Complex division, imag part

Percentage Accurate: 61.1% → 84.7%

Time: 10.3s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% 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: 84.7% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) INFINITY)
     (/ (/ t_0 (hypot c d)) (hypot c d))
     (/ (- (* c (/ b 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))) <= ((double) INFINITY)) {
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	} else {
		tmp = ((c * (b / 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))) <= Double.POSITIVE_INFINITY) {
		tmp = (t_0 / Math.hypot(c, d)) / Math.hypot(c, d);
	} else {
		tmp = ((c * (b / 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))) <= math.inf:
		tmp = (t_0 / math.hypot(c, d)) / math.hypot(c, d)
	else:
		tmp = ((c * (b / 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))) <= Inf)
		tmp = Float64(Float64(t_0 / hypot(c, d)) / hypot(c, d));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / 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))) <= Inf)
		tmp = (t_0 / hypot(c, d)) / hypot(c, d);
	else
		tmp = ((c * (b / 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], Infinity], N[(N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(b / 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))) < +inf.0

   1. Initial program 80.1%

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

      1. fma-define 80.1%

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

      2. add-sqr-sqrt 80.1%

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

      3. associate-/r* 80.3%

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

      4. fma-define 80.3%

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

      5. hypot-define 80.3%

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

      6. fma-define 80.3%

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

      7. hypot-define 95.7%

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

   4. Applied egg-rr 95.7%

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

   if +inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 0.0%

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

   3. Taylor expanded in c around 0 42.4%

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

      1. +-commutative 42.4%

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

      2. mul-1-neg 42.4%

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

      3. unsub-neg 42.4%

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

      4. unpow2 42.4%

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

      5. associate-/r* 42.6%

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

      6. div-sub 42.6%

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

      7. *-commutative 42.6%

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

      8. associate-/l* 54.9%

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

   5. Simplified 54.9%

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

Alternative 2: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -54000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-116}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 3 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{\frac{c \cdot c + d \cdot d}{b \cdot c - a \cdot d}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -54000000000.0)
     t_0
     (if (<= d 4.3e-116)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 3e+114)
         (/ 1.0 (/ (+ (* c c) (* d d)) (- (* b c) (* a d))))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -54000000000.0) {
		tmp = t_0;
	} else if (d <= 4.3e-116) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3e+114) {
		tmp = 1.0 / (((c * c) + (d * d)) / ((b * c) - (a * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * (b / d)) - a) / d
    if (d <= (-54000000000.0d0)) then
        tmp = t_0
    else if (d <= 4.3d-116) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 3d+114) then
        tmp = 1.0d0 / (((c * c) + (d * d)) / ((b * c) - (a * d)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -54000000000.0) {
		tmp = t_0;
	} else if (d <= 4.3e-116) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 3e+114) {
		tmp = 1.0 / (((c * c) + (d * d)) / ((b * c) - (a * d)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -54000000000.0:
		tmp = t_0
	elif d <= 4.3e-116:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 3e+114:
		tmp = 1.0 / (((c * c) + (d * d)) / ((b * c) - (a * d)))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -54000000000.0)
		tmp = t_0;
	elseif (d <= 4.3e-116)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 3e+114)
		tmp = Float64(1.0 / Float64(Float64(Float64(c * c) + Float64(d * d)) / Float64(Float64(b * c) - Float64(a * d))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -54000000000.0)
		tmp = t_0;
	elseif (d <= 4.3e-116)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 3e+114)
		tmp = 1.0 / (((c * c) + (d * d)) / ((b * c) - (a * d)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -54000000000.0], t$95$0, If[LessEqual[d, 4.3e-116], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3e+114], N[(1.0 / N[(N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.4e10 or 3e114 < d

   1. Initial program 47.2%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. unpow2 72.0%

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

      5. associate-/r* 74.8%

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

      6. div-sub 74.8%

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

      7. *-commutative 74.8%

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

      8. associate-/l* 81.6%

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

   5. Simplified 81.6%

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

   if -5.4e10 < d < 4.2999999999999997e-116

   1. Initial program 73.1%

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

      1. div-sub 65.8%

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

      2. *-commutative 65.8%

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

      3. fma-define 65.8%

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

      4. add-sqr-sqrt 65.8%

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

      5. times-frac 70.1%

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

      6. fma-neg 70.1%

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

      7. fma-define 70.1%

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

      8. hypot-define 70.1%

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

      9. fma-define 70.1%

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

      10. hypot-define 89.2%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in c around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

      4. *-lft-identity 90.6%

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

      5. times-frac 86.9%

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

      6. /-rgt-identity 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in d around 0 90.6%

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

   if 4.2999999999999997e-116 < d < 3e114

   1. Initial program 83.2%

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

      1. clear-num 83.3%

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

   4. Applied egg-rr 83.3%

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

Alternative 3: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -14600000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-125}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+114}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -14600000000.0)
     t_0
     (if (<= d 1.7e-125)
       (/ (- b (/ (* a d) c)) c)
       (if (<= d 1.18e+114)
         (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
         t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -14600000000.0) {
		tmp = t_0;
	} else if (d <= 1.7e-125) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.18e+114) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * (b / d)) - a) / d
    if (d <= (-14600000000.0d0)) then
        tmp = t_0
    else if (d <= 1.7d-125) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 1.18d+114) then
        tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -14600000000.0) {
		tmp = t_0;
	} else if (d <= 1.7e-125) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.18e+114) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -14600000000.0:
		tmp = t_0
	elif d <= 1.7e-125:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 1.18e+114:
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -14600000000.0)
		tmp = t_0;
	elseif (d <= 1.7e-125)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 1.18e+114)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -14600000000.0)
		tmp = t_0;
	elseif (d <= 1.7e-125)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 1.18e+114)
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -14600000000.0], t$95$0, If[LessEqual[d, 1.7e-125], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.18e+114], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.46e10 or 1.18000000000000005e114 < d

   1. Initial program 47.2%

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

   3. Taylor expanded in c around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. mul-1-neg 72.0%

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

      3. unsub-neg 72.0%

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

      4. unpow2 72.0%

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

      5. associate-/r* 74.8%

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

      6. div-sub 74.8%

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

      7. *-commutative 74.8%

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

      8. associate-/l* 81.6%

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

   5. Simplified 81.6%

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

   if -1.46e10 < d < 1.69999999999999988e-125

   1. Initial program 72.9%

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

      1. div-sub 65.4%

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

      2. *-commutative 65.4%

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

      3. fma-define 65.4%

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

      4. add-sqr-sqrt 65.4%

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

      5. times-frac 69.8%

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

      6. fma-neg 69.8%

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

      7. fma-define 69.8%

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

      8. hypot-define 69.8%

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

      9. fma-define 69.8%

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

      10. hypot-define 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in c around inf 90.6%

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

      1. mul-1-neg 90.6%

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

      2. unsub-neg 90.6%

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

      3. *-commutative 90.6%

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

      4. *-lft-identity 90.6%

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

      5. times-frac 86.7%

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

      6. /-rgt-identity 86.7%

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

   7. Simplified 86.7%

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

   8. Taylor expanded in d around 0 90.6%

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

   if 1.69999999999999988e-125 < d < 1.18000000000000005e114

   1. Initial program 83.5%

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

Alternative 4: 62.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-28} \lor \neg \left(d \leq 2.9 \cdot 10^{-41}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ a (- d))))
   (if (<= d -1.6e+89)
     t_0
     (if (<= d -8.5e+18)
       (/ (/ (* b c) d) d)
       (if (or (<= d -2.2e-28) (not (<= d 2.9e-41))) t_0 (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.6e+89) {
		tmp = t_0;
	} else if (d <= -8.5e+18) {
		tmp = ((b * c) / d) / d;
	} else if ((d <= -2.2e-28) || !(d <= 2.9e-41)) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = a / -d
    if (d <= (-1.6d+89)) then
        tmp = t_0
    else if (d <= (-8.5d+18)) then
        tmp = ((b * c) / d) / d
    else if ((d <= (-2.2d-28)) .or. (.not. (d <= 2.9d-41))) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = a / -d;
	double tmp;
	if (d <= -1.6e+89) {
		tmp = t_0;
	} else if (d <= -8.5e+18) {
		tmp = ((b * c) / d) / d;
	} else if ((d <= -2.2e-28) || !(d <= 2.9e-41)) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = a / -d
	tmp = 0
	if d <= -1.6e+89:
		tmp = t_0
	elif d <= -8.5e+18:
		tmp = ((b * c) / d) / d
	elif (d <= -2.2e-28) or not (d <= 2.9e-41):
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.6e+89)
		tmp = t_0;
	elseif (d <= -8.5e+18)
		tmp = Float64(Float64(Float64(b * c) / d) / d);
	elseif ((d <= -2.2e-28) || !(d <= 2.9e-41))
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = a / -d;
	tmp = 0.0;
	if (d <= -1.6e+89)
		tmp = t_0;
	elseif (d <= -8.5e+18)
		tmp = ((b * c) / d) / d;
	elseif ((d <= -2.2e-28) || ~((d <= 2.9e-41)))
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -1.6e+89], t$95$0, If[LessEqual[d, -8.5e+18], N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] / d), $MachinePrecision], If[Or[LessEqual[d, -2.2e-28], N[Not[LessEqual[d, 2.9e-41]], $MachinePrecision]], t$95$0, N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.59999999999999994e89 or -8.5e18 < d < -2.19999999999999996e-28 or 2.89999999999999977e-41 < d

   1. Initial program 53.8%

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

   3. Taylor expanded in c around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. distribute-neg-frac2 63.3%

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

   5. Simplified 63.3%

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

   if -1.59999999999999994e89 < d < -8.5e18

   1. Initial program 80.0%

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

   3. Taylor expanded in c around 0 68.2%

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

      1. +-commutative 68.2%

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

      2. mul-1-neg 68.2%

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

      3. unsub-neg 68.2%

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

      4. unpow2 68.2%

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

      5. associate-/r* 68.4%

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

      6. div-sub 68.4%

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

      7. *-commutative 68.4%

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

      8. associate-/l* 68.3%

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

   5. Simplified 68.3%

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

   6. Taylor expanded in c around inf 54.9%

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

   if -2.19999999999999996e-28 < d < 2.89999999999999977e-41

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf 68.6%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.6 \cdot 10^{+89}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -8.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d}}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-28} \lor \neg \left(d \leq 2.9 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -41000000000 \lor \neg \left(d \leq 5.5 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -41000000000.0) (not (<= d 5.5e+42)))
   (/ (- (* 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 <= -41000000000.0) || !(d <= 5.5e+42)) {
		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 <= (-41000000000.0d0)) .or. (.not. (d <= 5.5d+42))) 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 <= -41000000000.0) || !(d <= 5.5e+42)) {
		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 <= -41000000000.0) or not (d <= 5.5e+42):
		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 <= -41000000000.0) || !(d <= 5.5e+42))
		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 <= -41000000000.0) || ~((d <= 5.5e+42)))
		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, -41000000000.0], N[Not[LessEqual[d, 5.5e+42]], $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 < -4.1e10 or 5.50000000000000001e42 < d

   1. Initial program 52.7%

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

   3. Taylor expanded in c around 0 72.6%

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

      1. +-commutative 72.6%

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

      2. mul-1-neg 72.6%

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

      3. unsub-neg 72.6%

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

      4. unpow2 72.6%

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

      5. associate-/r* 75.1%

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

      6. div-sub 75.1%

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

      7. *-commutative 75.1%

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

      8. associate-/l* 80.9%

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

   5. Simplified 80.9%

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

   if -4.1e10 < d < 5.50000000000000001e42

   1. Initial program 75.2%

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

      1. div-sub 69.8%

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

      2. *-commutative 69.8%

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

      3. fma-define 69.8%

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

      4. add-sqr-sqrt 69.8%

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

      5. times-frac 73.1%

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

      6. fma-neg 73.1%

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

      7. fma-define 73.1%

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

      8. hypot-define 73.1%

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

      9. fma-define 73.1%

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

      10. hypot-define 91.3%

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

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in c around inf 83.4%

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

      1. mul-1-neg 83.4%

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

      2. unsub-neg 83.4%

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

      3. *-commutative 83.4%

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

      4. *-lft-identity 83.4%

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

      5. times-frac 80.6%

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

      6. /-rgt-identity 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in d around 0 83.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -41000000000 \lor \neg \left(d \leq 5.5 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+112} \lor \neg \left(d \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.4e+112) (not (<= d 7e+41)))
   (/ a (- d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.4e+112) || !(d <= 7e+41)) {
		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 <= (-5.4d+112)) .or. (.not. (d <= 7d+41))) 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 <= -5.4e+112) || !(d <= 7e+41)) {
		tmp = a / -d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.4e+112) or not (d <= 7e+41):
		tmp = a / -d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.4e+112) || !(d <= 7e+41))
		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 <= -5.4e+112) || ~((d <= 7e+41)))
		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, -5.4e+112], N[Not[LessEqual[d, 7e+41]], $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 < -5.4000000000000002e112 or 6.9999999999999998e41 < d

   1. Initial program 47.6%

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

   3. Taylor expanded in c around 0 66.4%

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

      1. mul-1-neg 66.4%

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

      2. distribute-neg-frac2 66.4%

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

   5. Simplified 66.4%

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

   if -5.4000000000000002e112 < d < 6.9999999999999998e41

   1. Initial program 75.0%

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

   3. Taylor expanded in c around inf 76.2%

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

      1. mul-1-neg 76.2%

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

      2. unsub-neg 76.2%

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

      3. associate-/l* 76.8%

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

   5. Simplified 76.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.4 \cdot 10^{+112} \lor \neg \left(d \leq 7 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+94} \lor \neg \left(d \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4e+94) (not (<= d 2.5e-41))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e+94) || !(d <= 2.5e-41)) {
		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 <= (-4d+94)) .or. (.not. (d <= 2.5d-41))) 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 <= -4e+94) || !(d <= 2.5e-41)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4e+94) or not (d <= 2.5e-41):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4e+94) || !(d <= 2.5e-41))
		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 <= -4e+94) || ~((d <= 2.5e-41)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4e+94], N[Not[LessEqual[d, 2.5e-41]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.0000000000000001e94 or 2.4999999999999998e-41 < d

   1. Initial program 52.0%

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

   3. Taylor expanded in c around 0 64.0%

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

      1. mul-1-neg 64.0%

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

      2. distribute-neg-frac2 64.0%

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

   5. Simplified 64.0%

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

   if -4.0000000000000001e94 < d < 2.4999999999999998e-41

   1. Initial program 74.5%

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

   3. Taylor expanded in c around inf 62.9%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{+94} \lor \neg \left(d \leq 2.5 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return b / 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 = b / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 inf 44.3%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Add Preprocessing

Developer target: 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 2024097 
(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))))