Complex division, imag part

Percentage Accurate: 61.7% → 82.5%

Time: 10.4s

Alternatives: 10

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: 61.7% 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: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - a \cdot d\\ \mathbf{if}\;c \leq -3 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 10^{+39}:\\ \;\;\;\;\frac{t\_0}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* a d))))
   (if (<= c -3e+52)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -2.3e-131)
       (/ t_0 (fma c c (* d d)))
       (if (<= c 3.9e-91)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 1e+39)
           (/ t_0 (+ (* d d) (* c c)))
           (/ (+ b (* a (/ -1.0 (/ c d)))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (a * d);
	double tmp;
	if (c <= -3e+52) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -2.3e-131) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (c <= 3.9e-91) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1e+39) {
		tmp = t_0 / ((d * d) + (c * c));
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(a * d))
	tmp = 0.0
	if (c <= -3e+52)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -2.3e-131)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (c <= 3.9e-91)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 1e+39)
		tmp = Float64(t_0 / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(-1.0 / Float64(c / d)))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3e+52], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -2.3e-131], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.9e-91], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1e+39], N[(t$95$0 / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(-1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -3e52

   1. Initial program 45.2%

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

   3. Taylor expanded in c around inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. remove-double-neg 82.4%

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

      3. mul-1-neg 82.4%

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

      4. distribute-neg-in 82.4%

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

      5. distribute-neg-in 82.4%

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

      6. mul-1-neg 82.4%

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

      7. remove-double-neg 82.4%

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

      8. unsub-neg 82.4%

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

      9. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

   if -3e52 < c < -2.30000000000000022e-131

   1. Initial program 84.5%

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

      1. fma-define 84.5%

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

   3. Simplified 84.5%

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

   if -2.30000000000000022e-131 < c < 3.89999999999999994e-91

   1. Initial program 68.1%

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

   3. Taylor expanded in d around -inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-neg-frac2 93.0%

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

      3. mul-1-neg 93.0%

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

      4. unsub-neg 93.0%

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

      5. *-commutative 93.0%

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

      6. associate-/l* 91.0%

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

   5. Simplified 91.0%

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

      1. associate-*r/ 93.0%

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

      2. *-commutative 93.0%

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

      3. clear-num 93.0%

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

      4. *-commutative 93.0%

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

   7. Applied egg-rr 93.0%

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

   8. Taylor expanded in a around 0 85.5%

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

      1. neg-mul-1 85.5%

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

      2. neg-sub0 85.5%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 85.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 85.5%

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

      5. associate-/r* 93.0%

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

      6. *-lft-identity 93.0%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 93.0%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 93.0%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 93.0%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 93.0%

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

      11. distribute-neg-frac 93.0%

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

   10. Simplified 92.0%

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

   11. Taylor expanded in b around 0 93.0%

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

   if 3.89999999999999994e-91 < c < 9.9999999999999994e38

   1. Initial program 89.1%

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

   if 9.9999999999999994e38 < c

   1. Initial program 35.1%

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

   3. Taylor expanded in c around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. remove-double-neg 73.4%

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

      3. mul-1-neg 73.4%

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

      4. distribute-neg-in 73.4%

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

      5. distribute-neg-in 73.4%

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

      6. mul-1-neg 73.4%

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

      7. remove-double-neg 73.4%

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

      8. unsub-neg 73.4%

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

      9. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

      1. clear-num 78.5%

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

      2. inv-pow 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. unpow-1 78.5%

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

   9. Simplified 78.5%

      \[\leadsto \frac{b - a \cdot \color{blue}{\frac{1}{\frac{c}{d}}}}{c} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-131}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 10^{+39}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* d d) (* c c)))))
   (if (<= c -1.65e+50)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -2.5e-131)
       t_0
       (if (<= c 3.4e-91)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 1e+39) t_0 (/ (+ b (* a (/ -1.0 (/ c d)))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -1.65e+50) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -2.5e-131) {
		tmp = t_0;
	} else if (c <= 3.4e-91) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1e+39) {
		tmp = t_0;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c))
    if (c <= (-1.65d+50)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= (-2.5d-131)) then
        tmp = t_0
    else if (c <= 3.4d-91) then
        tmp = (((c * b) / d) - a) / d
    else if (c <= 1d+39) then
        tmp = t_0
    else
        tmp = (b + (a * ((-1.0d0) / (c / d)))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -1.65e+50) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -2.5e-131) {
		tmp = t_0;
	} else if (c <= 3.4e-91) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1e+39) {
		tmp = t_0;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c))
	tmp = 0
	if c <= -1.65e+50:
		tmp = (b - (a * (d / c))) / c
	elif c <= -2.5e-131:
		tmp = t_0
	elif c <= 3.4e-91:
		tmp = (((c * b) / d) - a) / d
	elif c <= 1e+39:
		tmp = t_0
	else:
		tmp = (b + (a * (-1.0 / (c / d)))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -1.65e+50)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -2.5e-131)
		tmp = t_0;
	elseif (c <= 3.4e-91)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 1e+39)
		tmp = t_0;
	else
		tmp = Float64(Float64(b + Float64(a * Float64(-1.0 / Float64(c / d)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	tmp = 0.0;
	if (c <= -1.65e+50)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -2.5e-131)
		tmp = t_0;
	elseif (c <= 3.4e-91)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 1e+39)
		tmp = t_0;
	else
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+50], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -2.5e-131], t$95$0, If[LessEqual[c, 3.4e-91], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1e+39], t$95$0, N[(N[(b + N[(a * N[(-1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.65e50

   1. Initial program 45.2%

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

   3. Taylor expanded in c around inf 82.4%

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

      1. mul-1-neg 82.4%

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

      2. remove-double-neg 82.4%

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

      3. mul-1-neg 82.4%

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

      4. distribute-neg-in 82.4%

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

      5. distribute-neg-in 82.4%

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

      6. mul-1-neg 82.4%

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

      7. remove-double-neg 82.4%

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

      8. unsub-neg 82.4%

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

      9. associate-/l* 89.0%

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

   5. Simplified 89.0%

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

   if -1.65e50 < c < -2.5000000000000002e-131 or 3.40000000000000027e-91 < c < 9.9999999999999994e38

   1. Initial program 86.6%

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

   if -2.5000000000000002e-131 < c < 3.40000000000000027e-91

   1. Initial program 68.1%

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

   3. Taylor expanded in d around -inf 93.0%

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

      1. mul-1-neg 93.0%

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

      2. distribute-neg-frac2 93.0%

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

      3. mul-1-neg 93.0%

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

      4. unsub-neg 93.0%

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

      5. *-commutative 93.0%

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

      6. associate-/l* 91.0%

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

   5. Simplified 91.0%

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

      1. associate-*r/ 93.0%

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

      2. *-commutative 93.0%

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

      3. clear-num 93.0%

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

      4. *-commutative 93.0%

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

   7. Applied egg-rr 93.0%

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

   8. Taylor expanded in a around 0 85.5%

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

      1. neg-mul-1 85.5%

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

      2. neg-sub0 85.5%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 85.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 85.5%

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

      5. associate-/r* 93.0%

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

      6. *-lft-identity 93.0%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 93.0%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 93.0%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 93.0%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 93.0%

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

      11. distribute-neg-frac 93.0%

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

   10. Simplified 92.0%

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

   11. Taylor expanded in b around 0 93.0%

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

   if 9.9999999999999994e38 < c

   1. Initial program 35.1%

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

   3. Taylor expanded in c around inf 73.4%

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

      1. mul-1-neg 73.4%

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

      2. remove-double-neg 73.4%

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

      3. mul-1-neg 73.4%

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

      4. distribute-neg-in 73.4%

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

      5. distribute-neg-in 73.4%

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

      6. mul-1-neg 73.4%

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

      7. remove-double-neg 73.4%

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

      8. unsub-neg 73.4%

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

      9. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

      1. clear-num 78.5%

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

      2. inv-pow 78.5%

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

   7. Applied egg-rr 78.5%

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

      1. unpow-1 78.5%

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

   9. Simplified 78.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+50}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 10^{+39}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{\frac{d}{c \cdot b}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.6e-15)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 1.02e-57)
     (/ (- (/ 1.0 (/ d (* c b))) a) d)
     (/ (+ b (* a (/ -1.0 (/ c d)))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e-15) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.02e-57) {
		tmp = ((1.0 / (d / (c * b))) - a) / d;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-4.6d-15)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 1.02d-57) then
        tmp = ((1.0d0 / (d / (c * b))) - a) / d
    else
        tmp = (b + (a * ((-1.0d0) / (c / d)))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e-15) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.02e-57) {
		tmp = ((1.0 / (d / (c * b))) - a) / d;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.6e-15:
		tmp = (b - (a * (d / c))) / c
	elif c <= 1.02e-57:
		tmp = ((1.0 / (d / (c * b))) - a) / d
	else:
		tmp = (b + (a * (-1.0 / (c / d)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.6e-15)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 1.02e-57)
		tmp = Float64(Float64(Float64(1.0 / Float64(d / Float64(c * b))) - a) / d);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(-1.0 / Float64(c / d)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.6e-15)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 1.02e-57)
		tmp = ((1.0 / (d / (c * b))) - a) / d;
	else
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.6e-15], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.02e-57], N[(N[(N[(1.0 / N[(d / N[(c * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b + N[(a * N[(-1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.59999999999999981e-15

   1. Initial program 56.0%

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

   3. Taylor expanded in c around inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. remove-double-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. unsub-neg 80.4%

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

      9. associate-/l* 85.5%

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

   5. Simplified 85.5%

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

   if -4.59999999999999981e-15 < c < 1.02e-57

   1. Initial program 71.9%

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

   3. Taylor expanded in d around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. distribute-neg-frac2 87.6%

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

      3. mul-1-neg 87.6%

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

      4. unsub-neg 87.6%

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

      5. *-commutative 87.6%

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

      6. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

      1. associate-*r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. clear-num 87.6%

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

      4. *-commutative 87.6%

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

   7. Applied egg-rr 87.6%

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

   8. Taylor expanded in a around 0 81.8%

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

      1. neg-mul-1 81.8%

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

      2. neg-sub0 81.8%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 81.8%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 81.8%

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

      5. associate-/r* 87.5%

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

      6. *-lft-identity 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 87.5%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 87.6%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 87.6%

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

      11. distribute-neg-frac 87.6%

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

   10. Simplified 86.8%

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

       1. associate-*r/ 87.6%

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

       2. clear-num 87.6%

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

   12. Applied egg-rr 87.6%

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

   if 1.02e-57 < c

   1. Initial program 47.9%

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

   3. Taylor expanded in c around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. remove-double-neg 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-in 69.6%

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

      5. distribute-neg-in 69.6%

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

      6. mul-1-neg 69.6%

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

      7. remove-double-neg 69.6%

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

      8. unsub-neg 69.6%

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

      9. associate-/l* 73.3%

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

   5. Simplified 73.3%

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

      1. clear-num 73.4%

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

      2. inv-pow 73.4%

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

   7. Applied egg-rr 73.4%

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

      1. unpow-1 73.4%

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

   9. Simplified 73.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{1}{\frac{d}{c \cdot b}} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2e-14)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 1.02e-57)
     (/ (- (/ (* c b) d) a) d)
     (/ (+ b (* a (/ -1.0 (/ c d)))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2e-14) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.02e-57) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2d-14)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 1.02d-57) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = (b + (a * ((-1.0d0) / (c / d)))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2e-14) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 1.02e-57) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2e-14:
		tmp = (b - (a * (d / c))) / c
	elif c <= 1.02e-57:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = (b + (a * (-1.0 / (c / d)))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2e-14)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 1.02e-57)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b + Float64(a * Float64(-1.0 / Float64(c / d)))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2e-14)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 1.02e-57)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = (b + (a * (-1.0 / (c / d)))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2e-14], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.02e-57], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b + N[(a * N[(-1.0 / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2e-14

   1. Initial program 56.0%

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

   3. Taylor expanded in c around inf 80.4%

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

      1. mul-1-neg 80.4%

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

      2. remove-double-neg 80.4%

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

      3. mul-1-neg 80.4%

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

      4. distribute-neg-in 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. mul-1-neg 80.4%

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

      7. remove-double-neg 80.4%

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

      8. unsub-neg 80.4%

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

      9. associate-/l* 85.5%

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

   5. Simplified 85.5%

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

   if -2e-14 < c < 1.02e-57

   1. Initial program 71.9%

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

   3. Taylor expanded in d around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. distribute-neg-frac2 87.6%

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

      3. mul-1-neg 87.6%

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

      4. unsub-neg 87.6%

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

      5. *-commutative 87.6%

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

      6. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

      1. associate-*r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. clear-num 87.6%

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

      4. *-commutative 87.6%

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

   7. Applied egg-rr 87.6%

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

   8. Taylor expanded in a around 0 81.8%

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

      1. neg-mul-1 81.8%

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

      2. neg-sub0 81.8%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 81.8%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 81.8%

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

      5. associate-/r* 87.5%

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

      6. *-lft-identity 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 87.5%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 87.6%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 87.6%

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

      11. distribute-neg-frac 87.6%

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

   10. Simplified 86.8%

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

   11. Taylor expanded in b around 0 87.6%

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

   if 1.02e-57 < c

   1. Initial program 47.9%

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

   3. Taylor expanded in c around inf 69.6%

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

      1. mul-1-neg 69.6%

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

      2. remove-double-neg 69.6%

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

      3. mul-1-neg 69.6%

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

      4. distribute-neg-in 69.6%

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

      5. distribute-neg-in 69.6%

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

      6. mul-1-neg 69.6%

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

      7. remove-double-neg 69.6%

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

      8. unsub-neg 69.6%

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

      9. associate-/l* 73.3%

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

   5. Simplified 73.3%

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

      1. clear-num 73.4%

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

      2. inv-pow 73.4%

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

   7. Applied egg-rr 73.4%

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

      1. unpow-1 73.4%

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

   9. Simplified 73.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{-1}{\frac{c}{d}}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{-14} \lor \neg \left(c \leq 4.5 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.75e-14) (not (<= c 4.5e-58)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ (* c b) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.75e-14) || !(c <= 4.5e-58)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.75d-14)) .or. (.not. (c <= 4.5d-58))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = (((c * b) / d) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.75e-14) || !(c <= 4.5e-58)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.75e-14) or not (c <= 4.5e-58):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.75e-14) || !(c <= 4.5e-58))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.75e-14) || ~((c <= 4.5e-58)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.75e-14], N[Not[LessEqual[c, 4.5e-58]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.74999999999999996e-14 or 4.5000000000000003e-58 < c

   1. Initial program 51.3%

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

   3. Taylor expanded in c around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. remove-double-neg 74.1%

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

      3. mul-1-neg 74.1%

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

      4. distribute-neg-in 74.1%

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

      5. distribute-neg-in 74.1%

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

      6. mul-1-neg 74.1%

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

      7. remove-double-neg 74.1%

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

      8. unsub-neg 74.1%

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

      9. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   if -2.74999999999999996e-14 < c < 4.5000000000000003e-58

   1. Initial program 71.9%

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

   3. Taylor expanded in d around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. distribute-neg-frac2 87.6%

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

      3. mul-1-neg 87.6%

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

      4. unsub-neg 87.6%

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

      5. *-commutative 87.6%

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

      6. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

      1. associate-*r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. clear-num 87.6%

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

      4. *-commutative 87.6%

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

   7. Applied egg-rr 87.6%

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

   8. Taylor expanded in a around 0 81.8%

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

      1. neg-mul-1 81.8%

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

      2. neg-sub0 81.8%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 81.8%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 81.8%

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

      5. associate-/r* 87.5%

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

      6. *-lft-identity 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 87.5%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 87.6%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 87.6%

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

      11. distribute-neg-frac 87.6%

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

   10. Simplified 86.8%

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

   11. Taylor expanded in b around 0 87.6%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{-14} \lor \neg \left(c \leq 4.5 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-15} \lor \neg \left(c \leq 10^{-57}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.1e-15) (not (<= c 1e-57)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.1e-15) || !(c <= 1e-57)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.1d-15)) .or. (.not. (c <= 1d-57))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.1e-15) || !(c <= 1e-57)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.1e-15) or not (c <= 1e-57):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.1e-15) || !(c <= 1e-57))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.1e-15) || ~((c <= 1e-57)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.1e-15], N[Not[LessEqual[c, 1e-57]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.10000000000000036e-15 or 9.99999999999999955e-58 < c

   1. Initial program 51.3%

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

   3. Taylor expanded in c around inf 74.1%

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

      1. mul-1-neg 74.1%

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

      2. remove-double-neg 74.1%

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

      3. mul-1-neg 74.1%

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

      4. distribute-neg-in 74.1%

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

      5. distribute-neg-in 74.1%

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

      6. mul-1-neg 74.1%

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

      7. remove-double-neg 74.1%

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

      8. unsub-neg 74.1%

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

      9. associate-/l* 78.5%

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

   5. Simplified 78.5%

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

   if -4.10000000000000036e-15 < c < 9.99999999999999955e-58

   1. Initial program 71.9%

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

   3. Taylor expanded in d around -inf 87.6%

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

      1. mul-1-neg 87.6%

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

      2. distribute-neg-frac2 87.6%

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

      3. mul-1-neg 87.6%

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

      4. unsub-neg 87.6%

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

      5. *-commutative 87.6%

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

      6. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

      1. associate-*r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. clear-num 87.6%

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

      4. *-commutative 87.6%

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

   7. Applied egg-rr 87.6%

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

   8. Taylor expanded in a around 0 81.8%

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

      1. neg-mul-1 81.8%

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

      2. neg-sub0 81.8%

         \[\leadsto \color{blue}{\left(0 - \frac{a}{d}\right)} + \frac{b \cdot c}{{d}^{2}} \]

      3. *-commutative 81.8%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{c \cdot b}}{{d}^{2}} \]

      4. unpow2 81.8%

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

      5. associate-/r* 87.5%

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

      6. *-lft-identity 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\frac{\color{blue}{1 \cdot \left(c \cdot b\right)}}{d}}{d} \]

      7. associate-*l/ 87.5%

         \[\leadsto \left(0 - \frac{a}{d}\right) + \frac{\color{blue}{\frac{1}{d} \cdot \left(c \cdot b\right)}}{d} \]

      8. associate--r- 87.5%

         \[\leadsto \color{blue}{0 - \left(\frac{a}{d} - \frac{\frac{1}{d} \cdot \left(c \cdot b\right)}{d}\right)} \]

      9. div-sub 87.6%

         \[\leadsto 0 - \color{blue}{\frac{a - \frac{1}{d} \cdot \left(c \cdot b\right)}{d}} \]

      10. neg-sub0 87.6%

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

      11. distribute-neg-frac 87.6%

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

   10. Simplified 86.8%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-15} \lor \neg \left(c \leq 10^{-57}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{-14} \lor \neg \left(c \leq 4.9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.05e-14) (not (<= c 4.9e-107)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.05e-14) || !(c <= 4.9e-107)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.05d-14)) .or. (.not. (c <= 4.9d-107))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.05e-14) || !(c <= 4.9e-107)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.05e-14) or not (c <= 4.9e-107):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.05e-14) || !(c <= 4.9e-107))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.05e-14) || ~((c <= 4.9e-107)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.05e-14], N[Not[LessEqual[c, 4.9e-107]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.0500000000000001e-14 or 4.8999999999999998e-107 < c

   1. Initial program 54.3%

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

   3. Taylor expanded in c around inf 71.8%

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

      1. mul-1-neg 71.8%

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

      2. remove-double-neg 71.8%

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

      3. mul-1-neg 71.8%

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

      4. distribute-neg-in 71.8%

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

      5. distribute-neg-in 71.8%

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

      6. mul-1-neg 71.8%

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

      7. remove-double-neg 71.8%

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

      8. unsub-neg 71.8%

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

      9. associate-/l* 75.8%

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

   5. Simplified 75.8%

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

   if -2.0500000000000001e-14 < c < 4.8999999999999998e-107

   1. Initial program 70.3%

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

   3. Taylor expanded in c around 0 72.3%

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

      1. associate-*r/ 72.3%

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

      2. mul-1-neg 72.3%

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

   5. Simplified 72.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{-14} \lor \neg \left(c \leq 4.9 \cdot 10^{-107}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-14} \lor \neg \left(c \leq 6.5 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.7e-14) (not (<= c 6.5e-91))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.7e-14) || !(c <= 6.5e-91)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.7d-14)) .or. (.not. (c <= 6.5d-91))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.7e-14) || !(c <= 6.5e-91)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.7e-14) or not (c <= 6.5e-91):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.7e-14) || !(c <= 6.5e-91))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.7e-14) || ~((c <= 6.5e-91)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.7e-14], N[Not[LessEqual[c, 6.5e-91]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6999999999999999e-14 or 6.5000000000000001e-91 < c

   1. Initial program 54.0%

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

   3. Taylor expanded in c around inf 63.1%

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

   if -2.6999999999999999e-14 < c < 6.5000000000000001e-91

   1. Initial program 70.0%

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

   3. Taylor expanded in c around 0 71.0%

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

      1. associate-*r/ 71.0%

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

      2. mul-1-neg 71.0%

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

   5. Simplified 71.0%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{-14} \lor \neg \left(c \leq 6.5 \cdot 10^{-91}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+186} \lor \neg \left(d \leq 3.6 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.2e+186) (not (<= d 3.6e+135))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+186) || !(d <= 3.6e+135)) {
		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+186)) .or. (.not. (d <= 3.6d+135))) 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+186) || !(d <= 3.6e+135)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.2e+186) or not (d <= 3.6e+135):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.2e+186) || !(d <= 3.6e+135))
		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+186) || ~((d <= 3.6e+135)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.2e+186], N[Not[LessEqual[d, 3.6e+135]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.2000000000000003e186 or 3.5999999999999998e135 < d

   1. Initial program 34.8%

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

   3. Taylor expanded in c around 0 85.3%

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

      1. associate-*r/ 85.3%

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

      2. mul-1-neg 85.3%

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

   5. Simplified 85.3%

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

      1. add-sqr-sqrt 48.8%

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

      2. sqrt-unprod 56.3%

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

      3. sqr-neg 56.3%

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

      4. sqrt-unprod 12.2%

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

      5. add-sqr-sqrt 32.4%

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

      6. *-un-lft-identity 32.4%

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

      7. *-un-lft-identity 32.4%

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

      8. times-frac 32.4%

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

      9. metadata-eval 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. *-lft-identity 32.4%

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

   9. Simplified 32.4%

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

   if -7.2000000000000003e186 < d < 3.5999999999999998e135

   1. Initial program 69.8%

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

   3. Taylor expanded in c around inf 49.7%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+186} \lor \neg \left(d \leq 3.6 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 11.0% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in c around 0 43.9%

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

   1. associate-*r/ 43.9%

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

   2. mul-1-neg 43.9%

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

5. Simplified 43.9%

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

   1. add-sqr-sqrt 23.7%

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

   2. sqrt-unprod 23.4%

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

   3. sqr-neg 23.4%

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

   4. sqrt-unprod 4.9%

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

   5. add-sqr-sqrt 11.8%

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

   6. *-un-lft-identity 11.8%

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

   7. *-un-lft-identity 11.8%

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

   8. times-frac 11.8%

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

   9. metadata-eval 11.8%

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

7. Applied egg-rr 11.8%

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

   1. *-lft-identity 11.8%

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

9. Simplified 11.8%

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

Developer Target 1: 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 2024118 
(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))))