Complex division, imag part

Percentage Accurate: 61.2% → 80.0%

Time: 12.6s

Alternatives: 7

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: 61.2% 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: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;d \leq -3.25 \cdot 10^{+105}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 460:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= d -3.25e+105)
     (- (/ c (* d (/ d b))) (/ a d))
     (if (<= d -2.5e-47)
       t_0
       (if (<= d 460.0)
         (fma -1.0 (* (/ d c) (/ a c)) (/ b c))
         (if (<= d 4.2e+77) t_0 (- (* (/ c d) (/ b d)) (/ a d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (d <= -3.25e+105) {
		tmp = (c / (d * (d / b))) - (a / d);
	} else if (d <= -2.5e-47) {
		tmp = t_0;
	} else if (d <= 460.0) {
		tmp = fma(-1.0, ((d / c) * (a / c)), (b / c));
	} else if (d <= 4.2e+77) {
		tmp = t_0;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (d <= -3.25e+105)
		tmp = Float64(Float64(c / Float64(d * Float64(d / b))) - Float64(a / d));
	elseif (d <= -2.5e-47)
		tmp = t_0;
	elseif (d <= 460.0)
		tmp = fma(-1.0, Float64(Float64(d / c) * Float64(a / c)), Float64(b / c));
	elseif (d <= 4.2e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -3.25e+105], N[(N[(c / N[(d * N[(d / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.5e-47], t$95$0, If[LessEqual[d, 460.0], N[(-1.0 * N[(N[(d / c), $MachinePrecision] * N[(a / c), $MachinePrecision]), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.2e+77], t$95$0, N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.25000000000000024e105

   1. Initial program 33.7%

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

   3. Taylor expanded in c around 0 71.8%

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

      1. +-commutative 71.8%

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

      2. mul-1-neg 71.8%

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

      3. unsub-neg 71.8%

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

      4. *-commutative 71.8%

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

      5. associate-/l* 74.8%

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

   5. Simplified 74.8%

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

      1. pow2 74.8%

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

      2. *-un-lft-identity 74.8%

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

      3. times-frac 88.2%

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

   7. Applied egg-rr 88.2%

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

   if -3.25000000000000024e105 < d < -2.50000000000000006e-47 or 460 < d < 4.1999999999999997e77

   1. Initial program 86.2%

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

   if -2.50000000000000006e-47 < d < 460

   1. Initial program 65.6%

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

   3. Taylor expanded in c around inf 83.0%

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

      1. fma-def 83.0%

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

      2. associate-/l* 83.1%

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

   5. Simplified 83.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-1, \frac{a}{\frac{{c}^{2}}{d}}, \frac{b}{c}\right)} \]
   6. Step-by-step derivation

      1. pow2 83.1%

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

      2. *-un-lft-identity 83.1%

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

      3. times-frac 88.4%

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

   7. Applied egg-rr 88.4%

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

      1. /-rgt-identity 88.4%

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

      2. *-un-lft-identity 88.4%

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

      3. *-commutative 88.4%

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

      4. times-frac 89.2%

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

      5. clear-num 89.2%

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

   9. Applied egg-rr 89.2%

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

   if 4.1999999999999997e77 < d

   1. Initial program 38.1%

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

   3. Taylor expanded in c around 0 70.1%

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

      1. +-commutative 70.1%

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

      2. mul-1-neg 70.1%

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

      3. unsub-neg 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-/l* 74.9%

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

   5. Simplified 74.9%

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

      1. pow2 74.9%

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

      2. associate-/l* 70.1%

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

      3. times-frac 78.7%

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

   7. Applied egg-rr 78.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.25 \cdot 10^{+105}:\\ \;\;\;\;\frac{c}{d \cdot \frac{d}{b}} - \frac{a}{d}\\ \mathbf{elif}\;d \leq -2.5 \cdot 10^{-47}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 460:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{d}{c} \cdot \frac{a}{c}, \frac{b}{c}\right)\\ \mathbf{elif}\;d \leq 4.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+59}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (* d (/ (/ a c) c)))))
   (if (<= c -1.9e+89)
     t_1
     (if (<= c -1.7e-51)
       t_0
       (if (<= c 6.2e-114)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 3e+59) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b / c) - (d * ((a / c) / c));
	double tmp;
	if (c <= -1.9e+89) {
		tmp = t_1;
	} else if (c <= -1.7e-51) {
		tmp = t_0;
	} else if (c <= 6.2e-114) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 3e+59) {
		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 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    t_1 = (b / c) - (d * ((a / c) / c))
    if (c <= (-1.9d+89)) then
        tmp = t_1
    else if (c <= (-1.7d-51)) then
        tmp = t_0
    else if (c <= 6.2d-114) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 3d+59) 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 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b / c) - (d * ((a / c) / c));
	double tmp;
	if (c <= -1.9e+89) {
		tmp = t_1;
	} else if (c <= -1.7e-51) {
		tmp = t_0;
	} else if (c <= 6.2e-114) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 3e+59) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = (b / c) - (d * ((a / c) / c))
	tmp = 0
	if c <= -1.9e+89:
		tmp = t_1
	elif c <= -1.7e-51:
		tmp = t_0
	elif c <= 6.2e-114:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 3e+59:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)))
	tmp = 0.0
	if (c <= -1.9e+89)
		tmp = t_1;
	elseif (c <= -1.7e-51)
		tmp = t_0;
	elseif (c <= 6.2e-114)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 3e+59)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = (b / c) - (d * ((a / c) / c));
	tmp = 0.0;
	if (c <= -1.9e+89)
		tmp = t_1;
	elseif (c <= -1.7e-51)
		tmp = t_0;
	elseif (c <= 6.2e-114)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 3e+59)
		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[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.9e+89], t$95$1, If[LessEqual[c, -1.7e-51], t$95$0, If[LessEqual[c, 6.2e-114], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 3e+59], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.90000000000000012e89 or 3e59 < c

   1. Initial program 43.1%

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

   3. Taylor expanded in c around inf 79.4%

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

      1. +-commutative 79.4%

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

      2. mul-1-neg 79.4%

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

      3. unsub-neg 79.4%

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

      4. associate-/l* 78.0%

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

      5. associate-/r/ 79.8%

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

   5. Simplified 79.8%

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

      1. add-sqr-sqrt 35.2%

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

      2. pow2 35.2%

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

      3. times-frac 37.7%

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

   7. Applied egg-rr 37.7%

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

      1. associate-*l/ 37.7%

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

      2. associate-*r/ 37.7%

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

      3. rem-square-sqrt 87.5%

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

   9. Simplified 87.5%

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

   if -1.90000000000000012e89 < c < -1.70000000000000001e-51 or 6.2e-114 < c < 3e59

   1. Initial program 83.7%

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

   if -1.70000000000000001e-51 < c < 6.2e-114

   1. Initial program 65.9%

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

   3. Taylor expanded in c around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. mul-1-neg 78.5%

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

      3. unsub-neg 78.5%

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

      4. *-commutative 78.5%

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

      5. associate-/l* 76.6%

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

   5. Simplified 76.6%

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

      1. pow2 76.6%

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

      2. associate-/l* 78.5%

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

      3. times-frac 83.2%

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

   7. Applied egg-rr 83.2%

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

      1. associate-*r/ 85.2%

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

      2. sub-div 85.3%

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

      3. *-commutative 85.3%

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

   9. Applied egg-rr 85.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{+89}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+59}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-51} \lor \neg \left(c \leq 6.2 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.8e-51) (not (<= c 6.2e-52)))
   (- (/ b c) (* d (/ (/ a c) c)))
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-51) || !(c <= 6.2e-52)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.8d-51)) .or. (.not. (c <= 6.2d-52))) then
        tmp = (b / c) - (d * ((a / c) / c))
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-51) || !(c <= 6.2e-52)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.8e-51) or not (c <= 6.2e-52):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.8e-51) || !(c <= 6.2e-52))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.8e-51) || ~((c <= 6.2e-52)))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.8e-51], N[Not[LessEqual[c, 6.2e-52]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $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 c < -4.8e-51 or 6.1999999999999998e-52 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 76.8%

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

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. associate-/l* 75.8%

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

      5. associate-/r/ 75.8%

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

   5. Simplified 75.8%

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

      1. add-sqr-sqrt 30.6%

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

      2. pow2 30.6%

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

      3. times-frac 32.4%

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

   7. Applied egg-rr 32.4%

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

      1. associate-*l/ 32.4%

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

      2. associate-*r/ 32.4%

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

      3. rem-square-sqrt 81.2%

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

   9. Simplified 81.2%

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

   if -4.8e-51 < c < 6.1999999999999998e-52

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

      1. pow2 75.5%

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

      2. associate-/l* 77.3%

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

      3. times-frac 81.6%

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

   7. Applied egg-rr 81.6%

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

      1. associate-*r/ 83.5%

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

      2. sub-div 83.7%

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

      3. *-commutative 83.7%

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-51} \lor \neg \left(c \leq 6.2 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-51} \lor \neg \left(c \leq 7 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.2e-51) (not (<= c 7e-48)))
   (/ b c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.2e-51) || !(c <= 7e-48)) {
		tmp = b / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-7.2d-51)) .or. (.not. (c <= 7d-48))) then
        tmp = b / c
    else
        tmp = ((b * (c / d)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.2e-51) || !(c <= 7e-48)) {
		tmp = b / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -7.2e-51) or not (c <= 7e-48):
		tmp = b / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.2e-51) || !(c <= 7e-48))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -7.2e-51) || ~((c <= 7e-48)))
		tmp = b / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.2e-51], N[Not[LessEqual[c, 7e-48]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.2000000000000001e-51 or 6.99999999999999982e-48 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 67.8%

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

   if -7.2000000000000001e-51 < c < 6.99999999999999982e-48

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

      1. pow2 75.5%

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

      2. associate-/l* 77.3%

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

      3. times-frac 81.6%

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

   7. Applied egg-rr 81.6%

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

      1. associate-*r/ 83.5%

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

      2. sub-div 83.7%

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

      3. *-commutative 83.7%

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

   9. Applied egg-rr 83.7%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{-51} \lor \neg \left(c \leq 7 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{-51} \lor \neg \left(c \leq 5.5 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.2e-51) (not (<= c 5.5e-47))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.2e-51) || !(c <= 5.5e-47)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.2d-51)) .or. (.not. (c <= 5.5d-47))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.2e-51) || !(c <= 5.5e-47)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.2e-51) or not (c <= 5.5e-47):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.2e-51) || !(c <= 5.5e-47))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.2e-51) || ~((c <= 5.5e-47)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.2e-51], N[Not[LessEqual[c, 5.5e-47]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.20000000000000003e-51 or 5.5000000000000002e-47 < c

   1. Initial program 54.5%

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

   3. Taylor expanded in c around inf 67.8%

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

   if -4.20000000000000003e-51 < c < 5.5000000000000002e-47

   1. Initial program 68.5%

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

   3. Taylor expanded in c around 0 73.9%

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

      1. associate-*r/ 73.9%

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

      2. neg-mul-1 73.9%

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

   5. Simplified 73.9%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.2 \cdot 10^{-51} \lor \neg \left(c \leq 5.5 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 45.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+118} \lor \neg \left(d \leq 1.25 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.02e+118) (not (<= d 1.25e+156))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.02e+118) || !(d <= 1.25e+156)) {
		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 <= (-1.02d+118)) .or. (.not. (d <= 1.25d+156))) 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 <= -1.02e+118) || !(d <= 1.25e+156)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.02e+118) or not (d <= 1.25e+156):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.02e+118) || !(d <= 1.25e+156))
		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 <= -1.02e+118) || ~((d <= 1.25e+156)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.02e+118], N[Not[LessEqual[d, 1.25e+156]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.0199999999999999e118 or 1.24999999999999998e156 < d

   1. Initial program 32.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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%

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

      1. neg-sub0 73.4%

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

      2. sub-neg 73.4%

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

      3. add-sqr-sqrt 32.5%

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

      4. sqrt-unprod 39.0%

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

      5. sqr-neg 39.0%

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

      6. sqrt-unprod 16.0%

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

      7. add-sqr-sqrt 29.7%

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

   7. Applied egg-rr 29.7%

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

      1. +-lft-identity 29.7%

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

   9. Simplified 29.7%

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

   if -1.0199999999999999e118 < d < 1.24999999999999998e156

   1. Initial program 69.3%

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

   3. Taylor expanded in c around inf 56.7%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.02 \cdot 10^{+118} \lor \neg \left(d \leq 1.25 \cdot 10^{+156}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 11.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in c around 0 39.0%

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

   1. associate-*r/ 39.0%

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

   2. neg-mul-1 39.0%

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

5. Simplified 39.0%

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

   1. neg-sub0 39.0%

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

   2. sub-neg 39.0%

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

   3. add-sqr-sqrt 17.7%

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

   4. sqrt-unprod 21.6%

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

   5. sqr-neg 21.6%

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

   6. sqrt-unprod 5.5%

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

   7. add-sqr-sqrt 10.0%

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

7. Applied egg-rr 10.0%

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

   1. +-lft-identity 10.0%

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

9. Simplified 10.0%

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

10. Final simplification 10.0%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (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))))