Complex division, imag part

Percentage Accurate: 61.3% → 90.2%

Time: 10.8s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% 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: 90.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-72} \lor \neg \left(d \leq 3.4 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{d}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.9e-72) (not (<= d 3.4e-121)))
   (* (/ (- (* b (/ c d)) a) (hypot d c)) (/ d (hypot d c)))
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.9e-72) || !(d <= 3.4e-121)) {
		tmp = (((b * (c / d)) - a) / hypot(d, c)) * (d / hypot(d, c));
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.9e-72) || !(d <= 3.4e-121)) {
		tmp = (((b * (c / d)) - a) / Math.hypot(d, c)) * (d / Math.hypot(d, c));
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.9e-72) or not (d <= 3.4e-121):
		tmp = (((b * (c / d)) - a) / math.hypot(d, c)) * (d / math.hypot(d, c))
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.9e-72) || !(d <= 3.4e-121))
		tmp = Float64(Float64(Float64(Float64(b * Float64(c / d)) - a) / hypot(d, c)) * Float64(d / hypot(d, c)));
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.9e-72) || ~((d <= 3.4e-121)))
		tmp = (((b * (c / d)) - a) / hypot(d, c)) * (d / hypot(d, c));
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.9e-72], N[Not[LessEqual[d, 3.4e-121]], $MachinePrecision]], N[(N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.89999999999999998e-72 or 3.40000000000000001e-121 < d

   1. Initial program 55.7%

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

      1. fmm-def 55.7%

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

      2. distribute-rgt-neg-out 55.7%

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

      3. +-commutative 55.7%

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

      4. fma-define 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in d around inf 55.7%

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

      1. +-commutative 55.7%

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

      2. neg-mul-1 55.7%

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

      3. sub-neg 55.7%

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

      4. associate-/l* 55.2%

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

   7. Simplified 55.2%

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

      1. *-commutative 55.2%

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

      2. add-sqr-sqrt 55.2%

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

      3. times-frac 62.3%

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

      4. fma-undefine 62.3%

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

      5. hypot-define 62.3%

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

      6. fma-undefine 62.3%

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

      7. hypot-define 95.0%

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

   9. Applied egg-rr 95.0%

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

   if -2.89999999999999998e-72 < d < 3.40000000000000001e-121

   1. Initial program 68.5%

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

      1. fmm-def 68.5%

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

      2. distribute-rgt-neg-out 68.5%

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

      3. +-commutative 68.5%

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

      4. fma-define 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in c around inf 93.6%

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

      1. mul-1-neg 93.6%

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

      2. unsub-neg 93.6%

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

      3. *-commutative 93.6%

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

   7. Simplified 93.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.9 \cdot 10^{-72} \lor \neg \left(d \leq 3.4 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{d}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{if}\;d \leq -1.9 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+65}:\\ \;\;\;\;\frac{d \cdot \left(b \cdot \frac{c}{d} - a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ c (/ d b)) a) d)))
   (if (<= d -1.9e+23)
     t_0
     (if (<= d 5e-80)
       (/ (- b (/ (* d a) c)) c)
       (if (<= d 2.75e+65)
         (/ (* d (- (* b (/ c d)) a)) (fma d d (* c c)))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -1.9e+23) {
		tmp = t_0;
	} else if (d <= 5e-80) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 2.75e+65) {
		tmp = (d * ((b * (c / d)) - a)) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c / Float64(d / b)) - a) / d)
	tmp = 0.0
	if (d <= -1.9e+23)
		tmp = t_0;
	elseif (d <= 5e-80)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 2.75e+65)
		tmp = Float64(Float64(d * Float64(Float64(b * Float64(c / d)) - a)) / fma(d, d, Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.9e+23], t$95$0, If[LessEqual[d, 5e-80], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.75e+65], N[(N[(d * N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.89999999999999987e23 or 2.7499999999999998e65 < d

   1. Initial program 43.9%

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

      1. fmm-def 43.9%

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

      2. distribute-rgt-neg-out 43.9%

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

      3. +-commutative 43.9%

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

      4. fma-define 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in d around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. distribute-neg-frac2 79.2%

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

      3. mul-1-neg 79.2%

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

      4. unsub-neg 79.2%

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

      5. *-commutative 79.2%

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

      6. associate-/l* 86.7%

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

   7. Simplified 86.7%

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

      1. clear-num 86.8%

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

      2. un-div-inv 86.8%

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

   9. Applied egg-rr 86.8%

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

   if -1.89999999999999987e23 < d < 5e-80

   1. Initial program 72.2%

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

      1. fmm-def 72.2%

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

      2. distribute-rgt-neg-out 72.2%

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

      3. +-commutative 72.2%

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

      4. fma-define 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in c around inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

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

      3. *-commutative 89.2%

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

   7. Simplified 89.2%

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

   if 5e-80 < d < 2.7499999999999998e65

   1. Initial program 86.0%

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

      1. fmm-def 86.0%

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

      2. distribute-rgt-neg-out 86.0%

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

      3. +-commutative 86.0%

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

      4. fma-define 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in d around inf 86.0%

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

      1. +-commutative 86.0%

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

      2. neg-mul-1 86.0%

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

      3. sub-neg 86.0%

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

      4. associate-/l* 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+65}:\\ \;\;\;\;\frac{d \cdot \left(b \cdot \frac{c}{d} - a\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{if}\;d \leq -2.6 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (/ c (/ d b)) a) d)))
   (if (<= d -2.6e+22)
     t_0
     (if (<= d 4.8e-65)
       (/ (- b (/ (* d a) c)) c)
       (if (<= d 3.5e+65) (/ (- (* b c) (* d a)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -2.6e+22) {
		tmp = t_0;
	} else if (d <= 4.8e-65) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 3.5e+65) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c / (d / b)) - a) / d
    if (d <= (-2.6d+22)) then
        tmp = t_0
    else if (d <= 4.8d-65) then
        tmp = (b - ((d * a) / c)) / c
    else if (d <= 3.5d+65) then
        tmp = ((b * c) - (d * a)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c / (d / b)) - a) / d;
	double tmp;
	if (d <= -2.6e+22) {
		tmp = t_0;
	} else if (d <= 4.8e-65) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 3.5e+65) {
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c / (d / b)) - a) / d
	tmp = 0
	if d <= -2.6e+22:
		tmp = t_0
	elif d <= 4.8e-65:
		tmp = (b - ((d * a) / c)) / c
	elif d <= 3.5e+65:
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c / Float64(d / b)) - a) / d)
	tmp = 0.0
	if (d <= -2.6e+22)
		tmp = t_0;
	elseif (d <= 4.8e-65)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 3.5e+65)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c / (d / b)) - a) / d;
	tmp = 0.0;
	if (d <= -2.6e+22)
		tmp = t_0;
	elseif (d <= 4.8e-65)
		tmp = (b - ((d * a) / c)) / c;
	elseif (d <= 3.5e+65)
		tmp = ((b * c) - (d * a)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.6e+22], t$95$0, If[LessEqual[d, 4.8e-65], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.5e+65], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.6e22 or 3.5000000000000001e65 < d

   1. Initial program 43.9%

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

      1. fmm-def 43.9%

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

      2. distribute-rgt-neg-out 43.9%

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

      3. +-commutative 43.9%

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

      4. fma-define 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in d around -inf 79.2%

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

      1. mul-1-neg 79.2%

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

      2. distribute-neg-frac2 79.2%

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

      3. mul-1-neg 79.2%

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

      4. unsub-neg 79.2%

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

      5. *-commutative 79.2%

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

      6. associate-/l* 86.7%

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

   7. Simplified 86.7%

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

      1. clear-num 86.8%

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

      2. un-div-inv 86.8%

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

   9. Applied egg-rr 86.8%

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

   if -2.6e22 < d < 4.8000000000000003e-65

   1. Initial program 72.2%

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

      1. fmm-def 72.2%

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

      2. distribute-rgt-neg-out 72.2%

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

      3. +-commutative 72.2%

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

      4. fma-define 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in c around inf 89.2%

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

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

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

      3. *-commutative 89.2%

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

   7. Simplified 89.2%

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

   if 4.8000000000000003e-65 < d < 3.5000000000000001e65

   1. Initial program 86.0%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.6 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{elif}\;d \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 3.5 \cdot 10^{+65}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+19} \lor \neg \left(d \leq 9.5 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.2e+19) (not (<= d 9.5e-59)))
   (/ (- (/ c (/ d b)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e+19) || !(d <= 9.5e-59)) {
		tmp = ((c / (d / b)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-9.2d+19)) .or. (.not. (d <= 9.5d-59))) then
        tmp = ((c / (d / b)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.2e+19) || !(d <= 9.5e-59)) {
		tmp = ((c / (d / b)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9.2e+19) or not (d <= 9.5e-59):
		tmp = ((c / (d / b)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9.2e+19) || !(d <= 9.5e-59))
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -9.2e+19) || ~((d <= 9.5e-59)))
		tmp = ((c / (d / b)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9.2e+19], N[Not[LessEqual[d, 9.5e-59]], $MachinePrecision]], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.2e19 or 9.4999999999999994e-59 < d

   1. Initial program 50.1%

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

      1. fmm-def 50.0%

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

      2. distribute-rgt-neg-out 50.0%

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

      3. +-commutative 50.0%

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

      4. fma-define 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in d around -inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. distribute-neg-frac2 76.2%

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

      3. mul-1-neg 76.2%

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

      4. unsub-neg 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-/l* 82.6%

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

   7. Simplified 82.6%

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

      1. clear-num 82.6%

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

      2. un-div-inv 82.6%

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

   9. Applied egg-rr 82.6%

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

   if -9.2e19 < d < 9.4999999999999994e-59

   1. Initial program 72.5%

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

      1. fmm-def 72.5%

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

      2. distribute-rgt-neg-out 72.5%

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

      3. +-commutative 72.5%

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

      4. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in c around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. unsub-neg 89.3%

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

      3. *-commutative 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+19} \lor \neg \left(d \leq 9.5 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+21} \lor \neg \left(d \leq 1.3 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9e+21) (not (<= d 1.3e-58)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e+21) || !(d <= 1.3e-58)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-9d+21)) .or. (.not. (d <= 1.3d-58))) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9e+21) || !(d <= 1.3e-58)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -9e+21) or not (d <= 1.3e-58):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9e+21) || !(d <= 1.3e-58))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -9e+21) || ~((d <= 1.3e-58)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9e+21], N[Not[LessEqual[d, 1.3e-58]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9e21 or 1.30000000000000003e-58 < d

   1. Initial program 50.1%

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

      1. fmm-def 50.0%

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

      2. distribute-rgt-neg-out 50.0%

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

      3. +-commutative 50.0%

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

      4. fma-define 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in d around -inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. distribute-neg-frac2 76.2%

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

      3. mul-1-neg 76.2%

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

      4. unsub-neg 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-/l* 82.6%

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

   7. Simplified 82.6%

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

   if -9e21 < d < 1.30000000000000003e-58

   1. Initial program 72.5%

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

      1. fmm-def 72.5%

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

      2. distribute-rgt-neg-out 72.5%

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

      3. +-commutative 72.5%

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

      4. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in c around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. unsub-neg 89.3%

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

      3. *-commutative 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{+21} \lor \neg \left(d \leq 1.3 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+23} \lor \neg \left(d \leq 7.6 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.55e+23) (not (<= d 7.6e-59)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.55e+23) || !(d <= 7.6e-59)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.55d+23)) .or. (.not. (d <= 7.6d-59))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.55e+23) || !(d <= 7.6e-59)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.55e+23) or not (d <= 7.6e-59):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.55e+23) || !(d <= 7.6e-59))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.55e+23) || ~((d <= 7.6e-59)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.55e+23], N[Not[LessEqual[d, 7.6e-59]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.54999999999999985e23 or 7.59999999999999966e-59 < d

   1. Initial program 50.1%

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

      1. fmm-def 50.0%

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

      2. distribute-rgt-neg-out 50.0%

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

      3. +-commutative 50.0%

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

      4. fma-define 50.1%

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

   3. Simplified 50.1%

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

   5. Taylor expanded in d around -inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. distribute-neg-frac2 76.2%

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

      3. mul-1-neg 76.2%

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

      4. unsub-neg 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-/l* 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in d around inf 76.2%

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

      1. neg-mul-1 76.2%

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

      2. associate-*r/ 81.7%

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

      3. +-commutative 81.7%

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

      4. sub-neg 81.7%

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

   10. Simplified 81.7%

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

   if -1.54999999999999985e23 < d < 7.59999999999999966e-59

   1. Initial program 72.5%

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

      1. fmm-def 72.5%

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

      2. distribute-rgt-neg-out 72.5%

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

      3. +-commutative 72.5%

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

      4. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in c around inf 89.3%

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

      1. mul-1-neg 89.3%

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

      2. unsub-neg 89.3%

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

      3. *-commutative 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.55 \cdot 10^{+23} \lor \neg \left(d \leq 7.6 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+21} \lor \neg \left(d \leq 1.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.3e+21) (not (<= d 1.8e-26)))
   (/ a (- d))
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.3e+21) || !(d <= 1.8e-26)) {
		tmp = a / -d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.3d+21)) .or. (.not. (d <= 1.8d-26))) then
        tmp = a / -d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.3e+21) || !(d <= 1.8e-26)) {
		tmp = a / -d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.3e+21) or not (d <= 1.8e-26):
		tmp = a / -d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.3e+21) || !(d <= 1.8e-26))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.3e+21) || ~((d <= 1.8e-26)))
		tmp = a / -d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.3e+21], N[Not[LessEqual[d, 1.8e-26]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.3e21 or 1.8000000000000001e-26 < d

   1. Initial program 49.7%

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

      1. fmm-def 49.7%

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

      2. distribute-rgt-neg-out 49.7%

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

      3. +-commutative 49.7%

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

      4. fma-define 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in c around 0 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

   if -1.3e21 < d < 1.8000000000000001e-26

   1. Initial program 72.3%

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

      1. fmm-def 72.3%

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

      2. distribute-rgt-neg-out 72.3%

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

      3. +-commutative 72.3%

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

      4. fma-define 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in c around inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. *-commutative 88.0%

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

   7. Simplified 88.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.3 \cdot 10^{+21} \lor \neg \left(d \leq 1.8 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+20} \lor \neg \left(d \leq 1.75 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.2e+20) (not (<= d 1.75e-26)))
   (/ (- a) d)
   (/ (- b (* (/ d c) a)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.2e+20) || !(d <= 1.75e-26)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((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 <= (-4.2d+20)) .or. (.not. (d <= 1.75d-26))) then
        tmp = -a / d
    else
        tmp = (b - ((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 <= -4.2e+20) || !(d <= 1.75e-26)) {
		tmp = -a / d;
	} else {
		tmp = (b - ((d / c) * a)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.2e+20) or not (d <= 1.75e-26):
		tmp = -a / d
	else:
		tmp = (b - ((d / c) * a)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.2e+20) || !(d <= 1.75e-26))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d / c) * a)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.2e+20) || ~((d <= 1.75e-26)))
		tmp = -a / d;
	else
		tmp = (b - ((d / c) * a)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.2e+20], N[Not[LessEqual[d, 1.75e-26]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(N[(d / c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.2e20 or 1.74999999999999992e-26 < d

   1. Initial program 49.7%

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

      1. fmm-def 49.7%

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

      2. distribute-rgt-neg-out 49.7%

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

      3. +-commutative 49.7%

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

      4. fma-define 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in c around 0 67.0%

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

      1. associate-*r/ 67.0%

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

      2. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

   if -4.2e20 < d < 1.74999999999999992e-26

   1. Initial program 72.3%

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

      1. fmm-def 72.3%

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

      2. distribute-rgt-neg-out 72.3%

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

      3. +-commutative 72.3%

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

      4. fma-define 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in c around -inf 88.0%

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

      1. mul-1-neg 88.0%

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

      2. fma-define 88.0%

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

      3. associate-/l* 86.5%

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

   7. Simplified 86.5%

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

   8. Taylor expanded in b around 0 82.6%

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

      1. sub-neg 82.6%

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

      2. remove-double-neg 82.6%

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

      3. distribute-neg-frac 82.6%

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

      4. distribute-frac-neg2 82.6%

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

      5. *-rgt-identity 82.6%

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

      6. associate-*r/ 82.4%

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

      7. *-commutative 82.4%

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

      8. mul-1-neg 82.4%

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

      9. associate-*r/ 82.4%

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

      10. unpow2 82.4%

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

      11. times-frac 86.8%

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

      12. metadata-eval 86.8%

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

      13. distribute-neg-frac 86.8%

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

      14. distribute-frac-neg2 86.8%

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

      15. distribute-lft-in 87.8%

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

      16. neg-mul-1 87.8%

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

      17. fma-define 87.8%

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

      18. *-commutative 87.8%

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

      19. distribute-frac-neg2 87.8%

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

      20. distribute-rgt-neg-in 87.8%

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

      21. distribute-lft-neg-out 87.8%

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

      22. associate-*r/ 88.0%

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

   10. Simplified 86.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+20} \lor \neg \left(d \leq 1.75 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d}{c} \cdot a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.4 \cdot 10^{-16} \lor \neg \left(d \leq 5.4 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.4e-16) (not (<= d 5.4e-31))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.4e-16) || !(d <= 5.4e-31)) {
		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 <= (-8.4d-16)) .or. (.not. (d <= 5.4d-31))) 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 <= -8.4e-16) || !(d <= 5.4e-31)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.4e-16) or not (d <= 5.4e-31):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.4e-16) || !(d <= 5.4e-31))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.4e-16) || ~((d <= 5.4e-31)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.4e-16], N[Not[LessEqual[d, 5.4e-31]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.4000000000000004e-16 or 5.40000000000000027e-31 < d

   1. Initial program 51.1%

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

      1. fmm-def 51.1%

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

      2. distribute-rgt-neg-out 51.1%

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

      3. +-commutative 51.1%

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

      4. fma-define 51.1%

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

   3. Simplified 51.1%

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

   5. Taylor expanded in c around 0 65.6%

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

      1. associate-*r/ 65.6%

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

      2. neg-mul-1 65.6%

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

   7. Simplified 65.6%

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

   if -8.4000000000000004e-16 < d < 5.40000000000000027e-31

   1. Initial program 71.8%

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

      1. fmm-def 71.8%

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

      2. distribute-rgt-neg-out 71.8%

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

      3. +-commutative 71.8%

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

      4. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in c around inf 72.2%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.4 \cdot 10^{-16} \lor \neg \left(d \leq 5.4 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+190} \lor \neg \left(d \leq 2.35 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.5e+190) (not (<= d 2.35e+169))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.5e+190) || !(d <= 2.35e+169)) {
		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 <= (-6.5d+190)) .or. (.not. (d <= 2.35d+169))) 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 <= -6.5e+190) || !(d <= 2.35e+169)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.5e+190) or not (d <= 2.35e+169):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.5e+190) || !(d <= 2.35e+169))
		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 <= -6.5e+190) || ~((d <= 2.35e+169)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.5e+190], N[Not[LessEqual[d, 2.35e+169]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5000000000000001e190 or 2.3499999999999999e169 < d

   1. Initial program 31.1%

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

      1. fmm-def 31.1%

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

      2. distribute-rgt-neg-out 31.1%

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

      3. +-commutative 31.1%

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

      4. fma-define 31.1%

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

   3. Simplified 31.1%

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

   5. Taylor expanded in c around 0 81.5%

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

      1. associate-*r/ 81.5%

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

      2. neg-mul-1 81.5%

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

   7. Simplified 81.5%

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

      1. neg-sub0 81.5%

         \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

      2. sub-neg 81.5%

         \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

      3. add-sqr-sqrt 50.3%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

      4. sqrt-unprod 51.8%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

      5. sqr-neg 51.8%

         \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

      6. sqrt-unprod 16.2%

         \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

      7. add-sqr-sqrt 32.3%

         \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

   9. Applied egg-rr 32.3%

      \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
   10. Step-by-step derivation

       1. +-lft-identity 32.3%

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

   11. Simplified 32.3%

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

   if -6.5000000000000001e190 < d < 2.3499999999999999e169

   1. Initial program 68.8%

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

      1. fmm-def 68.8%

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

      2. distribute-rgt-neg-out 68.8%

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

      3. +-commutative 68.8%

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

      4. fma-define 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in c around inf 52.9%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.5 \cdot 10^{+190} \lor \neg \left(d \leq 2.35 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 10.5% 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 60.1%

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

   1. fmm-def 60.1%

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

   2. distribute-rgt-neg-out 60.1%

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

   3. +-commutative 60.1%

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

   4. fma-define 60.1%

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

3. Simplified 60.1%

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

5. Taylor expanded in c around 0 41.8%

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

   1. associate-*r/ 41.8%

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

   2. neg-mul-1 41.8%

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

7. Simplified 41.8%

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

   1. neg-sub0 41.8%

      \[\leadsto \frac{\color{blue}{0 - a}}{d} \]

   2. sub-neg 41.8%

      \[\leadsto \frac{\color{blue}{0 + \left(-a\right)}}{d} \]

   3. add-sqr-sqrt 20.8%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{-a} \cdot \sqrt{-a}}}{d} \]

   4. sqrt-unprod 22.6%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{\left(-a\right) \cdot \left(-a\right)}}}{d} \]

   5. sqr-neg 22.6%

      \[\leadsto \frac{0 + \sqrt{\color{blue}{a \cdot a}}}{d} \]

   6. sqrt-unprod 6.0%

      \[\leadsto \frac{0 + \color{blue}{\sqrt{a} \cdot \sqrt{a}}}{d} \]

   7. add-sqr-sqrt 11.4%

      \[\leadsto \frac{0 + \color{blue}{a}}{d} \]

9. Applied egg-rr 11.4%

   \[\leadsto \frac{\color{blue}{0 + a}}{d} \]
10. Step-by-step derivation

    1. +-lft-identity 11.4%

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

11. Simplified 11.4%

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

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024167 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d))))))

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