Complex division, imag part

Percentage Accurate: 62.4% → 91.5%

Time: 12.2s

Alternatives: 9

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: 62.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, d \cdot \frac{1}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+113}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{c \cdot b}{\mathsf{hypot}\left(c, d\right)} - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5e+102)
   (/ (fma a (* d (/ 1.0 c)) (- b)) (- c))
   (if (<= c 7.2e+113)
     (*
      (/ 1.0 (hypot c d))
      (- (/ (* c b) (hypot c d)) (* d (/ a (hypot c d)))))
     (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5e+102) {
		tmp = fma(a, (d * (1.0 / c)), -b) / -c;
	} else if (c <= 7.2e+113) {
		tmp = (1.0 / hypot(c, d)) * (((c * b) / hypot(c, d)) - (d * (a / hypot(c, d))));
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5e+102)
		tmp = Float64(fma(a, Float64(d * Float64(1.0 / c)), Float64(-b)) / Float64(-c));
	elseif (c <= 7.2e+113)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) / hypot(c, d)) - Float64(d * Float64(a / hypot(c, d)))));
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5e+102], N[(N[(a * N[(d * N[(1.0 / c), $MachinePrecision]), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, 7.2e+113], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] - N[(d * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -5e102

   1. Initial program 22.4%

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

      1. fmm-def 22.4%

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

      2. distribute-rgt-neg-out 22.4%

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

      3. +-commutative 22.4%

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

      4. fma-define 22.4%

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

   3. Simplified 22.4%

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

   5. Taylor expanded in c around -inf 80.2%

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

      1. mul-1-neg 80.2%

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

      2. distribute-neg-frac2 80.2%

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

      3. +-commutative 80.2%

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

      4. associate-/l* 86.2%

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

      5. fma-define 86.2%

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

      6. mul-1-neg 86.2%

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

   7. Simplified 86.2%

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

      1. div-inv 86.3%

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

   9. Applied egg-rr 86.3%

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

   if -5e102 < c < 7.19999999999999984e113

   1. Initial program 66.5%

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

      1. add-cube-cbrt 66.1%

         \[\leadsto \frac{b \cdot c - \color{blue}{\left(\sqrt[3]{a \cdot d} \cdot \sqrt[3]{a \cdot d}\right) \cdot \sqrt[3]{a \cdot d}}}{c \cdot c + d \cdot d} \]

      2. pow3 66.2%

         \[\leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^{3}}}{c \cdot c + d \cdot d} \]

   4. Applied egg-rr 66.2%

      \[\leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^{3}}}{c \cdot c + d \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 66.2%

         \[\leadsto \color{blue}{1 \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{c \cdot c + d \cdot d}} \]

      2. metadata-eval 66.2%

         \[\leadsto \color{blue}{\frac{1}{1}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{c \cdot c + d \cdot d} \]

      3. times-frac 66.2%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      4. *-un-lft-identity 66.2%

         \[\leadsto \frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{\color{blue}{c \cdot c + d \cdot d}} \]

      5. add-sqr-sqrt 66.2%

         \[\leadsto \frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      6. times-frac 66.1%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}}} \]

      7. hypot-define 66.1%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}} \]

      8. *-commutative 66.1%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{c \cdot b} - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}} \]

      9. rem-cube-cbrt 66.5%

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

      10. *-commutative 66.5%

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

      11. hypot-define 78.4%

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

   6. Applied egg-rr 78.4%

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

      1. div-sub 78.4%

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

      2. *-un-lft-identity 78.4%

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

      3. times-frac 93.7%

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

      4. /-rgt-identity 93.7%

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

   8. Applied egg-rr 93.7%

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

   if 7.19999999999999984e113 < c

   1. Initial program 31.6%

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

      1. fmm-def 31.6%

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

      2. distribute-rgt-neg-out 31.6%

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

      3. +-commutative 31.6%

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

      4. fma-define 31.6%

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

   3. Simplified 31.6%

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

   5. Taylor expanded in c around inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. unsub-neg 77.5%

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

      3. *-commutative 77.5%

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

      4. associate-/l* 87.1%

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

   7. Simplified 87.1%

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

Alternative 2: 86.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;d \leq -7.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-199}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (- (* c b) (* a d)) (hypot c d)))))
   (if (<= d -7.5e+129)
     (/ (- (/ b (/ d c)) a) d)
     (if (<= d -1.6e-54)
       t_0
       (if (<= d 5.4e-199)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 5e+83) t_0 (/ (- (* c (/ b d)) a) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (((c * b) - (a * d)) / hypot(c, d));
	double tmp;
	if (d <= -7.5e+129) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (d <= -1.6e-54) {
		tmp = t_0;
	} else if (d <= 5.4e-199) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 5e+83) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / Math.hypot(c, d)) * (((c * b) - (a * d)) / Math.hypot(c, d));
	double tmp;
	if (d <= -7.5e+129) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (d <= -1.6e-54) {
		tmp = t_0;
	} else if (d <= 5.4e-199) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 5e+83) {
		tmp = t_0;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / math.hypot(c, d)) * (((c * b) - (a * d)) / math.hypot(c, d))
	tmp = 0
	if d <= -7.5e+129:
		tmp = ((b / (d / c)) - a) / d
	elif d <= -1.6e-54:
		tmp = t_0
	elif d <= 5.4e-199:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 5e+83:
		tmp = t_0
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) - Float64(a * d)) / hypot(c, d)))
	tmp = 0.0
	if (d <= -7.5e+129)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (d <= -1.6e-54)
		tmp = t_0;
	elseif (d <= 5.4e-199)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 5e+83)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / hypot(c, d)) * (((c * b) - (a * d)) / hypot(c, d));
	tmp = 0.0;
	if (d <= -7.5e+129)
		tmp = ((b / (d / c)) - a) / d;
	elseif (d <= -1.6e-54)
		tmp = t_0;
	elseif (d <= 5.4e-199)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 5e+83)
		tmp = t_0;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7.5e+129], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -1.6e-54], t$95$0, If[LessEqual[d, 5.4e-199], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+83], t$95$0, N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -7.4999999999999998e129

   1. Initial program 30.4%

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

      1. fmm-def 30.4%

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

      2. distribute-rgt-neg-out 30.4%

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

      3. +-commutative 30.4%

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

      4. fma-define 30.4%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in d around -inf 77.5%

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

      1. mul-1-neg 77.5%

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

      2. distribute-neg-frac2 77.5%

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

      3. mul-1-neg 77.5%

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

      4. unsub-neg 77.5%

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

      5. *-commutative 77.5%

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

      6. associate-/l* 84.4%

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

   7. Simplified 84.4%

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

      1. clear-num 84.3%

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

      2. un-div-inv 84.3%

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

   9. Applied egg-rr 84.3%

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

   10. Taylor expanded in d around inf 77.5%

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

       1. +-commutative 77.5%

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

       2. neg-mul-1 77.5%

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

       3. sub-neg 77.5%

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

       4. associate-*r/ 84.3%

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

   12. Simplified 84.3%

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

       1. clear-num 84.3%

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

       2. un-div-inv 84.4%

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

   14. Applied egg-rr 84.4%

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

   if -7.4999999999999998e129 < d < -1.59999999999999999e-54 or 5.39999999999999979e-199 < d < 5.00000000000000029e83

   1. Initial program 74.4%

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

      1. add-cube-cbrt 73.9%

         \[\leadsto \frac{b \cdot c - \color{blue}{\left(\sqrt[3]{a \cdot d} \cdot \sqrt[3]{a \cdot d}\right) \cdot \sqrt[3]{a \cdot d}}}{c \cdot c + d \cdot d} \]

      2. pow3 73.9%

         \[\leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^{3}}}{c \cdot c + d \cdot d} \]

   4. Applied egg-rr 73.9%

      \[\leadsto \frac{b \cdot c - \color{blue}{{\left(\sqrt[3]{a \cdot d}\right)}^{3}}}{c \cdot c + d \cdot d} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 73.9%

         \[\leadsto \color{blue}{1 \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{c \cdot c + d \cdot d}} \]

      2. metadata-eval 73.9%

         \[\leadsto \color{blue}{\frac{1}{1}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{c \cdot c + d \cdot d} \]

      3. times-frac 73.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{1 \cdot \left(c \cdot c + d \cdot d\right)}} \]

      4. *-un-lft-identity 73.9%

         \[\leadsto \frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{\color{blue}{c \cdot c + d \cdot d}} \]

      5. add-sqr-sqrt 73.9%

         \[\leadsto \frac{1 \cdot \left(b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}\right)}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}} \]

      6. times-frac 73.7%

         \[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}}} \]

      7. hypot-define 73.8%

         \[\leadsto \frac{1}{\color{blue}{\mathsf{hypot}\left(c, d\right)}} \cdot \frac{b \cdot c - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}} \]

      8. *-commutative 73.8%

         \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\color{blue}{c \cdot b} - {\left(\sqrt[3]{a \cdot d}\right)}^{3}}{\sqrt{c \cdot c + d \cdot d}} \]

      9. rem-cube-cbrt 74.3%

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

      10. *-commutative 74.3%

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

      11. hypot-define 84.9%

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

   6. Applied egg-rr 84.9%

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

   if -1.59999999999999999e-54 < d < 5.39999999999999979e-199

   1. Initial program 57.5%

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

      1. fmm-def 57.5%

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

      2. distribute-rgt-neg-out 57.5%

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

      3. +-commutative 57.5%

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

      4. fma-define 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in c around inf 91.6%

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

      1. associate-*r/ 91.6%

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

      2. neg-mul-1 91.6%

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

      3. distribute-rgt-neg-in 91.6%

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

   7. Simplified 91.6%

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

   if 5.00000000000000029e83 < d

   1. Initial program 37.3%

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

      1. fmm-def 37.4%

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

      2. distribute-rgt-neg-out 37.4%

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

      3. +-commutative 37.4%

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

      4. fma-define 37.4%

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

   3. Simplified 37.4%

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

   5. Taylor expanded in d around -inf 81.3%

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

      1. mul-1-neg 81.3%

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

      2. distribute-neg-frac2 81.3%

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

      3. mul-1-neg 81.3%

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

      4. unsub-neg 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/l* 85.7%

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

   7. Simplified 85.7%

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

      1. clear-num 85.7%

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

      2. un-div-inv 85.7%

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

   9. Applied egg-rr 85.7%

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

   10. Taylor expanded in d around inf 81.3%

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

       1. +-commutative 81.3%

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

       2. neg-mul-1 81.3%

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

       3. sub-neg 81.3%

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

       4. associate-*r/ 85.6%

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

   12. Simplified 85.6%

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

       1. clear-num 85.7%

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

       2. un-div-inv 85.7%

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

   14. Applied egg-rr 85.7%

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

       1. associate-/r/ 85.7%

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

   16. Simplified 85.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.5 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;d \leq -1.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 5.4 \cdot 10^{-199}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.56 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-54}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -1.56e+25)
     t_1
     (if (<= d -3.3e-54)
       t_0
       (if (<= d 5.8e-130)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 5.2e+83) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.56e+25) {
		tmp = t_1;
	} else if (d <= -3.3e-54) {
		tmp = t_0;
	} else if (d <= 5.8e-130) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 5.2e+83) {
		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 = ((c * b) - (a * d)) / ((c * c) + (d * d))
    t_1 = ((c * (b / d)) - a) / d
    if (d <= (-1.56d+25)) then
        tmp = t_1
    else if (d <= (-3.3d-54)) then
        tmp = t_0
    else if (d <= 5.8d-130) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 5.2d+83) 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 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.56e+25) {
		tmp = t_1;
	} else if (d <= -3.3e-54) {
		tmp = t_0;
	} else if (d <= 5.8e-130) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 5.2e+83) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d))
	t_1 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -1.56e+25:
		tmp = t_1
	elif d <= -3.3e-54:
		tmp = t_0
	elif d <= 5.8e-130:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 5.2e+83:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -1.56e+25)
		tmp = t_1;
	elseif (d <= -3.3e-54)
		tmp = t_0;
	elseif (d <= 5.8e-130)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 5.2e+83)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (a * d)) / ((c * c) + (d * d));
	t_1 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -1.56e+25)
		tmp = t_1;
	elseif (d <= -3.3e-54)
		tmp = t_0;
	elseif (d <= 5.8e-130)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 5.2e+83)
		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[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.56e+25], t$95$1, If[LessEqual[d, -3.3e-54], t$95$0, If[LessEqual[d, 5.8e-130], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.2e+83], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.5599999999999999e25 or 5.2000000000000002e83 < d

   1. Initial program 38.2%

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

      1. fmm-def 38.2%

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

      2. distribute-rgt-neg-out 38.2%

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

      3. +-commutative 38.2%

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

      4. fma-define 38.2%

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

   3. Simplified 38.2%

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

   5. Taylor expanded in d around -inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. distribute-neg-frac2 77.2%

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

      3. mul-1-neg 77.2%

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

      4. unsub-neg 77.2%

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

      5. *-commutative 77.2%

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

      6. associate-/l* 82.0%

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

   7. Simplified 82.0%

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

      1. clear-num 82.0%

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

      2. un-div-inv 82.0%

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

   9. Applied egg-rr 82.0%

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

   10. Taylor expanded in d around inf 77.2%

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

       1. +-commutative 77.2%

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

       2. neg-mul-1 77.2%

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

       3. sub-neg 77.2%

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

       4. associate-*r/ 81.1%

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

   12. Simplified 81.1%

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

       1. clear-num 81.1%

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

       2. un-div-inv 81.1%

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

   14. Applied egg-rr 81.1%

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

       1. associate-/r/ 82.0%

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

   16. Simplified 82.0%

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

   if -1.5599999999999999e25 < d < -3.29999999999999993e-54 or 5.8e-130 < d < 5.2000000000000002e83

   1. Initial program 80.4%

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

   if -3.29999999999999993e-54 < d < 5.8e-130

   1. Initial program 58.5%

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

      1. fmm-def 58.5%

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

      2. distribute-rgt-neg-out 58.5%

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

      3. +-commutative 58.5%

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

      4. fma-define 58.5%

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

   3. Simplified 58.5%

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

   5. Taylor expanded in c around inf 90.8%

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

      1. associate-*r/ 90.8%

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

      2. neg-mul-1 90.8%

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

      3. distribute-rgt-neg-in 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.56 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-54}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 5.8 \cdot 10^{-130}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+28} \lor \neg \left(c \leq 1.32 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.25e+28) (not (<= c 1.32e+44)))
   (/ (- b (* d (/ a c))) c)
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.25e+28) || !(c <= 1.32e+44)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((c * (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 <= (-1.25d+28)) .or. (.not. (c <= 1.32d+44))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = ((c * (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 <= -1.25e+28) || !(c <= 1.32e+44)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.25e+28) or not (c <= 1.32e+44):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.25e+28) || !(c <= 1.32e+44))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.25e+28) || ~((c <= 1.32e+44)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.25e+28], N[Not[LessEqual[c, 1.32e+44]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.24999999999999989e28 or 1.3200000000000001e44 < c

   1. Initial program 39.2%

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

      1. fmm-def 39.2%

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

      2. distribute-rgt-neg-out 39.2%

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

      3. +-commutative 39.2%

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

      4. fma-define 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in c around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-/l* 79.9%

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

   7. Simplified 79.9%

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

   if -1.24999999999999989e28 < c < 1.3200000000000001e44

   1. Initial program 65.3%

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

      1. fmm-def 65.4%

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

      2. distribute-rgt-neg-out 65.4%

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

      3. +-commutative 65.4%

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

      4. fma-define 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in d around -inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-neg-frac2 81.7%

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

      3. mul-1-neg 81.7%

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

      4. unsub-neg 81.7%

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

      5. *-commutative 81.7%

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

      6. associate-/l* 81.7%

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

   7. Simplified 81.7%

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

      1. clear-num 81.7%

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

      2. un-div-inv 81.7%

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

   9. Applied egg-rr 81.7%

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

   10. Taylor expanded in d around inf 81.7%

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

       1. +-commutative 81.7%

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

       2. neg-mul-1 81.7%

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

       3. sub-neg 81.7%

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

       4. associate-*r/ 81.5%

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

   12. Simplified 81.5%

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

       1. clear-num 81.5%

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

       2. un-div-inv 81.5%

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

   14. Applied egg-rr 81.5%

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

       1. associate-/r/ 81.7%

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

   16. Simplified 81.7%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.25 \cdot 10^{+28} \lor \neg \left(c \leq 1.32 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+27} \lor \neg \left(c \leq 1.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.6e+27) (not (<= c 1.3e+44)))
   (/ (- b (* d (/ a c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.6e+27) || !(c <= 1.3e+44)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-7.6d+27)) .or. (.not. (c <= 1.3d+44))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.6e+27) || !(c <= 1.3e+44)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.6e+27) or not (c <= 1.3e+44):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.6e+27) || !(c <= 1.3e+44))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -7.6e+27) || ~((c <= 1.3e+44)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.6e+27], N[Not[LessEqual[c, 1.3e+44]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.60000000000000043e27 or 1.3e44 < c

   1. Initial program 39.2%

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

      1. fmm-def 39.2%

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

      2. distribute-rgt-neg-out 39.2%

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

      3. +-commutative 39.2%

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

      4. fma-define 39.2%

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

   3. Simplified 39.2%

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

   5. Taylor expanded in c around inf 73.9%

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

      1. mul-1-neg 73.9%

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

      2. unsub-neg 73.9%

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

      3. *-commutative 73.9%

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

      4. associate-/l* 79.9%

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

   7. Simplified 79.9%

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

   if -7.60000000000000043e27 < c < 1.3e44

   1. Initial program 65.3%

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

      1. fmm-def 65.4%

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

      2. distribute-rgt-neg-out 65.4%

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

      3. +-commutative 65.4%

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

      4. fma-define 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in d around -inf 81.7%

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

      1. mul-1-neg 81.7%

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

      2. distribute-neg-frac2 81.7%

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

      3. mul-1-neg 81.7%

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

      4. unsub-neg 81.7%

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

      5. *-commutative 81.7%

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

      6. associate-/l* 81.7%

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

   7. Simplified 81.7%

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

      1. clear-num 81.7%

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

      2. un-div-inv 81.7%

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

   9. Applied egg-rr 81.7%

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

   10. Taylor expanded in d around inf 81.7%

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

       1. +-commutative 81.7%

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

       2. neg-mul-1 81.7%

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

       3. sub-neg 81.7%

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

       4. associate-*r/ 81.5%

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

   12. Simplified 81.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+27} \lor \neg \left(c \leq 1.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+49} \lor \neg \left(d \leq 1.5 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.25e+49) (not (<= d 1.5e+76)))
   (/ (- a) d)
   (/ (- b (* d (/ a c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.25e+49) || !(d <= 1.5e+76)) {
		tmp = -a / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.25d+49)) .or. (.not. (d <= 1.5d+76))) then
        tmp = -a / d
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.25e+49) || !(d <= 1.5e+76)) {
		tmp = -a / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.25e+49) or not (d <= 1.5e+76):
		tmp = -a / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.25e+49) || !(d <= 1.5e+76))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.25e+49) || ~((d <= 1.5e+76)))
		tmp = -a / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.25e+49], N[Not[LessEqual[d, 1.5e+76]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.2500000000000001e49 or 1.4999999999999999e76 < d

   1. Initial program 41.5%

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

      1. fmm-def 41.5%

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

      2. distribute-rgt-neg-out 41.5%

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

      3. +-commutative 41.5%

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

      4. fma-define 41.5%

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

   3. Simplified 41.5%

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

   5. Taylor expanded in c around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   7. Simplified 71.2%

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

   if -1.2500000000000001e49 < d < 1.4999999999999999e76

   1. Initial program 63.7%

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

      1. fmm-def 63.7%

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

      2. distribute-rgt-neg-out 63.7%

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

      3. +-commutative 63.7%

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

      4. fma-define 63.7%

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

   3. Simplified 63.7%

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

   5. Taylor expanded in c around inf 77.1%

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

      1. mul-1-neg 77.1%

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

      2. unsub-neg 77.1%

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

      3. *-commutative 77.1%

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

      4. associate-/l* 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{+49} \lor \neg \left(d \leq 1.5 \cdot 10^{+76}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+49} \lor \neg \left(d \leq 1.25 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.6e+49) (not (<= d 1.25e-116))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.6e+49) || !(d <= 1.25e-116)) {
		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 <= (-2.6d+49)) .or. (.not. (d <= 1.25d-116))) 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 <= -2.6e+49) || !(d <= 1.25e-116)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.6e+49) or not (d <= 1.25e-116):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.6e+49) || !(d <= 1.25e-116))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.6e+49) || ~((d <= 1.25e-116)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.6e+49], N[Not[LessEqual[d, 1.25e-116]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.59999999999999989e49 or 1.2500000000000001e-116 < d

   1. Initial program 48.2%

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

      1. fmm-def 48.3%

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

      2. distribute-rgt-neg-out 48.3%

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

      3. +-commutative 48.3%

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

      4. fma-define 48.3%

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

   3. Simplified 48.3%

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

   5. Taylor expanded in c around 0 64.4%

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

      1. associate-*r/ 64.4%

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

      2. neg-mul-1 64.4%

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

   7. Simplified 64.4%

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

   if -2.59999999999999989e49 < d < 1.2500000000000001e-116

   1. Initial program 61.5%

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

      1. fmm-def 61.5%

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

      2. distribute-rgt-neg-out 61.5%

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

      3. +-commutative 61.5%

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

      4. fma-define 61.5%

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

   3. Simplified 61.5%

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

   5. Taylor expanded in c around inf 70.2%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+49} \lor \neg \left(d \leq 1.25 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+138} \lor \neg \left(d \leq 2.8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7e+138) (not (<= d 2.8e+184))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7e+138) || !(d <= 2.8e+184)) {
		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 <= (-7d+138)) .or. (.not. (d <= 2.8d+184))) 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 <= -7e+138) || !(d <= 2.8e+184)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7e+138) or not (d <= 2.8e+184):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7e+138) || !(d <= 2.8e+184))
		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 <= -7e+138) || ~((d <= 2.8e+184)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7e+138], N[Not[LessEqual[d, 2.8e+184]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.9999999999999996e138 or 2.7999999999999999e184 < d

   1. Initial program 28.3%

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

      1. fmm-def 28.3%

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

      2. distribute-rgt-neg-out 28.3%

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

      3. +-commutative 28.3%

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

      4. fma-define 28.3%

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

   3. Simplified 28.3%

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

      1. div-inv 28.3%

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

      2. fma-undefine 28.3%

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

      3. +-commutative 28.3%

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

      4. add-sqr-sqrt 16.5%

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

      5. sqrt-unprod 25.9%

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

      6. distribute-rgt-neg-out 25.9%

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

      7. distribute-rgt-neg-out 25.9%

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

      8. sqr-neg 25.9%

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

      9. sqrt-unprod 11.8%

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

      10. add-sqr-sqrt 27.5%

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

      11. add-sqr-sqrt 27.5%

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

      12. pow2 27.5%

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

      13. hypot-define 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in c around 0 28.7%

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

   if -6.9999999999999996e138 < d < 2.7999999999999999e184

   1. Initial program 62.7%

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

      1. fmm-def 62.7%

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

      2. distribute-rgt-neg-out 62.7%

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

      3. +-commutative 62.7%

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

      4. fma-define 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in c around inf 51.8%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+138} \lor \neg \left(d \leq 2.8 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 11.4% 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 54.2%

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

   1. fmm-def 54.2%

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

   2. distribute-rgt-neg-out 54.2%

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

   3. +-commutative 54.2%

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

   4. fma-define 54.2%

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

3. Simplified 54.2%

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

   1. div-inv 54.1%

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

   2. fma-undefine 54.1%

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

   3. +-commutative 54.1%

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

   4. add-sqr-sqrt 31.3%

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

   5. sqrt-unprod 38.4%

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

   6. distribute-rgt-neg-out 38.4%

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

   7. distribute-rgt-neg-out 38.4%

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

   8. sqr-neg 38.4%

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

   9. sqrt-unprod 20.0%

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

   10. add-sqr-sqrt 34.0%

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

   11. add-sqr-sqrt 34.0%

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

   12. pow2 34.0%

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

   13. hypot-define 34.0%

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

6. Applied egg-rr 34.0%

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

7. Taylor expanded in c around 0 10.0%

   \[\leadsto \color{blue}{\frac{a}{d}} \]
8. Add Preprocessing

Developer Target 1: 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 2024165 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))