Complex division, imag part

Percentage Accurate: 62.1% → 86.2%

Time: 9.5s

Alternatives: 9

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: 62.1% 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: 86.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 10^{+295}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+295)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (/ (- (* b (/ c d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+295) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+295) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+295:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+295)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+295)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+295], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 9.9999999999999998e294

   1. Initial program 81.0%

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

      1. *-un-lft-identity 81.0%

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

      2. add-sqr-sqrt 81.0%

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

      3. times-frac 81.0%

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

      4. hypot-define 81.0%

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

      5. hypot-define 97.1%

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

   4. Applied egg-rr 97.1%

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

   if 9.9999999999999998e294 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 21.8%

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

      1. *-un-lft-identity 21.8%

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

      2. add-sqr-sqrt 21.8%

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

      3. times-frac 21.9%

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

      4. hypot-define 21.9%

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

      5. hypot-define 25.4%

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

   4. Applied egg-rr 25.4%

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

   5. Taylor expanded in d around inf 62.4%

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

      1. +-commutative 62.4%

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

      2. neg-mul-1 62.4%

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

      3. sub-neg 62.4%

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

      4. associate-/l* 68.0%

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

   7. Simplified 68.0%

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

Alternative 2: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9.4 \cdot 10^{-104}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-156}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -1.6e+96)
     t_1
     (if (<= d -9.4e-104)
       t_0
       (if (<= d 3.3e-156)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 3.2e+115) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.6e+96) {
		tmp = t_1;
	} else if (d <= -9.4e-104) {
		tmp = t_0;
	} else if (d <= 3.3e-156) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.2e+115) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = ((c * (b / d)) - a) / d
    if (d <= (-1.6d+96)) then
        tmp = t_1
    else if (d <= (-9.4d-104)) then
        tmp = t_0
    else if (d <= 3.3d-156) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 3.2d+115) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.6e+96) {
		tmp = t_1;
	} else if (d <= -9.4e-104) {
		tmp = t_0;
	} else if (d <= 3.3e-156) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3.2e+115) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -1.6e+96:
		tmp = t_1
	elif d <= -9.4e-104:
		tmp = t_0
	elif d <= 3.3e-156:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 3.2e+115:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -1.6e+96)
		tmp = t_1;
	elseif (d <= -9.4e-104)
		tmp = t_0;
	elseif (d <= 3.3e-156)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3.2e+115)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -1.6e+96)
		tmp = t_1;
	elseif (d <= -9.4e-104)
		tmp = t_0;
	elseif (d <= 3.3e-156)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 3.2e+115)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.6e+96], t$95$1, If[LessEqual[d, -9.4e-104], t$95$0, If[LessEqual[d, 3.3e-156], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.2e+115], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.60000000000000003e96 or 3.2e115 < d

   1. Initial program 40.0%

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

   3. Taylor expanded in c around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. associate-/r* 83.3%

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

      6. div-sub 83.3%

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

      7. *-commutative 83.3%

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

      8. associate-/l* 86.1%

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

   5. Simplified 86.1%

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

   if -1.60000000000000003e96 < d < -9.4e-104 or 3.2999999999999999e-156 < d < 3.2e115

   1. Initial program 84.9%

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

   if -9.4e-104 < d < 3.2999999999999999e-156

   1. Initial program 76.5%

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

   3. Taylor expanded in c around inf 94.6%

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

      1. mul-1-neg 94.6%

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

      2. unsub-neg 94.6%

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

      3. *-commutative 94.6%

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

   5. Simplified 94.6%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+96}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -9.4 \cdot 10^{-104}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 3.3 \cdot 10^{-156}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3.2 \cdot 10^{+115}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97} \lor \neg \left(d \leq 9 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.6e+97) (not (<= d 9e-26)))
   (/ (- (* c (/ b d)) a) d)
   (/ (- b (/ (* a d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+97) || !(d <= 9e-26)) {
		tmp = ((c * (b / d)) - 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 <= (-1.6d+97)) .or. (.not. (d <= 9d-26))) then
        tmp = ((c * (b / d)) - 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 <= -1.6e+97) || !(d <= 9e-26)) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.6e+97) or not (d <= 9e-26):
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - ((a * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.6e+97) || !(d <= 9e-26))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.6e+97) || ~((d <= 9e-26)))
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - ((a * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.6e+97], N[Not[LessEqual[d, 9e-26]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.60000000000000008e97 or 8.9999999999999998e-26 < d

   1. Initial program 50.3%

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

   3. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. unpow2 75.1%

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

      5. associate-/r* 79.4%

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

      6. div-sub 79.4%

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

      7. *-commutative 79.4%

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

      8. associate-/l* 81.5%

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

   5. Simplified 81.5%

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

   if -1.60000000000000008e97 < d < 8.9999999999999998e-26

   1. Initial program 80.3%

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

   3. Taylor expanded in c around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. *-commutative 80.9%

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

   5. Simplified 80.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97} \lor \neg \left(d \leq 9 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97} \lor \neg \left(d \leq 1.45 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.6e+97) (not (<= d 1.45e-24)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ (* a d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.6e+97) || !(d <= 1.45e-24)) {
		tmp = ((b * (c / d)) - 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 <= (-1.6d+97)) .or. (.not. (d <= 1.45d-24))) then
        tmp = ((b * (c / d)) - 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 <= -1.6e+97) || !(d <= 1.45e-24)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.6e+97) or not (d <= 1.45e-24):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((a * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.6e+97) || !(d <= 1.45e-24))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.6e+97) || ~((d <= 1.45e-24)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((a * d) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.6e+97], N[Not[LessEqual[d, 1.45e-24]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.60000000000000008e97 or 1.4499999999999999e-24 < d

   1. Initial program 50.3%

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

      1. *-un-lft-identity 50.3%

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

      2. add-sqr-sqrt 50.3%

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

      3. times-frac 50.2%

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

      4. hypot-define 50.2%

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

      5. hypot-define 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in d around inf 79.4%

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

      1. +-commutative 79.4%

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

      2. neg-mul-1 79.4%

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

      3. sub-neg 79.4%

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

      4. associate-/l* 81.0%

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

   7. Simplified 81.0%

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

   if -1.60000000000000008e97 < d < 1.4499999999999999e-24

   1. Initial program 80.3%

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

   3. Taylor expanded in c around inf 80.9%

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

      1. mul-1-neg 80.9%

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

      2. unsub-neg 80.9%

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

      3. *-commutative 80.9%

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

   5. Simplified 80.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+97} \lor \neg \left(d \leq 1.45 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+106} \lor \neg \left(d \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3e+106) (not (<= d 5.8e-9)))
   (/ a (- d))
   (/ (- b (/ (* a d) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e+106) || !(d <= 5.8e-9)) {
		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 <= (-3d+106)) .or. (.not. (d <= 5.8d-9))) 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 <= -3e+106) || !(d <= 5.8e-9)) {
		tmp = a / -d;
	} else {
		tmp = (b - ((a * d) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3e+106) or not (d <= 5.8e-9):
		tmp = a / -d
	else:
		tmp = (b - ((a * d) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3e+106) || !(d <= 5.8e-9))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3e+106) || ~((d <= 5.8e-9)))
		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, -3e+106], N[Not[LessEqual[d, 5.8e-9]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.0000000000000001e106 or 5.79999999999999982e-9 < d

   1. Initial program 49.4%

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

   3. Taylor expanded in c around 0 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

   5. Simplified 73.4%

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

   if -3.0000000000000001e106 < d < 5.79999999999999982e-9

   1. Initial program 80.2%

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

   3. Taylor expanded in c around inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. unsub-neg 79.5%

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

      3. *-commutative 79.5%

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

   5. Simplified 79.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+106} \lor \neg \left(d \leq 5.8 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+106} \lor \neg \left(d \leq 2.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3e+106) (not (<= d 2.8e-15))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3e+106) || !(d <= 2.8e-15)) {
		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 <= (-3d+106)) .or. (.not. (d <= 2.8d-15))) 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 <= -3e+106) || !(d <= 2.8e-15)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3e+106) or not (d <= 2.8e-15):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3e+106) || !(d <= 2.8e-15))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3e+106) || ~((d <= 2.8e-15)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3e+106], N[Not[LessEqual[d, 2.8e-15]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.0000000000000001e106 or 2.80000000000000014e-15 < d

   1. Initial program 49.4%

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

   3. Taylor expanded in c around 0 73.0%

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

      1. associate-*r/ 73.0%

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

      2. neg-mul-1 73.0%

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

   5. Simplified 73.0%

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

   if -3.0000000000000001e106 < d < 2.80000000000000014e-15

   1. Initial program 80.6%

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

   3. Taylor expanded in c around inf 63.0%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3 \cdot 10^{+106} \lor \neg \left(d \leq 2.8 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 14.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+168} \lor \neg \left(d \leq 2.25 \cdot 10^{+147}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.6e+168) (not (<= d 2.25e+147))) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.6e+168) || !(d <= 2.25e+147)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-3.6d+168)) .or. (.not. (d <= 2.25d+147))) then
        tmp = a / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.6e+168) || !(d <= 2.25e+147)) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.6e+168) or not (d <= 2.25e+147):
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.6e+168) || !(d <= 2.25e+147))
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.6e+168) || ~((d <= 2.25e+147)))
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.6e+168], N[Not[LessEqual[d, 2.25e+147]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.5999999999999999e168 or 2.25000000000000004e147 < d

   1. Initial program 34.3%

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

      1. *-un-lft-identity 34.3%

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

      2. add-sqr-sqrt 34.3%

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

      3. times-frac 34.3%

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

      4. hypot-define 34.3%

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

      5. hypot-define 59.4%

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

   4. Applied egg-rr 59.4%

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

   5. Taylor expanded in d around -inf 48.2%

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

   6. Taylor expanded in c around 0 30.0%

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

   if -3.5999999999999999e168 < d < 2.25000000000000004e147

   1. Initial program 79.5%

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

      1. *-un-lft-identity 79.5%

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

      2. add-sqr-sqrt 79.5%

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

      3. times-frac 79.5%

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

      4. hypot-define 79.5%

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

      5. hypot-define 88.2%

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in d around -inf 20.4%

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

   6. Taylor expanded in c around inf 11.2%

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

4. Final simplification 16.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+168} \lor \neg \left(d \leq 2.25 \cdot 10^{+147}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{+138}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -2.3e+138) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -2.3e+138) {
		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 <= (-2.3d+138)) 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 <= -2.3e+138) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -2.3e+138:
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -2.3e+138)
		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 <= -2.3e+138)
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -2.3e+138], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.30000000000000008e138

   1. Initial program 34.5%

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

      1. *-un-lft-identity 34.5%

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

      2. add-sqr-sqrt 34.5%

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

      3. times-frac 34.5%

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

      4. hypot-define 34.5%

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

      5. hypot-define 53.8%

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

   4. Applied egg-rr 53.8%

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

   5. Taylor expanded in d around -inf 80.8%

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

   6. Taylor expanded in c around 0 30.1%

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

   if -2.30000000000000008e138 < d

   1. Initial program 72.5%

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

   3. Taylor expanded in c around inf 48.5%

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

Alternative 9: 9.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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 / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.2%

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

   1. *-un-lft-identity 67.2%

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

   2. add-sqr-sqrt 67.2%

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

   3. times-frac 67.1%

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

   4. hypot-define 67.1%

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

   5. hypot-define 80.3%

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

4. Applied egg-rr 80.3%

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

5. Taylor expanded in d around -inf 28.0%

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

6. Taylor expanded in c around inf 11.4%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
7. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024163 
(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))))