Complex division, imag part

Percentage Accurate: 62.0% → 83.4%

Time: 10.4s

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: 62.0% 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: 83.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(d, c\right)\right)}^{2}\\ t_1 := c \cdot \frac{b}{t\_0} - a \cdot \frac{d}{t\_0}\\ \mathbf{if}\;c \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.4 \cdot 10^{-91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (pow (hypot d c) 2.0)) (t_1 (- (* c (/ b t_0)) (* a (/ d t_0)))))
   (if (<= c -2e+109)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -1.4e-91)
       t_1
       (if (<= c 4.8e-23)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 7.2e+96) t_1 (/ (- b (* d (/ a c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = pow(hypot(d, c), 2.0);
	double t_1 = (c * (b / t_0)) - (a * (d / t_0));
	double tmp;
	if (c <= -2e+109) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -1.4e-91) {
		tmp = t_1;
	} else if (c <= 4.8e-23) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 7.2e+96) {
		tmp = t_1;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = Math.pow(Math.hypot(d, c), 2.0);
	double t_1 = (c * (b / t_0)) - (a * (d / t_0));
	double tmp;
	if (c <= -2e+109) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -1.4e-91) {
		tmp = t_1;
	} else if (c <= 4.8e-23) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 7.2e+96) {
		tmp = t_1;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = math.pow(math.hypot(d, c), 2.0)
	t_1 = (c * (b / t_0)) - (a * (d / t_0))
	tmp = 0
	if c <= -2e+109:
		tmp = (b - (a * (d / c))) / c
	elif c <= -1.4e-91:
		tmp = t_1
	elif c <= 4.8e-23:
		tmp = (((c * b) / d) - a) / d
	elif c <= 7.2e+96:
		tmp = t_1
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = hypot(d, c) ^ 2.0
	t_1 = Float64(Float64(c * Float64(b / t_0)) - Float64(a * Float64(d / t_0)))
	tmp = 0.0
	if (c <= -2e+109)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -1.4e-91)
		tmp = t_1;
	elseif (c <= 4.8e-23)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 7.2e+96)
		tmp = t_1;
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = hypot(d, c) ^ 2.0;
	t_1 = (c * (b / t_0)) - (a * (d / t_0));
	tmp = 0.0;
	if (c <= -2e+109)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -1.4e-91)
		tmp = t_1;
	elseif (c <= 4.8e-23)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 7.2e+96)
		tmp = t_1;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[Power[N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(a * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2e+109], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.4e-91], t$95$1, If[LessEqual[c, 4.8e-23], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.2e+96], t$95$1, N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.99999999999999996e109

   1. Initial program 41.1%

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

   3. Taylor expanded in c around inf 80.7%

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

      1. mul-1-neg 80.7%

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

      2. unsub-neg 80.7%

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

      3. associate-/l* 87.3%

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

   5. Simplified 87.3%

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

   if -1.99999999999999996e109 < c < -1.4e-91 or 4.79999999999999993e-23 < c < 7.20000000000000026e96

   1. Initial program 75.2%

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

   3. Taylor expanded in b around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. mul-1-neg 75.2%

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

      3. unsub-neg 75.2%

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

   5. Simplified 83.2%

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

   if -1.4e-91 < c < 4.79999999999999993e-23

   1. Initial program 63.4%

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

   3. Taylor expanded in c around 0 79.7%

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

      1. +-commutative 79.7%

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

      2. mul-1-neg 79.7%

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

      3. unsub-neg 79.7%

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

      4. unpow2 79.7%

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

      5. associate-/r* 89.0%

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

      6. div-sub 90.9%

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

      7. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

   6. Taylor expanded in b around 0 90.9%

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

   if 7.20000000000000026e96 < c

   1. Initial program 30.8%

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

   3. Taylor expanded in c around inf 81.4%

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

      1. remove-double-neg 81.4%

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

      2. mul-1-neg 81.4%

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

      3. neg-mul-1 81.4%

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

      4. distribute-lft-in 81.4%

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

      5. distribute-lft-in 81.4%

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

      6. neg-mul-1 81.4%

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

      7. mul-1-neg 81.4%

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

      8. remove-double-neg 81.4%

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

      9. associate-*r/ 81.4%

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

      10. associate-*r* 81.4%

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

      11. neg-mul-1 81.4%

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

   5. Simplified 81.4%

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

   6. Taylor expanded in a around 0 81.4%

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

      1. mul-1-neg 81.4%

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

      2. *-commutative 81.4%

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

      3. associate-*r/ 91.7%

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

      4. distribute-rgt-neg-in 91.7%

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

   8. Simplified 91.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -1.4 \cdot 10^{-91}:\\ \;\;\;\;c \cdot \frac{b}{{\left(\mathsf{hypot}\left(d, c\right)\right)}^{2}} - a \cdot \frac{d}{{\left(\mathsf{hypot}\left(d, c\right)\right)}^{2}}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{+96}:\\ \;\;\;\;c \cdot \frac{b}{{\left(\mathsf{hypot}\left(d, c\right)\right)}^{2}} - a \cdot \frac{d}{{\left(\mathsf{hypot}\left(d, c\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -1.75 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -6.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* a (/ d c))) c)))
   (if (<= c -1.75e+64)
     t_0
     (if (<= c -6.9e-74)
       (/ (* c b) (+ (* c c) (* d d)))
       (if (<= c -1.05e-84)
         t_0
         (if (<= c 3.4e-14)
           (/ (- (/ (* c b) d) a) d)
           (/ (- b (* d (/ a c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -1.75e+64) {
		tmp = t_0;
	} else if (c <= -6.9e-74) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= -1.05e-84) {
		tmp = t_0;
	} else if (c <= 3.4e-14) {
		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) :: t_0
    real(8) :: tmp
    t_0 = (b - (a * (d / c))) / c
    if (c <= (-1.75d+64)) then
        tmp = t_0
    else if (c <= (-6.9d-74)) then
        tmp = (c * b) / ((c * c) + (d * d))
    else if (c <= (-1.05d-84)) then
        tmp = t_0
    else if (c <= 3.4d-14) 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 t_0 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -1.75e+64) {
		tmp = t_0;
	} else if (c <= -6.9e-74) {
		tmp = (c * b) / ((c * c) + (d * d));
	} else if (c <= -1.05e-84) {
		tmp = t_0;
	} else if (c <= 3.4e-14) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -1.75e+64:
		tmp = t_0
	elif c <= -6.9e-74:
		tmp = (c * b) / ((c * c) + (d * d))
	elif c <= -1.05e-84:
		tmp = t_0
	elif c <= 3.4e-14:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = (b - (d * (a / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -1.75e+64)
		tmp = t_0;
	elseif (c <= -6.9e-74)
		tmp = Float64(Float64(c * b) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= -1.05e-84)
		tmp = t_0;
	elseif (c <= 3.4e-14)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -1.75e+64)
		tmp = t_0;
	elseif (c <= -6.9e-74)
		tmp = (c * b) / ((c * c) + (d * d));
	elseif (c <= -1.05e-84)
		tmp = t_0;
	elseif (c <= 3.4e-14)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = (b - (d * (a / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.75e+64], t$95$0, If[LessEqual[c, -6.9e-74], N[(N[(c * b), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.05e-84], t$95$0, If[LessEqual[c, 3.4e-14], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.7499999999999999e64 or -6.89999999999999981e-74 < c < -1.04999999999999999e-84

   1. Initial program 46.2%

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

   3. Taylor expanded in c around inf 74.7%

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

      1. mul-1-neg 74.7%

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

      2. unsub-neg 74.7%

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

      3. associate-/l* 81.7%

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

   5. Simplified 81.7%

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

   if -1.7499999999999999e64 < c < -6.89999999999999981e-74

   1. Initial program 88.2%

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

   3. Taylor expanded in b around inf 77.8%

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

      1. *-commutative 77.8%

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

   5. Simplified 77.8%

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

   if -1.04999999999999999e-84 < c < 3.40000000000000003e-14

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. mul-1-neg 79.8%

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

      3. unsub-neg 79.8%

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

      4. unpow2 79.8%

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

      5. associate-/r* 89.0%

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

      6. div-sub 90.9%

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

      7. associate-/l* 90.1%

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

   5. Simplified 90.1%

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

   6. Taylor expanded in b around 0 90.9%

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

   if 3.40000000000000003e-14 < c

   1. Initial program 45.1%

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

   3. Taylor expanded in c around inf 72.5%

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

      1. remove-double-neg 72.5%

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

      2. mul-1-neg 72.5%

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

      3. neg-mul-1 72.5%

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

      4. distribute-lft-in 72.5%

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

      5. distribute-lft-in 72.5%

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

      6. neg-mul-1 72.5%

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

      7. mul-1-neg 72.5%

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

      8. remove-double-neg 72.5%

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

      9. associate-*r/ 72.5%

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

      10. associate-*r* 72.5%

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

      11. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in a around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r/ 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.75 \cdot 10^{+64}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -6.9 \cdot 10^{-74}:\\ \;\;\;\;\frac{c \cdot b}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq -1.05 \cdot 10^{-84}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-119}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.2e+78)
   (/ (- b (* a (/ d c))) c)
   (if (<= c -6e-119)
     (/ (- (* c b) (* a d)) (+ (* c c) (* d d)))
     (if (<= c 1.8e-19) (/ (- (/ (* c b) d) a) d) (/ (- b (* d (/ a c))) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e+78) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -6e-119) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 1.8e-19) {
		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 (c <= (-1.2d+78)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= (-6d-119)) then
        tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
    else if (c <= 1.8d-19) 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 (c <= -1.2e+78) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -6e-119) {
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 1.8e-19) {
		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 c <= -1.2e+78:
		tmp = (b - (a * (d / c))) / c
	elif c <= -6e-119:
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d))
	elif c <= 1.8e-19:
		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 (c <= -1.2e+78)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -6e-119)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.8e-19)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.2e+78)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= -6e-119)
		tmp = ((c * b) - (a * d)) / ((c * c) + (d * d));
	elseif (c <= 1.8e-19)
		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[LessEqual[c, -1.2e+78], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -6e-119], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.8e-19], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.1999999999999999e78

   1. Initial program 39.9%

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

   3. Taylor expanded in c around inf 73.2%

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

      1. mul-1-neg 73.2%

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

      2. unsub-neg 73.2%

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

      3. associate-/l* 78.7%

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

   5. Simplified 78.7%

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

   if -1.1999999999999999e78 < c < -6.0000000000000004e-119

   1. Initial program 88.7%

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

   if -6.0000000000000004e-119 < c < 1.8000000000000001e-19

   1. Initial program 62.3%

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

   3. Taylor expanded in c around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. unsub-neg 79.0%

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

      4. unpow2 79.0%

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

      5. associate-/r* 88.6%

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

      6. div-sub 90.6%

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

      7. associate-/l* 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in b around 0 90.6%

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

   if 1.8000000000000001e-19 < c

   1. Initial program 45.1%

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

   3. Taylor expanded in c around inf 72.5%

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

      1. remove-double-neg 72.5%

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

      2. mul-1-neg 72.5%

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

      3. neg-mul-1 72.5%

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

      4. distribute-lft-in 72.5%

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

      5. distribute-lft-in 72.5%

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

      6. neg-mul-1 72.5%

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

      7. mul-1-neg 72.5%

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

      8. remove-double-neg 72.5%

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

      9. associate-*r/ 72.5%

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

      10. associate-*r* 72.5%

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

      11. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in a around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r/ 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -6 \cdot 10^{-119}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq 3.7 \cdot 10^{+17}:\\ \;\;\;\;d \cdot \left(\frac{a}{c} \cdot \frac{-1}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.5e-34)
   (/ b c)
   (if (<= c 8.6e-12)
     (/ a (- d))
     (if (<= c 3.7e+17) (* d (* (/ a c) (/ -1.0 c))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.5e-34) {
		tmp = b / c;
	} else if (c <= 8.6e-12) {
		tmp = a / -d;
	} else if (c <= 3.7e+17) {
		tmp = d * ((a / c) * (-1.0 / c));
	} 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 (c <= (-6.5d-34)) then
        tmp = b / c
    else if (c <= 8.6d-12) then
        tmp = a / -d
    else if (c <= 3.7d+17) then
        tmp = d * ((a / c) * ((-1.0d0) / c))
    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 (c <= -6.5e-34) {
		tmp = b / c;
	} else if (c <= 8.6e-12) {
		tmp = a / -d;
	} else if (c <= 3.7e+17) {
		tmp = d * ((a / c) * (-1.0 / c));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.5e-34:
		tmp = b / c
	elif c <= 8.6e-12:
		tmp = a / -d
	elif c <= 3.7e+17:
		tmp = d * ((a / c) * (-1.0 / c))
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.5e-34)
		tmp = Float64(b / c);
	elseif (c <= 8.6e-12)
		tmp = Float64(a / Float64(-d));
	elseif (c <= 3.7e+17)
		tmp = Float64(d * Float64(Float64(a / c) * Float64(-1.0 / c)));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6.5e-34)
		tmp = b / c;
	elseif (c <= 8.6e-12)
		tmp = a / -d;
	elseif (c <= 3.7e+17)
		tmp = d * ((a / c) * (-1.0 / c));
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.5e-34], N[(b / c), $MachinePrecision], If[LessEqual[c, 8.6e-12], N[(a / (-d)), $MachinePrecision], If[LessEqual[c, 3.7e+17], N[(d * N[(N[(a / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.49999999999999985e-34 or 3.7e17 < c

   1. Initial program 50.6%

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

   3. Taylor expanded in c around inf 69.3%

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

   if -6.49999999999999985e-34 < c < 8.59999999999999971e-12

   1. Initial program 64.8%

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

   3. Taylor expanded in c around 0 64.8%

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

      1. associate-*r/ 64.8%

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

      2. neg-mul-1 64.8%

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

   5. Simplified 64.8%

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

   if 8.59999999999999971e-12 < c < 3.7e17

   1. Initial program 86.0%

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

   3. Taylor expanded in c around inf 85.6%

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

      1. remove-double-neg 85.6%

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

      2. mul-1-neg 85.6%

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

      3. neg-mul-1 85.6%

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

      4. distribute-lft-in 85.6%

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

      5. distribute-lft-in 85.6%

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

      6. neg-mul-1 85.6%

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

      7. mul-1-neg 85.6%

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

      8. remove-double-neg 85.6%

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

      9. associate-*r/ 85.6%

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

      10. associate-*r* 85.6%

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

      11. neg-mul-1 85.6%

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

   5. Simplified 85.6%

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

   6. Taylor expanded in a around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. associate-*r/ 85.6%

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

      4. distribute-rgt-neg-in 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in b around 0 76.6%

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

       1. associate-*l/ 76.8%

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

       2. associate-*l* 76.8%

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

       3. *-commutative 76.8%

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

       4. associate-*r/ 76.8%

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

       5. mul-1-neg 76.8%

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

   11. Simplified 76.8%

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

       1. neg-mul-1 76.8%

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

       2. *-commutative 76.8%

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

       3. pow2 76.8%

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

       4. times-frac 76.6%

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

   13. Applied egg-rr 76.6%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-12}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq 3.7 \cdot 10^{+17}:\\ \;\;\;\;d \cdot \left(\frac{a}{c} \cdot \frac{-1}{c}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.02 \cdot 10^{+18} \lor \neg \left(d \leq 7 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.02e+18) (not (<= d 7e+61)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.02e+18) || !(d <= 7e+61)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.02d+18)) .or. (.not. (d <= 7d+61))) then
        tmp = a / -d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.02e+18) || !(d <= 7e+61)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.02e+18) or not (d <= 7e+61):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.02e+18) || !(d <= 7e+61))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.02e+18) || ~((d <= 7e+61)))
		tmp = a / -d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.02e+18], N[Not[LessEqual[d, 7e+61]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.02e18 or 7.00000000000000036e61 < d

   1. Initial program 35.7%

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

   3. Taylor expanded in c around 0 69.6%

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

      1. associate-*r/ 69.6%

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

      2. neg-mul-1 69.6%

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

   5. Simplified 69.6%

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

   if -2.02e18 < d < 7.00000000000000036e61

   1. Initial program 74.9%

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

   3. Taylor expanded in c around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. unsub-neg 75.8%

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

      3. associate-/l* 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.02 \cdot 10^{+18} \lor \neg \left(d \leq 7 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-43} \lor \neg \left(c \leq 1.04 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4e-43) (not (<= c 1.04e-13)))
   (/ (- 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 <= -4e-43) || !(c <= 1.04e-13)) {
		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 <= (-4d-43)) .or. (.not. (c <= 1.04d-13))) 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 <= -4e-43) || !(c <= 1.04e-13)) {
		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 <= -4e-43) or not (c <= 1.04e-13):
		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 <= -4e-43) || !(c <= 1.04e-13))
		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 <= -4e-43) || ~((c <= 1.04e-13)))
		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, -4e-43], N[Not[LessEqual[c, 1.04e-13]], $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 < -4.00000000000000031e-43 or 1.03999999999999999e-13 < c

   1. Initial program 52.4%

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

   3. Taylor expanded in c around inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

      3. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

   if -4.00000000000000031e-43 < c < 1.03999999999999999e-13

   1. Initial program 65.2%

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

   3. Taylor expanded in c around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. unpow2 77.4%

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

      5. associate-/r* 85.8%

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

      6. div-sub 88.4%

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

      7. associate-/l* 87.6%

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

   5. Simplified 87.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{-43} \lor \neg \left(c \leq 1.04 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{-42} \lor \neg \left(c \leq 8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.8e-42) (not (<= c 8e-12)))
   (/ (- 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 <= -2.8e-42) || !(c <= 8e-12)) {
		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 <= (-2.8d-42)) .or. (.not. (c <= 8d-12))) 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 <= -2.8e-42) || !(c <= 8e-12)) {
		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 <= -2.8e-42) or not (c <= 8e-12):
		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 <= -2.8e-42) || !(c <= 8e-12))
		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 <= -2.8e-42) || ~((c <= 8e-12)))
		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, -2.8e-42], N[Not[LessEqual[c, 8e-12]], $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 < -2.79999999999999998e-42 or 7.99999999999999984e-12 < c

   1. Initial program 52.4%

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

   3. Taylor expanded in c around inf 72.3%

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

      1. mul-1-neg 72.3%

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

      2. unsub-neg 72.3%

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

      3. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

   if -2.79999999999999998e-42 < c < 7.99999999999999984e-12

   1. Initial program 65.2%

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

   3. Taylor expanded in c around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. unpow2 77.4%

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

      5. associate-/r* 85.8%

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

      6. div-sub 88.4%

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

      7. associate-/l* 87.6%

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

   5. Simplified 87.6%

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

   6. Taylor expanded in b around 0 88.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{-42} \lor \neg \left(c \leq 8 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.4e-43)
   (/ (- b (* a (/ d c))) c)
   (if (<= c 5.3e-21) (/ (- (/ (* c b) d) a) d) (/ (- b (* d (/ a c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.4e-43) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 5.3e-21) {
		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 (c <= (-3.4d-43)) then
        tmp = (b - (a * (d / c))) / c
    else if (c <= 5.3d-21) 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 (c <= -3.4e-43) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= 5.3e-21) {
		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 c <= -3.4e-43:
		tmp = (b - (a * (d / c))) / c
	elif c <= 5.3e-21:
		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 (c <= -3.4e-43)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= 5.3e-21)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.4e-43)
		tmp = (b - (a * (d / c))) / c;
	elseif (c <= 5.3e-21)
		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[LessEqual[c, -3.4e-43], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 5.3e-21], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.4000000000000001e-43

   1. Initial program 61.3%

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

   3. Taylor expanded in c around inf 72.0%

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

      1. mul-1-neg 72.0%

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

      2. unsub-neg 72.0%

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

      3. associate-/l* 75.4%

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

   5. Simplified 75.4%

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

   if -3.4000000000000001e-43 < c < 5.2999999999999999e-21

   1. Initial program 65.2%

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

   3. Taylor expanded in c around 0 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. unpow2 77.4%

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

      5. associate-/r* 85.8%

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

      6. div-sub 88.4%

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

      7. associate-/l* 87.6%

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

   5. Simplified 87.6%

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

   6. Taylor expanded in b around 0 88.4%

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

   if 5.2999999999999999e-21 < c

   1. Initial program 45.1%

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

   3. Taylor expanded in c around inf 72.5%

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

      1. remove-double-neg 72.5%

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

      2. mul-1-neg 72.5%

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

      3. neg-mul-1 72.5%

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

      4. distribute-lft-in 72.5%

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

      5. distribute-lft-in 72.5%

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

      6. neg-mul-1 72.5%

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

      7. mul-1-neg 72.5%

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

      8. remove-double-neg 72.5%

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

      9. associate-*r/ 72.5%

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

      10. associate-*r* 72.5%

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

      11. neg-mul-1 72.5%

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

   5. Simplified 72.5%

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

   6. Taylor expanded in a around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. *-commutative 72.5%

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

      3. associate-*r/ 81.2%

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

      4. distribute-rgt-neg-in 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 5.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-33} \lor \neg \left(c \leq 2.55 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.45e-33) (not (<= c 2.55e+21))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.45e-33) || !(c <= 2.55e+21)) {
		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 <= (-1.45d-33)) .or. (.not. (c <= 2.55d+21))) 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 <= -1.45e-33) || !(c <= 2.55e+21)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.45e-33) or not (c <= 2.55e+21):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.45e-33) || !(c <= 2.55e+21))
		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 <= -1.45e-33) || ~((c <= 2.55e+21)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.45e-33], N[Not[LessEqual[c, 2.55e+21]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.45000000000000001e-33 or 2.55e21 < c

   1. Initial program 50.6%

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

   3. Taylor expanded in c around inf 69.3%

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

   if -1.45000000000000001e-33 < c < 2.55e21

   1. Initial program 66.0%

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

   3. Taylor expanded in c around 0 62.3%

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

      1. associate-*r/ 62.3%

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

      2. neg-mul-1 62.3%

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

   5. Simplified 62.3%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-33} \lor \neg \left(c \leq 2.55 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 9e+114) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 9e+114) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 9d+114) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 9e+114) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 9e+114:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 9e+114)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 9e+114)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 9e+114], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 9.0000000000000001e114

   1. Initial program 63.0%

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

   3. Taylor expanded in c around inf 48.6%

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

   if 9.0000000000000001e114 < d

   1. Initial program 30.1%

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

   3. Taylor expanded in b around 0 30.3%

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

      1. mul-1-neg 30.3%

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

      2. associate-/l* 36.9%

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

      3. distribute-rgt-neg-in 36.9%

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

      4. rem-square-sqrt 36.9%

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

      5. unpow2 36.9%

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

      6. unpow2 36.9%

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

      7. hypot-undefine 36.9%

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

      8. unpow2 36.9%

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

      9. unpow2 36.9%

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

      10. hypot-undefine 36.9%

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

      11. unpow2 36.9%

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

      12. distribute-frac-neg 36.9%

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

   5. Simplified 36.9%

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

   6. Taylor expanded in d around inf 80.8%

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

      1. associate-*r/ 80.9%

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

      2. *-commutative 80.9%

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

      3. neg-mul-1 80.9%

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

      4. add-sqr-sqrt 37.7%

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

      5. sqrt-unprod 41.3%

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

      6. sqr-neg 41.3%

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

      7. sqrt-unprod 14.1%

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

      8. add-sqr-sqrt 31.0%

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

   8. Applied egg-rr 31.0%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq 9 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 11.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in b around 0 32.8%

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

   1. mul-1-neg 32.8%

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

   2. associate-/l* 37.0%

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

   3. distribute-rgt-neg-in 37.0%

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

   4. rem-square-sqrt 37.0%

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

   5. unpow2 37.0%

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

   6. unpow2 37.0%

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

   7. hypot-undefine 37.0%

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

   8. unpow2 37.0%

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

   9. unpow2 37.0%

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

   10. hypot-undefine 37.0%

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

   11. unpow2 37.0%

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

   12. distribute-frac-neg 37.0%

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

5. Simplified 37.0%

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

6. Taylor expanded in d around inf 41.7%

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

   1. associate-*r/ 41.9%

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

   2. *-commutative 41.9%

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

   3. neg-mul-1 41.9%

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

   4. add-sqr-sqrt 22.4%

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

   5. sqrt-unprod 21.6%

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

   6. sqr-neg 21.6%

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

   7. sqrt-unprod 3.6%

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

   8. add-sqr-sqrt 9.8%

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

8. Applied egg-rr 9.8%

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

9. Final simplification 9.8%

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

Developer target: 99.2% 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 2024071 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))