (/ c d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.32e+174) {
		tmp = (b - (a * (d / c))) / c;
	} else if ((c <= -3.1e-153) || !(c <= 3.4e-136)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (a * (-d / pow(hypot(c, d), 2.0))));
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.32e+174)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif ((c <= -3.1e-153) || !(c <= 3.4e-136))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(-d) / (hypot(c, d) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.32e+174], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[c, -3.1e-153], N[Not[LessEqual[c, 3.4e-136]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a * N[((-d) / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.31999999999999999e174

   1. Initial program 32.2%

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

   3. Taylor expanded in c around inf 70.6%

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

      1. remove-double-neg 70.6%

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

      2. mul-1-neg 70.6%

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

      3. neg-mul-1 70.6%

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

      4. distribute-lft-in 70.6%

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

      5. mul-1-neg 70.6%

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

      6. distribute-neg-in 70.6%

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

      7. mul-1-neg 70.6%

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

      8. remove-double-neg 70.6%

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

      9. unsub-neg 70.6%

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

      10. associate-/l* 96.3%

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

   5. Simplified 96.3%

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

   if -1.31999999999999999e174 < c < -3.09999999999999995e-153 or 3.4e-136 < c

   1. Initial program 67.9%

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

      1. div-sub 67.9%

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

      2. *-commutative 67.9%

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

      3. add-sqr-sqrt 67.9%

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

      4. times-frac 73.4%

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

      5. fma-neg 73.4%

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

      6. hypot-define 73.4%

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

      7. hypot-define 87.3%

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

      8. associate-/l* 87.5%

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

      9. add-sqr-sqrt 87.5%

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

      10. pow2 87.5%

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

      11. hypot-define 87.5%

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

   4. Applied egg-rr 87.5%

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

   if -3.09999999999999995e-153 < c < 3.4e-136

   1. Initial program 73.9%

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

      1. div-sub 68.1%

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

      2. *-commutative 68.1%

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

      3. add-sqr-sqrt 68.1%

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

      4. times-frac 66.4%

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

      5. fma-neg 66.4%

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

      6. hypot-define 66.4%

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

      7. hypot-define 67.8%

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

      8. associate-/l* 69.3%

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

      9. add-sqr-sqrt 69.3%

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

      10. pow2 69.3%

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

      11. hypot-define 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in d around inf 93.7%

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

      1. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.32 \cdot 10^{+174}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-153} \lor \neg \left(c \leq 3.4 \cdot 10^{-136}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-136}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.2e-47)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 1.7e-136)
     (/ (- (* b (/ c d)) a) d)
     (if (<= c 1.7e+74)
       (/ (- (* c b) (* a d)) (+ (* c c) (* d d)))
       (/ (- b (/ a (/ c d))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-47) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.7e-136) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.7e+74) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b - (a / (c / d))) / 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 (c <= (-6.2d-47)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 1.7d-136) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 1.7d+74) then
        tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-47) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.7e-136) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.7e+74) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.2e-47:
		tmp = (b - (a * (d / c))) / c
	elif c <= 1.7e-136:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 1.7e+74:
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.2e-47)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 1.7e-136)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 1.7e+74)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6.2e-47)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 1.7e-136)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 1.7e+74)
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.2e-47], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.7e-136], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.7e+74], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.1999999999999996e-47

   1. Initial program 53.1%

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

   3. Taylor expanded in c around inf 71.7%

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

      1. remove-double-neg 71.7%

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

      2. mul-1-neg 71.7%

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

      3. neg-mul-1 71.7%

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

      4. distribute-lft-in 71.7%

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

      5. mul-1-neg 71.7%

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

      6. distribute-neg-in 71.7%

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

      7. mul-1-neg 71.7%

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

      8. remove-double-neg 71.7%

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

      9. unsub-neg 71.7%

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

      10. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -6.1999999999999996e-47 < c < 1.7e-136

   1. Initial program 71.9%

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

      1. div-sub 67.4%

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

      2. *-commutative 67.4%

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

      3. add-sqr-sqrt 67.4%

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

      4. times-frac 67.1%

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

      5. fma-neg 67.1%

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

      6. hypot-define 67.1%

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

      7. hypot-define 69.8%

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

      8. associate-/l* 71.9%

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

      9. add-sqr-sqrt 71.9%

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

      10. pow2 71.9%

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

      11. hypot-define 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in d around inf 90.2%

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

      1. associate-/l* 90.8%

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

   7. Simplified 90.8%

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

   if 1.7e-136 < c < 1.7e74

   1. Initial program 85.7%

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

   if 1.7e74 < c

   1. Initial program 49.8%

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

   3. Taylor expanded in c around inf 83.9%

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

      1. remove-double-neg 83.9%

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

      2. mul-1-neg 83.9%

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

      3. neg-mul-1 83.9%

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

      4. distribute-lft-in 83.9%

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

      5. mul-1-neg 83.9%

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

      6. distribute-neg-in 83.9%

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

      7. mul-1-neg 83.9%

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

      8. remove-double-neg 83.9%

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

      9. unsub-neg 83.9%

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

      10. associate-/l* 88.5%

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

   5. Simplified 88.5%

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

      1. clear-num 88.5%

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

      2. un-div-inv 88.6%

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-136}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{a}{-c}\\ \mathbf{elif}\;c \leq -1.45 \cdot 10^{-64} \lor \neg \left(c \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.75e+215)
   (* (/ d c) (/ a (- c)))
   (if (or (<= c -1.45e-64) (not (<= c 8.4e-57))) (/ b c) (/ a (- d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.75e+215) {
		tmp = (d / c) * (a / -c);
	} else if ((c <= -1.45e-64) || !(c <= 8.4e-57)) {
		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 <= (-2.75d+215)) then
        tmp = (d / c) * (a / -c)
    else if ((c <= (-1.45d-64)) .or. (.not. (c <= 8.4d-57))) 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 <= -2.75e+215) {
		tmp = (d / c) * (a / -c);
	} else if ((c <= -1.45e-64) || !(c <= 8.4e-57)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.75e+215:
		tmp = (d / c) * (a / -c)
	elif (c <= -1.45e-64) or not (c <= 8.4e-57):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.75e+215)
		tmp = Float64(Float64(d / c) * Float64(a / Float64(-c)));
	elseif ((c <= -1.45e-64) || !(c <= 8.4e-57))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.75e+215)
		tmp = (d / c) * (a / -c);
	elseif ((c <= -1.45e-64) || ~((c <= 8.4e-57)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.75e+215], N[(N[(d / c), $MachinePrecision] * N[(a / (-c)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c, -1.45e-64], N[Not[LessEqual[c, 8.4e-57]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.75e215

   1. Initial program 45.7%

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

   3. Taylor expanded in c around inf 62.0%

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

      1. remove-double-neg 62.0%

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

      2. mul-1-neg 62.0%

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

      3. neg-mul-1 62.0%

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

      4. distribute-lft-in 62.0%

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

      5. distribute-lft-in 62.0%

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

      6. neg-mul-1 62.0%

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

      7. mul-1-neg 62.0%

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

      8. remove-double-neg 62.0%

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

      9. associate-*r/ 62.0%

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

      10. associate-*r* 62.0%

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

      11. neg-mul-1 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in d around inf 95.6%

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

      1. +-commutative 95.6%

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

      2. mul-1-neg 95.6%

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

      3. unsub-neg 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in d around inf 53.8%

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

       1. associate-*r/ 53.8%

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

       2. mul-1-neg 53.8%

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

       3. *-commutative 53.8%

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

   11. Simplified 53.8%

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

       1. associate-/l/ 45.9%

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

       2. distribute-lft-neg-in 45.9%

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

       3. times-frac 83.4%

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

   13. Applied egg-rr 83.4%

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

   if -2.75e215 < c < -1.4499999999999999e-64 or 8.3999999999999998e-57 < c

   1. Initial program 61.1%

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

   3. Taylor expanded in c around inf 62.7%

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

   if -1.4499999999999999e-64 < c < 8.3999999999999998e-57

   1. Initial program 73.2%

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

   3. Taylor expanded in c around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{+215}:\\ \;\;\;\;\frac{d}{c} \cdot \frac{a}{-c}\\ \mathbf{elif}\;c \leq -1.45 \cdot 10^{-64} \lor \neg \left(c \leq 8.4 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{a \cdot \frac{-d}{c}}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-65} \lor \neg \left(c \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.8e+215)
   (/ (* a (/ (- d) c)) c)
   (if (or (<= c -6e-65) (not (<= c 1.4e-56))) (/ b c) (/ a (- d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.8e+215) {
		tmp = (a * (-d / c)) / c;
	} else if ((c <= -6e-65) || !(c <= 1.4e-56)) {
		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 <= (-2.8d+215)) then
        tmp = (a * (-d / c)) / c
    else if ((c <= (-6d-65)) .or. (.not. (c <= 1.4d-56))) 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 <= -2.8e+215) {
		tmp = (a * (-d / c)) / c;
	} else if ((c <= -6e-65) || !(c <= 1.4e-56)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.8e+215:
		tmp = (a * (-d / c)) / c
	elif (c <= -6e-65) or not (c <= 1.4e-56):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.8e+215)
		tmp = Float64(Float64(a * Float64(Float64(-d) / c)) / c);
	elseif ((c <= -6e-65) || !(c <= 1.4e-56))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.8e+215)
		tmp = (a * (-d / c)) / c;
	elseif ((c <= -6e-65) || ~((c <= 1.4e-56)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.8e+215], N[(N[(a * N[((-d) / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[Or[LessEqual[c, -6e-65], N[Not[LessEqual[c, 1.4e-56]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.8e215

   1. Initial program 45.7%

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

   3. Taylor expanded in c around inf 62.0%

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

      1. remove-double-neg 62.0%

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

      2. mul-1-neg 62.0%

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

      3. neg-mul-1 62.0%

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

      4. distribute-lft-in 62.0%

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

      5. distribute-lft-in 62.0%

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

      6. neg-mul-1 62.0%

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

      7. mul-1-neg 62.0%

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

      8. remove-double-neg 62.0%

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

      9. associate-*r/ 62.0%

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

      10. associate-*r* 62.0%

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

      11. neg-mul-1 62.0%

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

   5. Simplified 62.0%

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

   6. Taylor expanded in d around inf 95.6%

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

      1. +-commutative 95.6%

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

      2. mul-1-neg 95.6%

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

      3. unsub-neg 95.6%

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

   8. Simplified 95.6%

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

   9. Taylor expanded in d around inf 53.8%

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

       1. mul-1-neg 53.8%

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

       2. associate-*r/ 83.5%

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

       3. distribute-rgt-neg-in 83.5%

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

       4. distribute-neg-frac2 83.5%

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

   11. Simplified 83.5%

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

   if -2.8e215 < c < -5.99999999999999996e-65 or 1.39999999999999997e-56 < c

   1. Initial program 61.1%

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

   3. Taylor expanded in c around inf 62.7%

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

   if -5.99999999999999996e-65 < c < 1.39999999999999997e-56

   1. Initial program 73.2%

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

   3. Taylor expanded in c around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{+215}:\\ \;\;\;\;\frac{a \cdot \frac{-d}{c}}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-65} \lor \neg \left(c \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-96} \lor \neg \left(c \leq 2.3 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.5e-96) (not (<= c 2.3e-63)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e-96) || !(c <= 2.3e-63)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.5d-96)) .or. (.not. (c <= 2.3d-63))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.5e-96) || !(c <= 2.3e-63)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.5e-96) or not (c <= 2.3e-63):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.5e-96) || !(c <= 2.3e-63))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.5e-96) || ~((c <= 2.3e-63)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.5e-96], N[Not[LessEqual[c, 2.3e-63]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.5e-96 or 2.3e-63 < c

   1. Initial program 60.1%

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

   3. Taylor expanded in c around inf 74.4%

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

      1. remove-double-neg 74.4%

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

      2. mul-1-neg 74.4%

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

      3. neg-mul-1 74.4%

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

      4. distribute-lft-in 74.4%

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

      5. mul-1-neg 74.4%

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

      6. distribute-neg-in 74.4%

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

      7. mul-1-neg 74.4%

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

      8. remove-double-neg 74.4%

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

      9. unsub-neg 74.4%

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

      10. associate-/l* 81.6%

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

   5. Simplified 81.6%

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

   if -4.5e-96 < c < 2.3e-63

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   5. Simplified 72.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.5 \cdot 10^{-96} \lor \neg \left(c \leq 2.3 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.8e-96)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 4.9e-59) (/ a (- d)) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8e-96) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 4.9e-59) {
		tmp = a / -d;
	} else {
		tmp = (b - (a / (c / d))) / 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 (c <= (-5.8d-96)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 4.9d-59) then
        tmp = a / -d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8e-96) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 4.9e-59) {
		tmp = a / -d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -5.8e-96:
		tmp = (b - (a * (d / c))) / c
	elif c <= 4.9e-59:
		tmp = a / -d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.8e-96)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 4.9e-59)
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -5.8e-96)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 4.9e-59)
		tmp = a / -d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.8e-96], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 4.9e-59], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.79999999999999987e-96

   1. Initial program 56.3%

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

   3. Taylor expanded in c around inf 69.8%

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

      1. remove-double-neg 69.8%

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

      2. mul-1-neg 69.8%

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

      3. neg-mul-1 69.8%

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

      4. distribute-lft-in 69.8%

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

      5. mul-1-neg 69.8%

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

      6. distribute-neg-in 69.8%

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

      7. mul-1-neg 69.8%

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

      8. remove-double-neg 69.8%

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

      9. unsub-neg 69.8%

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

      10. associate-/l* 80.5%

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

   5. Simplified 80.5%

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

   if -5.79999999999999987e-96 < c < 4.89999999999999977e-59

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   5. Simplified 72.4%

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

   if 4.89999999999999977e-59 < c

   1. Initial program 64.7%

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

   3. Taylor expanded in c around inf 80.0%

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

      1. remove-double-neg 80.0%

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

      2. mul-1-neg 80.0%

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

      3. neg-mul-1 80.0%

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

      4. distribute-lft-in 80.0%

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

      5. mul-1-neg 80.0%

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

      6. distribute-neg-in 80.0%

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

      7. mul-1-neg 80.0%

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

      8. remove-double-neg 80.0%

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

      9. unsub-neg 80.0%

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

      10. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.9%

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 4.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.2e-47)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 1.6e-58) (/ (- (* b (/ c d)) a) d) (/ (- b (/ a (/ c d))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-47) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.6e-58) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / 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 (c <= (-6.2d-47)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 1.6d-58) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a / (c / d))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.2e-47) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.6e-58) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a / (c / d))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.2e-47:
		tmp = (b - (a * (d / c))) / c
	elif c <= 1.6e-58:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a / (c / d))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.2e-47)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 1.6e-58)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a / Float64(c / d))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6.2e-47)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 1.6e-58)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a / (c / d))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.2e-47], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.6e-58], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.1999999999999996e-47

   1. Initial program 53.1%

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

   3. Taylor expanded in c around inf 71.7%

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

      1. remove-double-neg 71.7%

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

      2. mul-1-neg 71.7%

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

      3. neg-mul-1 71.7%

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

      4. distribute-lft-in 71.7%

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

      5. mul-1-neg 71.7%

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

      6. distribute-neg-in 71.7%

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

      7. mul-1-neg 71.7%

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

      8. remove-double-neg 71.7%

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

      9. unsub-neg 71.7%

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

      10. associate-/l* 83.5%

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

   5. Simplified 83.5%

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

   if -6.1999999999999996e-47 < c < 1.6e-58

   1. Initial program 73.0%

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

      1. div-sub 69.1%

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

      2. *-commutative 69.1%

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

      3. add-sqr-sqrt 69.1%

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

      4. times-frac 69.7%

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

      5. fma-neg 69.7%

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

      6. hypot-define 69.7%

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

      7. hypot-define 72.0%

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

      8. associate-/l* 73.9%

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

      9. add-sqr-sqrt 73.9%

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

      10. pow2 73.9%

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

      11. hypot-define 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in d around inf 88.2%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

   if 1.6e-58 < c

   1. Initial program 64.7%

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

   3. Taylor expanded in c around inf 80.0%

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

      1. remove-double-neg 80.0%

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

      2. mul-1-neg 80.0%

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

      3. neg-mul-1 80.0%

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

      4. distribute-lft-in 80.0%

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

      5. mul-1-neg 80.0%

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

      6. distribute-neg-in 80.0%

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

      7. mul-1-neg 80.0%

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

      8. remove-double-neg 80.0%

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

      9. unsub-neg 80.0%

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

      10. associate-/l* 82.9%

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

   5. Simplified 82.9%

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

      1. clear-num 82.8%

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

      2. un-div-inv 82.9%

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

   7. Applied egg-rr 82.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{-65} \lor \neg \left(c \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e-65) (not (<= c 1.4e-56))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e-65) || !(c <= 1.4e-56)) {
		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 <= (-2.4d-65)) .or. (.not. (c <= 1.4d-56))) 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 <= -2.4e-65) || !(c <= 1.4e-56)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e-65) or not (c <= 1.4e-56):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e-65) || !(c <= 1.4e-56))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.4e-65) || ~((c <= 1.4e-56)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e-65], N[Not[LessEqual[c, 1.4e-56]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.4000000000000002e-65 or 1.39999999999999997e-56 < c

   1. Initial program 58.7%

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

   3. Taylor expanded in c around inf 62.2%

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

   if -2.4000000000000002e-65 < c < 1.39999999999999997e-56

   1. Initial program 73.2%

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

   3. Taylor expanded in c around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. neg-mul-1 70.2%

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

   5. Simplified 70.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{-65} \lor \neg \left(c \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+117} \lor \neg \left(d \leq 7.2 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.45e+117) (not (<= d 7.2e+95))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.45e+117) || !(d <= 7.2e+95)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.45d+117)) .or. (.not. (d <= 7.2d+95))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.45e+117) || !(d <= 7.2e+95)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.45e+117) or not (d <= 7.2e+95):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.45e+117) || !(d <= 7.2e+95))
		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 <= -1.45e+117) || ~((d <= 7.2e+95)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.45e+117], N[Not[LessEqual[d, 7.2e+95]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.45000000000000014e117 or 7.19999999999999955e95 < d

   1. Initial program 42.7%

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

   3. Taylor expanded in c around 0 73.4%

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

      1. +-commutative 73.4%

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

      2. mul-1-neg 73.4%

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

      3. unsub-neg 73.4%

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

      4. unpow2 73.4%

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

      5. associate-/r* 82.7%

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

      6. div-sub 82.7%

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

      7. *-commutative 82.7%

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

      8. associate-/l* 85.4%

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

   5. Simplified 85.4%

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

      1. clear-num 85.1%

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

      2. inv-pow 85.1%

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

      3. fma-neg 85.1%

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

      4. add-sqr-sqrt 40.1%

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

      5. sqrt-unprod 56.3%

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

      6. sqr-neg 56.3%

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

      7. sqrt-unprod 20.0%

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

      8. add-sqr-sqrt 42.7%

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

   7. Applied egg-rr 42.7%

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

      1. unpow-1 42.7%

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

   9. Simplified 42.7%

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

   10. Taylor expanded in d around inf 30.8%

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

   if -1.45000000000000014e117 < d < 7.19999999999999955e95

   1. Initial program 74.6%

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

   3. Taylor expanded in c around inf 53.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+117} \lor \neg \left(d \leq 7.2 \cdot 10^{+95}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 10.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in c around 0 44.9%

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

   1. +-commutative 44.9%

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

   2. mul-1-neg 44.9%

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

   3. unsub-neg 44.9%

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

   4. unpow2 44.9%

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

   5. associate-/r* 49.8%

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

   6. div-sub 49.9%

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

   7. *-commutative 49.9%

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

   8. associate-/l* 50.4%

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

5. Simplified 50.4%

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

   1. clear-num 50.3%

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

   2. inv-pow 50.3%

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

   3. fma-neg 50.3%

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

   4. add-sqr-sqrt 23.5%

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

   5. sqrt-unprod 30.9%

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

   6. sqr-neg 30.9%

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

   7. sqrt-unprod 10.8%

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

   8. add-sqr-sqrt 22.9%

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

7. Applied egg-rr 22.9%

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

   1. unpow-1 22.9%

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

9. Simplified 22.9%

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

10. Taylor expanded in d around inf 11.6%

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

11. Final simplification 11.6%

    \[\leadsto \frac{a}{d} \]
12. Add Preprocessing

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024059 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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