Complex division, imag part

Percentage Accurate: 61.5% → 78.3%

Time: 8.9s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% 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: 78.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 1.72 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ c (* d (/ d b))) (/ a d))))
   (if (<= d -4.8e+91)
     t_0
     (if (<= d 1.72e-69)
       (- (/ b c) (/ (/ a (/ c d)) c))
       (if (<= d 1.9e+45) (/ (fma b c (* d (- a))) (fma c c (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c / (d * (d / b))) - (a / d);
	double tmp;
	if (d <= -4.8e+91) {
		tmp = t_0;
	} else if (d <= 1.72e-69) {
		tmp = (b / c) - ((a / (c / d)) / c);
	} else if (d <= 1.9e+45) {
		tmp = fma(b, c, (d * -a)) / fma(c, c, (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c / Float64(d * Float64(d / b))) - Float64(a / d))
	tmp = 0.0
	if (d <= -4.8e+91)
		tmp = t_0;
	elseif (d <= 1.72e-69)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / Float64(c / d)) / c));
	elseif (d <= 1.9e+45)
		tmp = Float64(fma(b, c, Float64(d * Float64(-a))) / fma(c, c, Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[(d * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -4.8e+91], t$95$0, If[LessEqual[d, 1.72e-69], N[(N[(b / c), $MachinePrecision] - N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e+45], N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.79999999999999966e91 or 1.9000000000000001e45 < d

   1. Initial program 41.8%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

      1. pow2 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

   if -4.79999999999999966e91 < d < 1.72e-69

   1. Initial program 66.7%

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

   3. Taylor expanded in c around inf 72.7%

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

      1. +-commutative 72.7%

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

      2. mul-1-neg 72.7%

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

      3. unsub-neg 72.7%

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

      4. associate-/l* 72.1%

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

      5. associate-/r/ 65.4%

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

   5. Simplified 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. pow2 65.4%

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

      3. times-frac 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-un-lft-identity 77.5%

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

      4. associate-/r/ 80.1%

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

   9. Applied egg-rr 80.1%

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

   if 1.72e-69 < d < 1.9000000000000001e45

   1. Initial program 76.7%

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

      1. fma-neg 76.8%

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

      2. distribute-lft-neg-out 76.8%

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

      3. *-commutative 76.8%

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

      4. fma-def 76.8%

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

   3. Simplified 76.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.72 \cdot 10^{-69}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+45}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ c (* d (/ d b))) (/ a d))))
   (if (<= d -5e+91)
     t_0
     (if (<= d 4.5e-72)
       (- (/ b c) (/ (/ a (/ c d)) c))
       (if (<= d 1.55e+44) (/ (- (* c b) (* d a)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c / (d * (d / b))) - (a / d);
	double tmp;
	if (d <= -5e+91) {
		tmp = t_0;
	} else if (d <= 4.5e-72) {
		tmp = (b / c) - ((a / (c / d)) / c);
	} else if (d <= 1.55e+44) {
		tmp = ((c * b) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (c / (d * (d / b))) - (a / d)
    if (d <= (-5d+91)) then
        tmp = t_0
    else if (d <= 4.5d-72) then
        tmp = (b / c) - ((a / (c / d)) / c)
    else if (d <= 1.55d+44) then
        tmp = ((c * b) - (d * a)) / ((d * d) + (c * c))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (c / (d * (d / b))) - (a / d);
	double tmp;
	if (d <= -5e+91) {
		tmp = t_0;
	} else if (d <= 4.5e-72) {
		tmp = (b / c) - ((a / (c / d)) / c);
	} else if (d <= 1.55e+44) {
		tmp = ((c * b) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c / (d * (d / b))) - (a / d)
	tmp = 0
	if d <= -5e+91:
		tmp = t_0
	elif d <= 4.5e-72:
		tmp = (b / c) - ((a / (c / d)) / c)
	elif d <= 1.55e+44:
		tmp = ((c * b) - (d * a)) / ((d * d) + (c * c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c / Float64(d * Float64(d / b))) - Float64(a / d))
	tmp = 0.0
	if (d <= -5e+91)
		tmp = t_0;
	elseif (d <= 4.5e-72)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / Float64(c / d)) / c));
	elseif (d <= 1.55e+44)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (c / (d * (d / b))) - (a / d);
	tmp = 0.0;
	if (d <= -5e+91)
		tmp = t_0;
	elseif (d <= 4.5e-72)
		tmp = (b / c) - ((a / (c / d)) / c);
	elseif (d <= 1.55e+44)
		tmp = ((c * b) - (d * a)) / ((d * d) + (c * c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[(d * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e+91], t$95$0, If[LessEqual[d, 4.5e-72], N[(N[(b / c), $MachinePrecision] - N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.55e+44], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.0000000000000002e91 or 1.54999999999999998e44 < d

   1. Initial program 41.8%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

      1. pow2 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

   if -5.0000000000000002e91 < d < 4.5e-72

   1. Initial program 66.7%

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

   3. Taylor expanded in c around inf 72.7%

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

      1. +-commutative 72.7%

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

      2. mul-1-neg 72.7%

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

      3. unsub-neg 72.7%

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

      4. associate-/l* 72.1%

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

      5. associate-/r/ 65.4%

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

   5. Simplified 65.4%

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

      1. *-un-lft-identity 65.4%

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

      2. pow2 65.4%

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

      3. times-frac 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. associate-*l* 77.5%

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

      2. associate-*l/ 77.5%

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

      3. *-un-lft-identity 77.5%

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

      4. associate-/r/ 80.1%

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

   9. Applied egg-rr 80.1%

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

   if 4.5e-72 < d < 1.54999999999999998e44

   1. Initial program 76.7%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \mathbf{elif}\;d \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -0.000235 \lor \neg \left(c \leq -3.55 \cdot 10^{-168}\right) \land \left(c \leq -6.2 \cdot 10^{-202} \lor \neg \left(c \leq 360000000000\right)\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -0.000235)
         (and (not (<= c -3.55e-168))
              (or (<= c -6.2e-202) (not (<= c 360000000000.0)))))
   (- (/ b c) (* d (/ (/ a c) c)))
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -0.000235) || (!(c <= -3.55e-168) && ((c <= -6.2e-202) || !(c <= 360000000000.0)))) {
		tmp = (b / c) - (d * ((a / 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 <= (-0.000235d0)) .or. (.not. (c <= (-3.55d-168))) .and. (c <= (-6.2d-202)) .or. (.not. (c <= 360000000000.0d0))) then
        tmp = (b / c) - (d * ((a / 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 <= -0.000235) || (!(c <= -3.55e-168) && ((c <= -6.2e-202) || !(c <= 360000000000.0)))) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -0.000235) or (not (c <= -3.55e-168) and ((c <= -6.2e-202) or not (c <= 360000000000.0))):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -0.000235) || (!(c <= -3.55e-168) && ((c <= -6.2e-202) || !(c <= 360000000000.0))))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -0.000235) || (~((c <= -3.55e-168)) && ((c <= -6.2e-202) || ~((c <= 360000000000.0)))))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -0.000235], And[N[Not[LessEqual[c, -3.55e-168]], $MachinePrecision], Or[LessEqual[c, -6.2e-202], N[Not[LessEqual[c, 360000000000.0]], $MachinePrecision]]]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.34999999999999993e-4 or -3.55000000000000009e-168 < c < -6.2e-202 or 3.6e11 < c

   1. Initial program 47.6%

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

   3. Taylor expanded in c around inf 62.5%

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

      1. +-commutative 62.5%

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

      2. mul-1-neg 62.5%

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

      3. unsub-neg 62.5%

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

      4. associate-/l* 62.2%

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

      5. associate-/r/ 64.7%

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

   5. Simplified 64.7%

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

      1. *-un-lft-identity 64.7%

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

      2. pow2 64.7%

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

      3. times-frac 73.9%

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

   7. Applied egg-rr 73.9%

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

      1. associate-*l/ 73.9%

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

      2. *-lft-identity 73.9%

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

   9. Simplified 73.9%

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

   if -2.34999999999999993e-4 < c < -3.55000000000000009e-168 or -6.2e-202 < c < 3.6e11

   1. Initial program 70.5%

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

   3. Taylor expanded in c around 0 68.9%

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

      1. associate-*r/ 68.9%

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

      2. neg-mul-1 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.000235 \lor \neg \left(c \leq -3.55 \cdot 10^{-168}\right) \land \left(c \leq -6.2 \cdot 10^{-202} \lor \neg \left(c \leq 360000000000\right)\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+93} \lor \neg \left(d \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.9e+93) (not (<= d 9e+43)))
   (/ (- a) d)
   (- (/ b c) (/ (/ a (/ c d)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e+93) || !(d <= 9e+43)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((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 ((d <= (-1.9d+93)) .or. (.not. (d <= 9d+43))) then
        tmp = -a / d
    else
        tmp = (b / c) - ((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 ((d <= -1.9e+93) || !(d <= 9e+43)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((a / (c / d)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.9e+93) or not (d <= 9e+43):
		tmp = -a / d
	else:
		tmp = (b / c) - ((a / (c / d)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.9e+93) || !(d <= 9e+43))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / Float64(c / d)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.9e+93) || ~((d <= 9e+43)))
		tmp = -a / d;
	else
		tmp = (b / c) - ((a / (c / d)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.9e+93], N[Not[LessEqual[d, 9e+43]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.8999999999999999e93 or 9e43 < d

   1. Initial program 41.3%

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

   3. Taylor expanded in c around 0 71.2%

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

      1. associate-*r/ 71.2%

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

      2. neg-mul-1 71.2%

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

   5. Simplified 71.2%

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

   if -1.8999999999999999e93 < d < 9e43

   1. Initial program 69.3%

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

   3. Taylor expanded in c around inf 69.5%

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

      1. +-commutative 69.5%

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

      2. mul-1-neg 69.5%

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

      3. unsub-neg 69.5%

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

      4. associate-/l* 69.0%

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

      5. associate-/r/ 63.9%

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

   5. Simplified 63.9%

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

      1. *-un-lft-identity 63.9%

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

      2. pow2 63.9%

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

      3. times-frac 68.5%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*l* 73.8%

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

      2. associate-*l/ 73.8%

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

      3. *-un-lft-identity 73.8%

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

      4. associate-/r/ 75.8%

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

   9. Applied egg-rr 75.8%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+93} \lor \neg \left(d \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+91} \lor \neg \left(d \leq 5.2 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.8e+91) (not (<= d 5.2e+43)))
   (- (/ c (* d (/ d b))) (/ a d))
   (- (/ b c) (/ (/ a (/ c d)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.8e+91) || !(d <= 5.2e+43)) {
		tmp = (c / (d * (d / b))) - (a / d);
	} else {
		tmp = (b / c) - ((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 ((d <= (-4.8d+91)) .or. (.not. (d <= 5.2d+43))) then
        tmp = (c / (d * (d / b))) - (a / d)
    else
        tmp = (b / c) - ((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 ((d <= -4.8e+91) || !(d <= 5.2e+43)) {
		tmp = (c / (d * (d / b))) - (a / d);
	} else {
		tmp = (b / c) - ((a / (c / d)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.8e+91) or not (d <= 5.2e+43):
		tmp = (c / (d * (d / b))) - (a / d)
	else:
		tmp = (b / c) - ((a / (c / d)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.8e+91) || !(d <= 5.2e+43))
		tmp = Float64(Float64(c / Float64(d * Float64(d / b))) - Float64(a / d));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / Float64(c / d)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.8e+91) || ~((d <= 5.2e+43)))
		tmp = (c / (d * (d / b))) - (a / d);
	else
		tmp = (b / c) - ((a / (c / d)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.8e+91], N[Not[LessEqual[d, 5.2e+43]], $MachinePrecision]], N[(N[(c / N[(d * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.79999999999999966e91 or 5.20000000000000042e43 < d

   1. Initial program 41.8%

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

   3. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. *-commutative 74.5%

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

      5. associate-/l* 76.7%

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

   5. Simplified 76.7%

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

      1. pow2 76.7%

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

      2. *-un-lft-identity 76.7%

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

      3. times-frac 80.2%

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

   7. Applied egg-rr 80.2%

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

   if -4.79999999999999966e91 < d < 5.20000000000000042e43

   1. Initial program 69.1%

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

   3. Taylor expanded in c around inf 69.9%

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

      1. +-commutative 69.9%

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

      2. mul-1-neg 69.9%

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

      3. unsub-neg 69.9%

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

      4. associate-/l* 69.5%

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

      5. associate-/r/ 64.3%

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

   5. Simplified 64.3%

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

      1. *-un-lft-identity 64.3%

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

      2. pow2 64.3%

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

      3. times-frac 68.9%

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

   7. Applied egg-rr 68.9%

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

      1. associate-*l* 74.3%

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

      2. associate-*l/ 74.3%

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

      3. *-un-lft-identity 74.3%

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

      4. associate-/r/ 76.3%

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

   9. Applied egg-rr 76.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+91} \lor \neg \left(d \leq 5.2 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+44} \lor \neg \left(d \leq 3.1 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.7e+44) (not (<= d 3.1e+41))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.7e+44) || !(d <= 3.1e+41)) {
		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 <= (-1.7d+44)) .or. (.not. (d <= 3.1d+41))) 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 <= -1.7e+44) || !(d <= 3.1e+41)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.7e+44) or not (d <= 3.1e+41):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.7e+44) || !(d <= 3.1e+41))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.7e+44) || ~((d <= 3.1e+41)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.7e+44], N[Not[LessEqual[d, 3.1e+41]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.7e44 or 3.1e41 < d

   1. Initial program 44.1%

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

   3. Taylor expanded in c around 0 68.6%

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

      1. associate-*r/ 68.6%

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

      2. neg-mul-1 68.6%

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

   5. Simplified 68.6%

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

   if -1.7e44 < d < 3.1e41

   1. Initial program 68.7%

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

   3. Taylor expanded in c around inf 62.9%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.7 \cdot 10^{+44} \lor \neg \left(d \leq 3.1 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 12.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -1.2e+108) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.2e+108) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.2d+108)) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.2e+108) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.2e+108:
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.2e+108)
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.2e+108)
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.2e+108], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.20000000000000009e108

   1. Initial program 39.7%

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

      1. fma-neg 39.7%

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

      2. *-commutative 39.7%

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

      3. distribute-rgt-neg-out 39.7%

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

      4. add-sqr-sqrt 39.7%

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

      5. *-un-lft-identity 39.7%

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

      6. times-frac 39.7%

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

      7. hypot-def 39.7%

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

      8. add-sqr-sqrt 24.8%

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

      9. sqrt-unprod 37.5%

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

      10. sqr-neg 37.5%

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

      11. sqrt-unprod 12.7%

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

      12. add-sqr-sqrt 33.3%

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

      13. *-commutative 33.3%

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

      14. hypot-def 37.3%

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

   4. Applied egg-rr 37.3%

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

   5. Taylor expanded in c around 0 32.0%

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

   if -1.20000000000000009e108 < d

   1. Initial program 61.3%

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

      1. fma-neg 61.3%

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

      2. *-commutative 61.3%

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

      3. distribute-rgt-neg-out 61.3%

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

      4. add-sqr-sqrt 61.3%

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

      5. *-un-lft-identity 61.3%

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

      6. times-frac 61.2%

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

      7. hypot-def 61.2%

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

      8. add-sqr-sqrt 28.8%

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

      9. sqrt-unprod 42.1%

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

      10. sqr-neg 42.1%

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

      11. sqrt-unprod 19.7%

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

      12. add-sqr-sqrt 34.1%

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

      13. *-commutative 34.1%

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

      14. hypot-def 41.7%

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

   4. Applied egg-rr 41.7%

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

   5. Taylor expanded in c around inf 35.8%

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

   6. Taylor expanded in c around 0 9.4%

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

4. Final simplification 13.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -1.05e+114) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.05e+114) {
		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 <= (-1.05d+114)) 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 <= -1.05e+114) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.05e+114:
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.05e+114)
		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 <= -1.05e+114)
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.05e+114], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.05e114

   1. Initial program 40.5%

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

      1. fma-neg 40.5%

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

      2. *-commutative 40.5%

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

      3. distribute-rgt-neg-out 40.5%

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

      4. add-sqr-sqrt 40.5%

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

      5. *-un-lft-identity 40.5%

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

      6. times-frac 40.5%

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

      7. hypot-def 40.5%

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

      8. add-sqr-sqrt 25.3%

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

      9. sqrt-unprod 38.2%

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

      10. sqr-neg 38.2%

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

      11. sqrt-unprod 12.9%

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

      12. add-sqr-sqrt 34.0%

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

      13. *-commutative 34.0%

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

      14. hypot-def 38.1%

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

   4. Applied egg-rr 38.1%

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

   5. Taylor expanded in c around 0 32.7%

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

   if -1.05e114 < d

   1. Initial program 61.0%

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

   3. Taylor expanded in c around inf 46.7%

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

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.05 \cdot 10^{+114}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.3%

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

   1. fma-neg 57.3%

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

   2. *-commutative 57.3%

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

   3. distribute-rgt-neg-out 57.3%

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

   4. add-sqr-sqrt 57.3%

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

   5. *-un-lft-identity 57.3%

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

   6. times-frac 57.3%

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

   7. hypot-def 57.3%

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

   8. add-sqr-sqrt 28.0%

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

   9. sqrt-unprod 41.2%

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

   10. sqr-neg 41.2%

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

   11. sqrt-unprod 18.4%

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

   12. add-sqr-sqrt 34.0%

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

   13. *-commutative 34.0%

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

   14. hypot-def 40.9%

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

4. Applied egg-rr 40.9%

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

5. Taylor expanded in c around inf 32.0%

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

6. Taylor expanded in c around 0 10.2%

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

7. Final simplification 10.2%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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