Complex division, imag part

Percentage Accurate: 61.8% → 77.7%

Time: 8.8s

Alternatives: 7

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))))
   (if (<= c -2.15e+80)
     (- (/ b c) (* (/ d c) (/ a c)))
     (if (<= c -2.1e-298)
       t_0
       (if (<= c 2.35e-222)
         (/ (- a) d)
         (if (<= c 8.8e+26) t_0 (+ (/ b c) (* (* d (/ a c)) (/ -1.0 c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -2.15e+80) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else if (c <= -2.1e-298) {
		tmp = t_0;
	} else if (c <= 2.35e-222) {
		tmp = -a / d;
	} else if (c <= 8.8e+26) {
		tmp = t_0;
	} else {
		tmp = (b / c) + ((d * (a / c)) * (-1.0 / 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) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
    if (c <= (-2.15d+80)) then
        tmp = (b / c) - ((d / c) * (a / c))
    else if (c <= (-2.1d-298)) then
        tmp = t_0
    else if (c <= 2.35d-222) then
        tmp = -a / d
    else if (c <= 8.8d+26) then
        tmp = t_0
    else
        tmp = (b / c) + ((d * (a / c)) * ((-1.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -2.15e+80) {
		tmp = (b / c) - ((d / c) * (a / c));
	} else if (c <= -2.1e-298) {
		tmp = t_0;
	} else if (c <= 2.35e-222) {
		tmp = -a / d;
	} else if (c <= 8.8e+26) {
		tmp = t_0;
	} else {
		tmp = (b / c) + ((d * (a / c)) * (-1.0 / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c))
	tmp = 0
	if c <= -2.15e+80:
		tmp = (b / c) - ((d / c) * (a / c))
	elif c <= -2.1e-298:
		tmp = t_0
	elif c <= 2.35e-222:
		tmp = -a / d
	elif c <= 8.8e+26:
		tmp = t_0
	else:
		tmp = (b / c) + ((d * (a / c)) * (-1.0 / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -2.15e+80)
		tmp = Float64(Float64(b / c) - Float64(Float64(d / c) * Float64(a / c)));
	elseif (c <= -2.1e-298)
		tmp = t_0;
	elseif (c <= 2.35e-222)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 8.8e+26)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) + Float64(Float64(d * Float64(a / c)) * Float64(-1.0 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (d * a)) / ((d * d) + (c * c));
	tmp = 0.0;
	if (c <= -2.15e+80)
		tmp = (b / c) - ((d / c) * (a / c));
	elseif (c <= -2.1e-298)
		tmp = t_0;
	elseif (c <= 2.35e-222)
		tmp = -a / d;
	elseif (c <= 8.8e+26)
		tmp = t_0;
	else
		tmp = (b / c) + ((d * (a / c)) * (-1.0 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.15e+80], N[(N[(b / c), $MachinePrecision] - N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.1e-298], t$95$0, If[LessEqual[c, 2.35e-222], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 8.8e+26], t$95$0, N[(N[(b / c), $MachinePrecision] + N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.15000000000000002e80

   1. Initial program 36.5%

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

   2. Taylor expanded in c around inf 65.8%

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

      1. +-commutative 65.8%

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

      2. mul-1-neg 65.8%

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

      3. unsub-neg 65.8%

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

      4. *-commutative 65.8%

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

      5. associate-/l* 76.3%

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

   4. Simplified 76.3%

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

      1. div-inv 76.3%

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

      2. clear-num 76.3%

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

      3. *-un-lft-identity 76.3%

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

      4. unpow2 76.3%

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

      5. frac-times 79.2%

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

      6. associate-*r* 82.8%

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

      7. div-inv 82.8%

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

   6. Applied egg-rr 82.8%

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

   if -2.15000000000000002e80 < c < -2.10000000000000005e-298 or 2.3499999999999999e-222 < c < 8.80000000000000028e26

   1. Initial program 80.0%

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

   if -2.10000000000000005e-298 < c < 2.3499999999999999e-222

   1. Initial program 72.7%

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

   2. Taylor expanded in c around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. neg-mul-1 100.0%

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

   4. Simplified 100.0%

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

   if 8.80000000000000028e26 < c

   1. Initial program 51.3%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. *-commutative 73.5%

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

      5. associate-/l* 74.5%

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

   4. Simplified 74.5%

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

      1. div-inv 74.4%

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

      2. clear-num 74.4%

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

      3. *-un-lft-identity 74.4%

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

      4. unpow2 74.4%

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

      5. frac-times 81.5%

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

      6. *-commutative 81.5%

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

      7. associate-*l* 84.6%

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

   6. Applied egg-rr 84.6%

      \[\leadsto \frac{b}{c} - \color{blue}{\frac{1}{c} \cdot \left(\frac{a}{c} \cdot d\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c} \cdot \frac{a}{c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-298}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.35 \cdot 10^{-222}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \end{array} \]

Alternative 2: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* (/ b (pow d 2.0)) c) (/ a d))))
   (if (<= d -5.8e+29)
     t_0
     (if (<= d 2.9e-121)
       (/ (- b (* a (/ d c))) c)
       (if (<= d 2e+50) (/ (- (* 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 = ((b / pow(d, 2.0)) * c) - (a / d);
	double tmp;
	if (d <= -5.8e+29) {
		tmp = t_0;
	} else if (d <= 2.9e-121) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 2e+50) {
		tmp = ((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(b / (d ^ 2.0)) * c) - Float64(a / d))
	tmp = 0.0
	if (d <= -5.8e+29)
		tmp = t_0;
	elseif (d <= 2.9e-121)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 2e+50)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * 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[(b / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.8e+29], t$95$0, If[LessEqual[d, 2.9e-121], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2e+50], N[(N[(N[(b * c), $MachinePrecision] - 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 < -5.7999999999999999e29 or 2.0000000000000002e50 < d

   1. Initial program 43.7%

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

   2. Taylor expanded in c around 0 69.8%

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

      1. +-commutative 69.8%

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

      2. mul-1-neg 69.8%

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

      3. unsub-neg 69.8%

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

      4. associate-/l* 70.4%

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

      5. associate-/r/ 73.2%

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

   4. Simplified 73.2%

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

   if -5.7999999999999999e29 < d < 2.9e-121

   1. Initial program 77.4%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/l* 75.9%

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

   4. Simplified 75.9%

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

      1. div-inv 75.9%

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

      2. clear-num 75.9%

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

      3. *-un-lft-identity 75.9%

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

      4. unpow2 75.9%

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

      5. frac-times 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-*l* 85.6%

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

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in b around 0 77.3%

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

      1. neg-mul-1 77.3%

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

      2. +-commutative 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 86.3%

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

      6. div-sub 88.9%

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

      7. *-lft-identity 88.9%

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

      8. times-frac 88.9%

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

      9. /-rgt-identity 88.9%

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

   9. Simplified 88.9%

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

   if 2.9e-121 < d < 2.0000000000000002e50

   1. Initial program 83.7%

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

      1. fma-def 83.7%

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

   3. Simplified 83.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+29}:\\ \;\;\;\;\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \mathbf{elif}\;d \leq 2.9 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{+50}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \end{array} \]

Alternative 3: 78.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \mathbf{if}\;d \leq -3.4 \cdot 10^{+30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* (/ b (pow d 2.0)) c) (/ a d))))
   (if (<= d -3.4e+30)
     t_0
     (if (<= d 2.4e-121)
       (/ (- b (* a (/ d c))) c)
       (if (<= d 2.1e+50) (/ (- (* b c) (* d a)) (+ (* d d) (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b / pow(d, 2.0)) * c) - (a / d);
	double tmp;
	if (d <= -3.4e+30) {
		tmp = t_0;
	} else if (d <= 2.4e-121) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 2.1e+50) {
		tmp = ((b * c) - (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 = ((b / (d ** 2.0d0)) * c) - (a / d)
    if (d <= (-3.4d+30)) then
        tmp = t_0
    else if (d <= 2.4d-121) then
        tmp = (b - (a * (d / c))) / c
    else if (d <= 2.1d+50) then
        tmp = ((b * c) - (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 = ((b / Math.pow(d, 2.0)) * c) - (a / d);
	double tmp;
	if (d <= -3.4e+30) {
		tmp = t_0;
	} else if (d <= 2.4e-121) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 2.1e+50) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b / math.pow(d, 2.0)) * c) - (a / d)
	tmp = 0
	if d <= -3.4e+30:
		tmp = t_0
	elif d <= 2.4e-121:
		tmp = (b - (a * (d / c))) / c
	elif d <= 2.1e+50:
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b / (d ^ 2.0)) * c) - Float64(a / d))
	tmp = 0.0
	if (d <= -3.4e+30)
		tmp = t_0;
	elseif (d <= 2.4e-121)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 2.1e+50)
		tmp = Float64(Float64(Float64(b * c) - 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 = ((b / (d ^ 2.0)) * c) - (a / d);
	tmp = 0.0;
	if (d <= -3.4e+30)
		tmp = t_0;
	elseif (d <= 2.4e-121)
		tmp = (b - (a * (d / c))) / c;
	elseif (d <= 2.1e+50)
		tmp = ((b * c) - (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[(N[(b / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.4e+30], t$95$0, If[LessEqual[d, 2.4e-121], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.1e+50], N[(N[(N[(b * c), $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 < -3.4000000000000002e30 or 2.1e50 < d

   1. Initial program 43.7%

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

   2. Taylor expanded in c around 0 69.8%

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

      1. +-commutative 69.8%

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

      2. mul-1-neg 69.8%

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

      3. unsub-neg 69.8%

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

      4. associate-/l* 70.4%

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

      5. associate-/r/ 73.2%

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

   4. Simplified 73.2%

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

   if -3.4000000000000002e30 < d < 2.40000000000000003e-121

   1. Initial program 77.4%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/l* 75.9%

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

   4. Simplified 75.9%

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

      1. div-inv 75.9%

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

      2. clear-num 75.9%

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

      3. *-un-lft-identity 75.9%

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

      4. unpow2 75.9%

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

      5. frac-times 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-*l* 85.6%

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

   6. Applied egg-rr 85.6%

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

   7. Taylor expanded in b around 0 77.3%

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

      1. neg-mul-1 77.3%

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

      2. +-commutative 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. associate-/r* 86.3%

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

      6. div-sub 88.9%

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

      7. *-lft-identity 88.9%

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

      8. times-frac 88.9%

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

      9. /-rgt-identity 88.9%

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

   9. Simplified 88.9%

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

   if 2.40000000000000003e-121 < d < 2.1e50

   1. Initial program 83.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.4 \cdot 10^{+30}:\\ \;\;\;\;\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \mathbf{elif}\;d \leq 2.4 \cdot 10^{-121}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+50}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{{d}^{2}} \cdot c - \frac{a}{d}\\ \end{array} \]

Alternative 4: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+31} \lor \neg \left(d \leq 9.5 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.1e+31) (not (<= d 9.5e-29)))
   (/ (- a) d)
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.1e+31) || !(d <= 9.5e-29)) {
		tmp = -a / d;
	} else {
		tmp = (b - (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 <= (-2.1d+31)) .or. (.not. (d <= 9.5d-29))) then
        tmp = -a / d
    else
        tmp = (b - (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 <= -2.1e+31) || !(d <= 9.5e-29)) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.1e+31) or not (d <= 9.5e-29):
		tmp = -a / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.1e+31) || !(d <= 9.5e-29))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.1e+31) || ~((d <= 9.5e-29)))
		tmp = -a / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.1e+31], N[Not[LessEqual[d, 9.5e-29]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.09999999999999979e31 or 9.50000000000000023e-29 < d

   1. Initial program 49.5%

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

   2. Taylor expanded in c around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   4. Simplified 68.2%

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

   if -2.09999999999999979e31 < d < 9.50000000000000023e-29

   1. Initial program 77.5%

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

   2. Taylor expanded in c around inf 74.7%

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

      1. +-commutative 74.7%

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

      2. mul-1-neg 74.7%

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

      3. unsub-neg 74.7%

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

      4. *-commutative 74.7%

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

      5. associate-/l* 73.5%

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

   4. Simplified 73.5%

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

      1. div-inv 73.5%

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

      2. clear-num 73.5%

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

      3. *-un-lft-identity 73.5%

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

      4. unpow2 73.5%

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

      5. frac-times 76.7%

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

      6. *-commutative 76.7%

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

      7. associate-*l* 82.3%

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

   6. Applied egg-rr 82.3%

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

   7. Taylor expanded in b around 0 74.7%

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

      1. neg-mul-1 74.7%

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

      2. +-commutative 74.7%

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

      3. unsub-neg 74.7%

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

      4. unpow2 74.7%

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

      5. associate-/r* 82.9%

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

      6. div-sub 85.1%

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

      7. *-lft-identity 85.1%

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

      8. times-frac 85.1%

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

      9. /-rgt-identity 85.1%

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

   9. Simplified 85.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.1 \cdot 10^{+31} \lor \neg \left(d \leq 9.5 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 5: 63.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+31} \lor \neg \left(d \leq 7.2 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.8e+31) (not (<= d 7.2e-40))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.8e+31) || !(d <= 7.2e-40)) {
		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 <= (-3.8d+31)) .or. (.not. (d <= 7.2d-40))) 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 <= -3.8e+31) || !(d <= 7.2e-40)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.8e+31) or not (d <= 7.2e-40):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.8e+31) || !(d <= 7.2e-40))
		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 <= -3.8e+31) || ~((d <= 7.2e-40)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.8e+31], N[Not[LessEqual[d, 7.2e-40]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.8000000000000001e31 or 7.2e-40 < d

   1. Initial program 49.9%

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

   2. Taylor expanded in c around 0 67.9%

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

      1. associate-*r/ 67.9%

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

      2. neg-mul-1 67.9%

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

   4. Simplified 67.9%

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

   if -3.8000000000000001e31 < d < 7.2e-40

   1. Initial program 77.6%

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

   2. Taylor expanded in c around inf 61.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.8 \cdot 10^{+31} \lor \neg \left(d \leq 7.2 \cdot 10^{-40}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 6: 46.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+222} \lor \neg \left(d \leq 1.25 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.2e+222) (not (<= d 1.25e+203))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+222) || !(d <= 1.25e+203)) {
		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 <= (-7.2d+222)) .or. (.not. (d <= 1.25d+203))) 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 <= -7.2e+222) || !(d <= 1.25e+203)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.2e+222) or not (d <= 1.25e+203):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.2e+222) || !(d <= 1.25e+203))
		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 <= -7.2e+222) || ~((d <= 1.25e+203)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.2e+222], N[Not[LessEqual[d, 1.25e+203]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.2000000000000003e222 or 1.24999999999999999e203 < d

   1. Initial program 43.4%

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

      1. cancel-sign-sub-inv 43.4%

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

      2. *-commutative 43.4%

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

      3. *-commutative 43.4%

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

      4. add-sqr-sqrt 22.3%

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

      5. sqrt-unprod 43.4%

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

      6. sqr-neg 43.4%

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

      7. sqrt-unprod 21.1%

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

      8. add-sqr-sqrt 43.4%

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

   3. Applied egg-rr 43.4%

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

   4. Taylor expanded in c around 0 43.9%

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

   if -7.2000000000000003e222 < d < 1.24999999999999999e203

   1. Initial program 68.8%

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

   2. Taylor expanded in c around inf 47.4%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+222} \lor \neg \left(d \leq 1.25 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 7: 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 64.9%

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

   1. cancel-sign-sub-inv 64.9%

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

   2. *-commutative 64.9%

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

   3. *-commutative 64.9%

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

   4. add-sqr-sqrt 34.3%

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

   5. sqrt-unprod 44.9%

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

   6. sqr-neg 44.9%

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

   7. sqrt-unprod 17.3%

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

   8. add-sqr-sqrt 38.4%

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

3. Applied egg-rr 38.4%

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

4. Taylor expanded in c around 0 11.1%

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

5. Final simplification 11.1%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023322 
(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))))