Complex division, imag part

Percentage Accurate: 61.3% → 81.6%

Time: 8.9s

Alternatives: 10

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: 61.3% 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: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -7.8 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -7.8e+150)
     t_0
     (if (<= d -7.5e-131)
       (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))
       (if (<= d 4.1e-25)
         (- (/ b c) (/ (* a (/ d c)) c))
         (if (<= d 1.2e+133)
           (/ (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 * (b / d)) - a) / d;
	double tmp;
	if (d <= -7.8e+150) {
		tmp = t_0;
	} else if (d <= -7.5e-131) {
		tmp = ((c * b) - (d * a)) / ((c * c) + (d * d));
	} else if (d <= 4.1e-25) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else if (d <= 1.2e+133) {
		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(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -7.8e+150)
		tmp = t_0;
	elseif (d <= -7.5e-131)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 4.1e-25)
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	elseif (d <= 1.2e+133)
		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[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -7.8e+150], t$95$0, If[LessEqual[d, -7.5e-131], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.1e-25], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.2e+133], 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 4 regimes

2. if d < -7.79999999999999981e150 or 1.1999999999999999e133 < d

   1. Initial program 27.2%

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

   2. Taylor expanded in c around 0 77.0%

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

      1. +-commutative 77.0%

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

      2. mul-1-neg 77.0%

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

      3. unsub-neg 77.0%

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

      4. *-commutative 77.0%

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

   4. Simplified 77.0%

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

      1. pow2 77.0%

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

      2. times-frac 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. associate-*l/ 88.5%

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

      2. sub-div 88.5%

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

   8. Applied egg-rr 88.5%

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

   if -7.79999999999999981e150 < d < -7.49999999999999964e-131

   1. Initial program 81.9%

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

   if -7.49999999999999964e-131 < d < 4.09999999999999987e-25

   1. Initial program 63.1%

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

   2. Taylor expanded in c around inf 79.6%

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

      1. *-commutative 79.6%

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

      2. pow2 79.6%

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

      3. times-frac 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-*r/ 86.8%

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

   6. Applied egg-rr 86.8%

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

   if 4.09999999999999987e-25 < d < 1.1999999999999999e133

   1. Initial program 79.5%

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

      1. fma-neg 79.6%

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

      2. distribute-lft-neg-out 79.6%

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

      3. *-commutative 79.6%

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

      4. fma-def 79.6%

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

   3. Simplified 79.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.8 \cdot 10^{+150}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -7.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.1 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 2: 81.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 3.85 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -1.9e+151)
     t_1
     (if (<= d -1.45e-130)
       t_0
       (if (<= d 3.85e-25)
         (- (/ b c) (/ (* a (/ d c)) c))
         (if (<= d 2.8e+132) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.9e+151) {
		tmp = t_1;
	} else if (d <= -1.45e-130) {
		tmp = t_0;
	} else if (d <= 3.85e-25) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else if (d <= 2.8e+132) {
		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) - (d * a)) / ((c * c) + (d * d))
    t_1 = ((c * (b / d)) - a) / d
    if (d <= (-1.9d+151)) then
        tmp = t_1
    else if (d <= (-1.45d-130)) then
        tmp = t_0
    else if (d <= 3.85d-25) then
        tmp = (b / c) - ((a * (d / c)) / c)
    else if (d <= 2.8d+132) 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) - (d * a)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.9e+151) {
		tmp = t_1;
	} else if (d <= -1.45e-130) {
		tmp = t_0;
	} else if (d <= 3.85e-25) {
		tmp = (b / c) - ((a * (d / c)) / c);
	} else if (d <= 2.8e+132) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -1.9e+151:
		tmp = t_1
	elif d <= -1.45e-130:
		tmp = t_0
	elif d <= 3.85e-25:
		tmp = (b / c) - ((a * (d / c)) / c)
	elif d <= 2.8e+132:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -1.9e+151)
		tmp = t_1;
	elseif (d <= -1.45e-130)
		tmp = t_0;
	elseif (d <= 3.85e-25)
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	elseif (d <= 2.8e+132)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -1.9e+151)
		tmp = t_1;
	elseif (d <= -1.45e-130)
		tmp = t_0;
	elseif (d <= 3.85e-25)
		tmp = (b / c) - ((a * (d / c)) / c);
	elseif (d <= 2.8e+132)
		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[(d * a), $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, -1.9e+151], t$95$1, If[LessEqual[d, -1.45e-130], t$95$0, If[LessEqual[d, 3.85e-25], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.8e+132], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.9e151 or 2.7999999999999999e132 < d

   1. Initial program 27.2%

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

   2. Taylor expanded in c around 0 77.0%

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

      1. +-commutative 77.0%

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

      2. mul-1-neg 77.0%

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

      3. unsub-neg 77.0%

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

      4. *-commutative 77.0%

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

   4. Simplified 77.0%

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

      1. pow2 77.0%

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

      2. times-frac 88.5%

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

   6. Applied egg-rr 88.5%

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

      1. associate-*l/ 88.5%

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

      2. sub-div 88.5%

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

   8. Applied egg-rr 88.5%

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

   if -1.9e151 < d < -1.45e-130 or 3.8500000000000001e-25 < d < 2.7999999999999999e132

   1. Initial program 80.7%

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

   if -1.45e-130 < d < 3.8500000000000001e-25

   1. Initial program 63.1%

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

   2. Taylor expanded in c around inf 79.6%

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

      1. *-commutative 79.6%

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

      2. pow2 79.6%

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

      3. times-frac 82.9%

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

   4. Applied egg-rr 82.9%

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

      1. associate-*r/ 86.8%

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

   6. Applied egg-rr 86.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+151}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.45 \cdot 10^{-130}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.85 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.8 \cdot 10^{+132}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 3: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+73} \lor \neg \left(d \leq 5.3 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.22e+73) (not (<= d 5.3e-25)))
   (/ (- (* c (/ b d)) a) d)
   (- (/ b c) (* (/ d c) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.22e+73) || !(d <= 5.3e-25)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b / c) - ((d / c) * (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 <= (-1.22d+73)) .or. (.not. (d <= 5.3d-25))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b / c) - ((d / c) * (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 <= -1.22e+73) || !(d <= 5.3e-25)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b / c) - ((d / c) * (a / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.22e+73) or not (d <= 5.3e-25):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b / c) - ((d / c) * (a / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.22e+73) || !(d <= 5.3e-25))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.22e+73) || ~((d <= 5.3e-25)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b / c) - ((d / c) * (a / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.22e+73], N[Not[LessEqual[d, 5.3e-25]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.21999999999999998e73 or 5.2999999999999997e-25 < d

   1. Initial program 50.3%

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

   2. Taylor expanded in c around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. mul-1-neg 72.8%

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

      3. unsub-neg 72.8%

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

      4. *-commutative 72.8%

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

   4. Simplified 72.8%

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

      1. pow2 72.8%

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

      2. times-frac 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-*l/ 79.3%

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

      2. sub-div 79.3%

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

   8. Applied egg-rr 79.3%

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

   if -1.21999999999999998e73 < d < 5.2999999999999997e-25

   1. Initial program 67.5%

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

   2. Taylor expanded in c around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 78.0%

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

   4. Applied egg-rr 78.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+73} \lor \neg \left(d \leq 5.3 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \end{array} \]

Alternative 4: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+73} \lor \neg \left(d \leq 5.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.22e+73) (not (<= d 5.2e-25)))
   (/ (- (* c (/ b d)) a) d)
   (- (/ b c) (/ (* a (/ d c)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.22e+73) || !(d <= 5.2e-25)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b / c) - ((a * (d / 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 <= (-1.22d+73)) .or. (.not. (d <= 5.2d-25))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b / c) - ((a * (d / 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 <= -1.22e+73) || !(d <= 5.2e-25)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b / c) - ((a * (d / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.22e+73) or not (d <= 5.2e-25):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b / c) - ((a * (d / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.22e+73) || !(d <= 5.2e-25))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a * Float64(d / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.22e+73) || ~((d <= 5.2e-25)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b / c) - ((a * (d / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.22e+73], N[Not[LessEqual[d, 5.2e-25]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.21999999999999998e73 or 5.2e-25 < d

   1. Initial program 50.3%

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

   2. Taylor expanded in c around 0 72.8%

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

      1. +-commutative 72.8%

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

      2. mul-1-neg 72.8%

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

      3. unsub-neg 72.8%

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

      4. *-commutative 72.8%

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

   4. Simplified 72.8%

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

      1. pow2 72.8%

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

      2. times-frac 77.9%

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

   6. Applied egg-rr 77.9%

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

      1. associate-*l/ 79.3%

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

      2. sub-div 79.3%

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

   8. Applied egg-rr 79.3%

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

   if -1.21999999999999998e73 < d < 5.2e-25

   1. Initial program 67.5%

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

   2. Taylor expanded in c around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. pow2 74.1%

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

      3. times-frac 78.0%

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

   4. Applied egg-rr 78.0%

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

      1. associate-*r/ 81.0%

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

   6. Applied egg-rr 81.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.22 \cdot 10^{+73} \lor \neg \left(d \leq 5.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 5: 72.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+49} \lor \neg \left(c \leq 8.2 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e+49) (not (<= c 8.2e+116)))
   (/ b c)
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+49) || !(c <= 8.2e+116)) {
		tmp = b / 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 <= (-2.4d+49)) .or. (.not. (c <= 8.2d+116))) then
        tmp = b / 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 <= -2.4e+49) || !(c <= 8.2e+116)) {
		tmp = b / c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e+49) or not (c <= 8.2e+116):
		tmp = b / c
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e+49) || !(c <= 8.2e+116))
		tmp = Float64(b / 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 <= -2.4e+49) || ~((c <= 8.2e+116)))
		tmp = b / c;
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e+49], N[Not[LessEqual[c, 8.2e+116]], $MachinePrecision]], N[(b / 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 < -2.4e49 or 8.1999999999999996e116 < c

   1. Initial program 39.7%

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

   2. Taylor expanded in c around inf 73.8%

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

   if -2.4e49 < c < 8.1999999999999996e116

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 68.0%

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

      1. +-commutative 68.0%

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

      2. mul-1-neg 68.0%

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

      3. unsub-neg 68.0%

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

      4. *-commutative 68.0%

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

   4. Simplified 68.0%

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

      1. pow2 68.0%

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

      2. times-frac 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. associate-*l/ 71.8%

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

      2. sub-div 71.9%

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

   8. Applied egg-rr 71.9%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+49} \lor \neg \left(c \leq 8.2 \cdot 10^{+116}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 6: 73.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+50} \lor \neg \left(c \leq 10^{+117}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e+50) (not (<= c 1e+117)))
   (/ b c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+50) || !(c <= 1e+117)) {
		tmp = b / 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 <= (-2.4d+50)) .or. (.not. (c <= 1d+117))) then
        tmp = b / 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 <= -2.4e+50) || !(c <= 1e+117)) {
		tmp = b / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e+50) or not (c <= 1e+117):
		tmp = b / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e+50) || !(c <= 1e+117))
		tmp = Float64(b / 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 <= -2.4e+50) || ~((c <= 1e+117)))
		tmp = b / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e+50], N[Not[LessEqual[c, 1e+117]], $MachinePrecision]], N[(b / 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 < -2.4000000000000002e50 or 1.00000000000000005e117 < c

   1. Initial program 39.7%

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

   2. Taylor expanded in c around inf 73.8%

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

   if -2.4000000000000002e50 < c < 1.00000000000000005e117

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 68.0%

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

      1. +-commutative 68.0%

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

      2. mul-1-neg 68.0%

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

      3. unsub-neg 68.0%

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

      4. *-commutative 68.0%

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

   4. Simplified 68.0%

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

      1. pow2 68.0%

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

      2. times-frac 70.7%

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

   6. Applied egg-rr 70.7%

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

      1. associate-*r/ 73.5%

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

      2. sub-div 73.6%

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

   8. Applied egg-rr 73.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+50} \lor \neg \left(c \leq 10^{+117}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]

Alternative 7: 63.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+18} \lor \neg \left(d \leq 15800\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.2e+18) (not (<= d 15800.0))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.2e+18) || !(d <= 15800.0)) {
		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 <= (-3.2d+18)) .or. (.not. (d <= 15800.0d0))) 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 <= -3.2e+18) || !(d <= 15800.0)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.2e+18) or not (d <= 15800.0):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.2e+18) || !(d <= 15800.0))
		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 <= -3.2e+18) || ~((d <= 15800.0)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.2e+18], N[Not[LessEqual[d, 15800.0]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.2e18 or 15800 < d

   1. Initial program 50.1%

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

   2. Taylor expanded in c around 0 63.1%

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

      1. associate-*r/ 63.1%

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

      2. neg-mul-1 63.1%

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

   4. Simplified 63.1%

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

   if -3.2e18 < d < 15800

   1. Initial program 68.2%

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

   2. Taylor expanded in c around inf 63.0%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+18} \lor \neg \left(d \leq 15800\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8: 44.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+203}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.55e+203) (/ b d) (if (<= d 1.4e+178) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+203) {
		tmp = b / d;
	} else if (d <= 1.4e+178) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.55d+203)) then
        tmp = b / d
    else if (d <= 1.4d+178) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.55e+203) {
		tmp = b / d;
	} else if (d <= 1.4e+178) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.55e+203:
		tmp = b / d
	elif d <= 1.4e+178:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.55e+203)
		tmp = Float64(b / d);
	elseif (d <= 1.4e+178)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.55e+203)
		tmp = b / d;
	elseif (d <= 1.4e+178)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.55e+203], N[(b / d), $MachinePrecision], If[LessEqual[d, 1.4e+178], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.55e203

   1. Initial program 28.2%

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

      1. fma-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-out 28.2%

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

      4. add-sqr-sqrt 28.2%

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

      5. *-un-lft-identity 28.2%

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

      6. times-frac 28.2%

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

      7. hypot-def 28.2%

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

      8. add-sqr-sqrt 18.8%

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

      9. sqrt-unprod 28.2%

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

      10. sqr-neg 28.2%

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

      11. sqrt-unprod 9.4%

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

      12. add-sqr-sqrt 28.2%

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

      13. *-commutative 28.2%

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

      14. hypot-def 36.1%

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

   3. Applied egg-rr 36.1%

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

   4. Taylor expanded in b around inf 37.6%

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

   5. Taylor expanded in d around -inf 34.4%

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

   6. Taylor expanded in c around -inf 26.5%

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

   if -1.55e203 < d < 1.39999999999999997e178

   1. Initial program 67.5%

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

   2. Taylor expanded in c around inf 48.1%

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

   if 1.39999999999999997e178 < d

   1. Initial program 23.9%

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

      1. fma-neg 23.9%

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

      2. *-commutative 23.9%

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

      3. distribute-rgt-neg-out 23.9%

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

      4. add-sqr-sqrt 23.9%

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

      5. *-un-lft-identity 23.9%

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

      6. times-frac 23.9%

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

      7. hypot-def 23.9%

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

      8. add-sqr-sqrt 17.8%

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

      9. sqrt-unprod 23.9%

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

      10. sqr-neg 23.9%

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

      11. sqrt-unprod 6.2%

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

      12. add-sqr-sqrt 23.9%

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

      13. *-commutative 23.9%

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

      14. hypot-def 27.4%

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

   3. Applied egg-rr 27.4%

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

   4. Taylor expanded in c around 0 25.4%

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

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+203}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 9: 43.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 2e+178) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 2e+178) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 2d+178) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 2e+178) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 2e+178:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 2e+178)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 2e+178)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 2e+178], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 2.0000000000000001e178

   1. Initial program 63.5%

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

   2. Taylor expanded in c around inf 44.0%

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

   if 2.0000000000000001e178 < d

   1. Initial program 23.9%

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

      1. fma-neg 23.9%

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

      2. *-commutative 23.9%

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

      3. distribute-rgt-neg-out 23.9%

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

      4. add-sqr-sqrt 23.9%

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

      5. *-un-lft-identity 23.9%

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

      6. times-frac 23.9%

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

      7. hypot-def 23.9%

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

      8. add-sqr-sqrt 17.8%

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

      9. sqrt-unprod 23.9%

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

      10. sqr-neg 23.9%

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

      11. sqrt-unprod 6.2%

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

      12. add-sqr-sqrt 23.9%

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

      13. *-commutative 23.9%

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

      14. hypot-def 27.4%

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

   3. Applied egg-rr 27.4%

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

   4. Taylor expanded in c around 0 25.4%

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

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 2 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 10: 11.0% 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 58.9%

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

   1. fma-neg 58.9%

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

   2. *-commutative 58.9%

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

   3. distribute-rgt-neg-out 58.9%

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

   4. add-sqr-sqrt 58.9%

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

   5. *-un-lft-identity 58.9%

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

   6. times-frac 58.8%

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

   7. hypot-def 58.8%

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

   8. add-sqr-sqrt 34.0%

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

   9. sqrt-unprod 43.1%

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

   10. sqr-neg 43.1%

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

   11. sqrt-unprod 14.8%

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

   12. add-sqr-sqrt 36.0%

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

   13. *-commutative 36.0%

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

   14. hypot-def 44.1%

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

3. Applied egg-rr 44.1%

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

4. Taylor expanded in c around 0 8.8%

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

5. Final simplification 8.8%

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

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 2023330 
(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))))