Complex division, imag part

Percentage Accurate: 61.2% → 97.7%

Time: 10.5s

Alternatives: 10

Speedup: 0.8×

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: 97.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)} - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(a, b, c, d):
	return ((c * (b / math.hypot(c, d))) - (d * (a / math.hypot(c, d)))) / math.hypot(c, d)

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(c * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. hypot-define 66.5%

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

   5. hypot-define 78.0%

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

4. Applied egg-rr 78.0%

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

   1. associate-*l/ 78.2%

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

   2. *-un-lft-identity 78.2%

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

   3. *-commutative 78.2%

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

   4. *-commutative 78.2%

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

6. Applied egg-rr 78.2%

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

   1. div-sub 78.2%

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

8. Applied egg-rr 78.2%

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

   1. associate-/l* 86.3%

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

   2. associate-/l* 97.7%

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

10. Simplified 97.7%

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

Alternative 2: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t\_0}{c \cdot c + d \cdot d} \leq 4 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{t\_0}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 4e+261)
     (/ (/ t_0 (hypot c d)) (hypot c d))
     (/ (- (* b (/ c d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 4e+261) {
		tmp = (t_0 / hypot(c, d)) / 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 = (c * b) - (d * a);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 4e+261) {
		tmp = (t_0 / Math.hypot(c, d)) / Math.hypot(c, d);
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (c * b) - (d * a)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 4e+261:
		tmp = (t_0 / math.hypot(c, d)) / math.hypot(c, d)
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 4e+261)
		tmp = Float64(Float64(t_0 / hypot(c, d)) / 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 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 4e+261)
		tmp = (t_0 / hypot(c, d)) / 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[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4e+261], N[(N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $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))) < 3.9999999999999997e261

   1. Initial program 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. add-sqr-sqrt 83.6%

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

      3. times-frac 83.5%

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

      4. hypot-define 83.5%

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

      5. hypot-define 97.0%

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

   4. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.3%

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

      2. *-un-lft-identity 97.3%

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

      3. *-commutative 97.3%

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

      4. *-commutative 97.3%

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

   6. Applied egg-rr 97.3%

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

   if 3.9999999999999997e261 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 10.7%

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

      1. *-un-lft-identity 10.7%

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

      2. add-sqr-sqrt 10.7%

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

      3. times-frac 10.7%

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

      4. hypot-define 10.7%

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

      5. hypot-define 15.8%

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

   4. Applied egg-rr 15.8%

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

   5. Taylor expanded in d around inf 48.7%

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

      1. neg-mul-1 48.7%

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

      2. +-commutative 48.7%

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

      3. sub-neg 48.7%

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

      4. associate-/l* 56.9%

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

   7. Simplified 56.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 4 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ t_1 := c \cdot \frac{b}{d} - a\\ t_2 := c \cdot c + d \cdot d\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{t\_0 + 2 \cdot \mathsf{fma}\left(a, -d, d \cdot a\right)}{t\_2}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{t\_0}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a)))
        (t_1 (- (* c (/ b d)) a))
        (t_2 (+ (* c c) (* d d))))
   (if (<= d -2.8e+49)
     (/ t_1 d)
     (if (<= d -1.15e-87)
       (/ (+ t_0 (* 2.0 (fma a (- d) (* d a)))) t_2)
       (if (<= d 1.9e-158)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 5.2e+122) (/ t_0 t_2) (/ t_1 (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double t_1 = (c * (b / d)) - a;
	double t_2 = (c * c) + (d * d);
	double tmp;
	if (d <= -2.8e+49) {
		tmp = t_1 / d;
	} else if (d <= -1.15e-87) {
		tmp = (t_0 + (2.0 * fma(a, -d, (d * a)))) / t_2;
	} else if (d <= 1.9e-158) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 5.2e+122) {
		tmp = t_0 / t_2;
	} else {
		tmp = t_1 / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	t_1 = Float64(Float64(c * Float64(b / d)) - a)
	t_2 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (d <= -2.8e+49)
		tmp = Float64(t_1 / d);
	elseif (d <= -1.15e-87)
		tmp = Float64(Float64(t_0 + Float64(2.0 * fma(a, Float64(-d), Float64(d * a)))) / t_2);
	elseif (d <= 1.9e-158)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 5.2e+122)
		tmp = Float64(t_0 / t_2);
	else
		tmp = Float64(t_1 / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.8e+49], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -1.15e-87], N[(N[(t$95$0 + N[(2.0 * N[(a * (-d) + N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[d, 1.9e-158], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5.2e+122], N[(t$95$0 / t$95$2), $MachinePrecision], N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -2.7999999999999998e49

   1. Initial program 50.1%

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

   3. Taylor expanded in c around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. mul-1-neg 76.2%

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

      3. unsub-neg 76.2%

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

      4. unpow2 76.2%

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

      5. associate-/r* 78.0%

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

      6. div-sub 78.0%

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

      7. *-commutative 78.0%

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

      8. associate-/l* 80.0%

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

   5. Simplified 80.0%

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

   if -2.7999999999999998e49 < d < -1.1500000000000001e-87

   1. Initial program 91.4%

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

      1. prod-diff 91.4%

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

      2. *-commutative 91.4%

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

      3. fmm-def 91.4%

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

      4. prod-diff 91.4%

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

      5. *-commutative 91.4%

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

      6. fmm-def 91.4%

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

      7. associate-+l+ 91.5%

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

      8. *-commutative 91.5%

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

      9. fma-undefine 91.4%

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

      10. distribute-lft-neg-in 91.4%

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

      11. *-commutative 91.4%

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

      12. distribute-rgt-neg-in 91.4%

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

      13. fma-define 91.5%

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

      14. *-commutative 91.5%

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

      15. fma-undefine 91.4%

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

      16. distribute-lft-neg-in 91.4%

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

      17. *-commutative 91.4%

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

      18. distribute-rgt-neg-in 91.4%

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

      19. fma-define 91.5%

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

   4. Applied egg-rr 91.5%

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

      1. *-commutative 91.5%

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

      2. *-commutative 91.5%

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

      3. count-2 91.5%

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

      4. *-commutative 91.5%

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

   6. Simplified 91.5%

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

   if -1.1500000000000001e-87 < d < 1.8999999999999999e-158

   1. Initial program 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. add-sqr-sqrt 76.3%

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

      3. times-frac 76.2%

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

      4. hypot-define 76.2%

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

      5. hypot-define 85.4%

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in c around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. associate-/l* 93.9%

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

   7. Simplified 93.9%

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

   if 1.8999999999999999e-158 < d < 5.20000000000000015e122

   1. Initial program 85.9%

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

   if 5.20000000000000015e122 < d

   1. Initial program 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. add-sqr-sqrt 37.3%

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

      3. times-frac 37.3%

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

      4. hypot-define 37.3%

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

      5. hypot-define 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. associate-*l/ 63.4%

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

      2. *-un-lft-identity 63.4%

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

      3. *-commutative 63.4%

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

      4. *-commutative 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. neg-mul-1 79.8%

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

      3. sub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r/ 88.1%

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

   9. Simplified 88.1%

      \[\leadsto \frac{\color{blue}{c \cdot \frac{b}{d} - a}}{\mathsf{hypot}\left(c, d\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -1.15 \cdot 10^{-87}:\\ \;\;\;\;\frac{\left(c \cdot b - d \cdot a\right) + 2 \cdot \mathsf{fma}\left(a, -d, d \cdot a\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := c \cdot \frac{b}{d} - a\\ \mathbf{if}\;d \leq -3.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{t\_1}{d}\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+123}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (- (* c (/ b d)) a)))
   (if (<= d -3.1e+45)
     (/ t_1 d)
     (if (<= d -6.5e-88)
       t_0
       (if (<= d 4.3e-159)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 5e+123) t_0 (/ t_1 (hypot c d))))))))

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 = (c * (b / d)) - a;
	double tmp;
	if (d <= -3.1e+45) {
		tmp = t_1 / d;
	} else if (d <= -6.5e-88) {
		tmp = t_0;
	} else if (d <= 4.3e-159) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 5e+123) {
		tmp = t_0;
	} else {
		tmp = t_1 / hypot(c, d);
	}
	return tmp;
}

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 = (c * (b / d)) - a;
	double tmp;
	if (d <= -3.1e+45) {
		tmp = t_1 / d;
	} else if (d <= -6.5e-88) {
		tmp = t_0;
	} else if (d <= 4.3e-159) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 5e+123) {
		tmp = t_0;
	} else {
		tmp = t_1 / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = (c * (b / d)) - a
	tmp = 0
	if d <= -3.1e+45:
		tmp = t_1 / d
	elif d <= -6.5e-88:
		tmp = t_0
	elif d <= 4.3e-159:
		tmp = (b - (a * (d / c))) / c
	elif d <= 5e+123:
		tmp = t_0
	else:
		tmp = t_1 / math.hypot(c, 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)))
	t_1 = Float64(Float64(c * Float64(b / d)) - a)
	tmp = 0.0
	if (d <= -3.1e+45)
		tmp = Float64(t_1 / d);
	elseif (d <= -6.5e-88)
		tmp = t_0;
	elseif (d <= 4.3e-159)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 5e+123)
		tmp = t_0;
	else
		tmp = Float64(t_1 / hypot(c, d));
	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 = (c * (b / d)) - a;
	tmp = 0.0;
	if (d <= -3.1e+45)
		tmp = t_1 / d;
	elseif (d <= -6.5e-88)
		tmp = t_0;
	elseif (d <= 4.3e-159)
		tmp = (b - (a * (d / c))) / c;
	elseif (d <= 5e+123)
		tmp = t_0;
	else
		tmp = t_1 / hypot(c, d);
	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[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]}, If[LessEqual[d, -3.1e+45], N[(t$95$1 / d), $MachinePrecision], If[LessEqual[d, -6.5e-88], t$95$0, If[LessEqual[d, 4.3e-159], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+123], t$95$0, N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -3.09999999999999988e45

   1. Initial program 50.9%

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

   3. Taylor expanded in c around 0 76.7%

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

      1. +-commutative 76.7%

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

      2. mul-1-neg 76.7%

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

      3. unsub-neg 76.7%

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

      4. unpow2 76.7%

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

      5. associate-/r* 78.4%

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

      6. div-sub 78.4%

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

      7. *-commutative 78.4%

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

      8. associate-/l* 80.3%

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

   5. Simplified 80.3%

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

   if -3.09999999999999988e45 < d < -6.50000000000000006e-88 or 4.3e-159 < d < 4.99999999999999974e123

   1. Initial program 87.5%

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

   if -6.50000000000000006e-88 < d < 4.3e-159

   1. Initial program 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. add-sqr-sqrt 76.3%

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

      3. times-frac 76.2%

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

      4. hypot-define 76.2%

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

      5. hypot-define 85.4%

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in c around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. associate-/l* 93.9%

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

   7. Simplified 93.9%

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

   if 4.99999999999999974e123 < d

   1. Initial program 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. add-sqr-sqrt 37.3%

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

      3. times-frac 37.3%

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

      4. hypot-define 37.3%

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

      5. hypot-define 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. associate-*l/ 63.4%

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

      2. *-un-lft-identity 63.4%

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

      3. *-commutative 63.4%

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

      4. *-commutative 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in c around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. neg-mul-1 79.8%

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

      3. sub-neg 79.8%

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

      4. *-commutative 79.8%

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

      5. associate-*r/ 88.1%

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

   9. Simplified 88.1%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.1 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{-159}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+122}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- (* c (/ b d)) a) d)))
   (if (<= d -1.5e+44)
     t_1
     (if (<= d -3.5e-86)
       t_0
       (if (<= d 1.75e-158)
         (/ (- b (* a (/ d c))) c)
         (if (<= d 6.5e+122) 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 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.5e+44) {
		tmp = t_1;
	} else if (d <= -3.5e-86) {
		tmp = t_0;
	} else if (d <= 1.75e-158) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 6.5e+122) {
		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 = ((c * (b / d)) - a) / d
    if (d <= (-1.5d+44)) then
        tmp = t_1
    else if (d <= (-3.5d-86)) then
        tmp = t_0
    else if (d <= 1.75d-158) then
        tmp = (b - (a * (d / c))) / c
    else if (d <= 6.5d+122) 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 = ((c * (b / d)) - a) / d;
	double tmp;
	if (d <= -1.5e+44) {
		tmp = t_1;
	} else if (d <= -3.5e-86) {
		tmp = t_0;
	} else if (d <= 1.75e-158) {
		tmp = (b - (a * (d / c))) / c;
	} else if (d <= 6.5e+122) {
		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 = ((c * (b / d)) - a) / d
	tmp = 0
	if d <= -1.5e+44:
		tmp = t_1
	elif d <= -3.5e-86:
		tmp = t_0
	elif d <= 1.75e-158:
		tmp = (b - (a * (d / c))) / c
	elif d <= 6.5e+122:
		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(Float64(c * Float64(b / d)) - a) / d)
	tmp = 0.0
	if (d <= -1.5e+44)
		tmp = t_1;
	elseif (d <= -3.5e-86)
		tmp = t_0;
	elseif (d <= 1.75e-158)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (d <= 6.5e+122)
		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 = ((c * (b / d)) - a) / d;
	tmp = 0.0;
	if (d <= -1.5e+44)
		tmp = t_1;
	elseif (d <= -3.5e-86)
		tmp = t_0;
	elseif (d <= 1.75e-158)
		tmp = (b - (a * (d / c))) / c;
	elseif (d <= 6.5e+122)
		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[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.5e+44], t$95$1, If[LessEqual[d, -3.5e-86], t$95$0, If[LessEqual[d, 1.75e-158], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 6.5e+122], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.49999999999999993e44 or 6.49999999999999963e122 < d

   1. Initial program 44.8%

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

   3. Taylor expanded in c around 0 76.2%

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

      1. +-commutative 76.2%

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

      2. mul-1-neg 76.2%

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

      3. unsub-neg 76.2%

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

      4. unpow2 76.2%

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

      5. associate-/r* 78.1%

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

      6. div-sub 78.1%

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

      7. *-commutative 78.1%

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

      8. associate-/l* 83.7%

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

   5. Simplified 83.7%

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

   if -1.49999999999999993e44 < d < -3.50000000000000021e-86 or 1.75000000000000006e-158 < d < 6.49999999999999963e122

   1. Initial program 87.5%

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

   if -3.50000000000000021e-86 < d < 1.75000000000000006e-158

   1. Initial program 76.3%

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

      1. *-un-lft-identity 76.3%

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

      2. add-sqr-sqrt 76.3%

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

      3. times-frac 76.2%

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

      4. hypot-define 76.2%

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

      5. hypot-define 85.4%

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

   4. Applied egg-rr 85.4%

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

   5. Taylor expanded in c around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. sub-neg 93.9%

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

      3. associate-/l* 93.9%

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

   7. Simplified 93.9%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;d \leq -3.5 \cdot 10^{-86}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;d \leq 6.5 \cdot 10^{+122}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -0.0021 \lor \neg \left(d \leq 1.4 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -0.0021) (not (<= d 1.4e-24)))
   (/ (- (* 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 <= -0.0021) || !(d <= 1.4e-24)) {
		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 <= (-0.0021d0)) .or. (.not. (d <= 1.4d-24))) 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 <= -0.0021) || !(d <= 1.4e-24)) {
		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 <= -0.0021) or not (d <= 1.4e-24):
		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 <= -0.0021) || !(d <= 1.4e-24))
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / 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 <= -0.0021) || ~((d <= 1.4e-24)))
		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, -0.0021], N[Not[LessEqual[d, 1.4e-24]], $MachinePrecision]], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / 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 < -0.00209999999999999987 or 1.4000000000000001e-24 < d

   1. Initial program 54.7%

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

   3. Taylor expanded in c around 0 73.6%

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

      1. +-commutative 73.6%

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

      2. mul-1-neg 73.6%

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

      3. unsub-neg 73.6%

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

      4. unpow2 73.6%

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

      5. associate-/r* 75.0%

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

      6. div-sub 75.0%

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

      7. *-commutative 75.0%

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

      8. associate-/l* 79.2%

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

   5. Simplified 79.2%

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

   if -0.00209999999999999987 < d < 1.4000000000000001e-24

   1. Initial program 81.4%

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

      1. *-un-lft-identity 81.4%

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

      2. add-sqr-sqrt 81.4%

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

      3. times-frac 81.3%

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

      4. hypot-define 81.3%

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

      5. hypot-define 87.5%

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

   4. Applied egg-rr 87.5%

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

   5. Taylor expanded in c around inf 84.3%

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

      1. mul-1-neg 84.3%

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

      2. sub-neg 84.3%

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

      3. associate-/l* 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -0.0021 \lor \neg \left(d \leq 1.4 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-14} \lor \neg \left(d \leq 1.26 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.95e-14) (not (<= d 1.26e-23)))
   (/ (- (* 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 <= -1.95e-14) || !(d <= 1.26e-23)) {
		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 <= (-1.95d-14)) .or. (.not. (d <= 1.26d-23))) 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 <= -1.95e-14) || !(d <= 1.26e-23)) {
		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 <= -1.95e-14) or not (d <= 1.26e-23):
		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 <= -1.95e-14) || !(d <= 1.26e-23))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / 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 <= -1.95e-14) || ~((d <= 1.26e-23)))
		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, -1.95e-14], N[Not[LessEqual[d, 1.26e-23]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / 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 < -1.9499999999999999e-14 or 1.25999999999999996e-23 < d

   1. Initial program 55.3%

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

      1. *-un-lft-identity 55.3%

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

      2. add-sqr-sqrt 55.4%

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

      3. times-frac 55.4%

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

      4. hypot-define 55.4%

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

      5. hypot-define 70.9%

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

   4. Applied egg-rr 70.9%

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

   5. Taylor expanded in d around inf 74.7%

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

      1. neg-mul-1 74.7%

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

      2. +-commutative 74.7%

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

      3. sub-neg 74.7%

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

      4. associate-/l* 76.9%

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

   7. Simplified 76.9%

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

   if -1.9499999999999999e-14 < d < 1.25999999999999996e-23

   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 80.9%

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

      4. hypot-define 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 87.3%

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

   4. Applied egg-rr 87.3%

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

   5. Taylor expanded in c around inf 84.8%

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

      1. mul-1-neg 84.8%

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

      2. sub-neg 84.8%

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

      3. associate-/l* 84.8%

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

   7. Simplified 84.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.95 \cdot 10^{-14} \lor \neg \left(d \leq 1.26 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 7.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 7.8 \cdot 10^{-283}:\\ \;\;\;\;\frac{b}{d} - \frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} - \frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d 7.8e-283) (- (/ b d) (/ a c)) (- (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 7.8e-283) {
		tmp = (b / d) - (a / c);
	} else {
		tmp = (a / c) - (b / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 7.8d-283) then
        tmp = (b / d) - (a / c)
    else
        tmp = (a / c) - (b / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 7.8e-283) {
		tmp = (b / d) - (a / c);
	} else {
		tmp = (a / c) - (b / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 7.8e-283:
		tmp = (b / d) - (a / c)
	else:
		tmp = (a / c) - (b / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 7.8e-283)
		tmp = Float64(Float64(b / d) - Float64(a / c));
	else
		tmp = Float64(Float64(a / c) - Float64(b / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 7.8e-283)
		tmp = (b / d) - (a / c);
	else
		tmp = (a / c) - (b / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 7.8e-283], N[(N[(b / d), $MachinePrecision] - N[(a / c), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] - N[(b / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 7.8000000000000004e-283

   1. Initial program 66.7%

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

      1. *-un-lft-identity 66.7%

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

      2. add-sqr-sqrt 66.7%

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

      3. times-frac 66.7%

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

      4. hypot-define 66.7%

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

      5. hypot-define 77.1%

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

   4. Applied egg-rr 77.1%

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

   5. Taylor expanded in c around -inf 31.2%

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

      1. +-commutative 31.2%

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

      2. neg-mul-1 31.2%

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

      3. sub-neg 31.2%

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

      4. associate-/l* 33.4%

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

   7. Simplified 33.4%

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

   8. Taylor expanded in d around -inf 8.8%

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

      1. +-commutative 8.8%

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

      2. mul-1-neg 8.8%

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

      3. sub-neg 8.8%

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

   10. Simplified 8.8%

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

   if 7.8000000000000004e-283 < d

   1. Initial program 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. add-sqr-sqrt 66.2%

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

      3. times-frac 66.2%

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

      4. hypot-define 66.2%

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

      5. hypot-define 78.9%

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

   4. Applied egg-rr 78.9%

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

   5. Taylor expanded in c around -inf 33.6%

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

      1. +-commutative 33.6%

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

      2. neg-mul-1 33.6%

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

      3. sub-neg 33.6%

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

      4. associate-/l* 34.5%

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

   7. Simplified 34.5%

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

   8. Taylor expanded in d around inf 10.3%

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

      1. +-commutative 10.3%

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

      2. neg-mul-1 10.3%

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

      3. sub-neg 10.3%

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

   10. Simplified 10.3%

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

Alternative 9: 53.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. hypot-define 66.5%

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

   5. hypot-define 78.0%

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

4. Applied egg-rr 78.0%

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

5. Taylor expanded in c around inf 48.3%

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

   1. mul-1-neg 48.3%

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

   2. sub-neg 48.3%

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

   3. associate-/l* 51.0%

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

7. Simplified 51.0%

   \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]
8. Add Preprocessing

Alternative 10: 7.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.5%

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

   1. *-un-lft-identity 66.5%

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

   2. add-sqr-sqrt 66.5%

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

   3. times-frac 66.5%

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

   4. hypot-define 66.5%

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

   5. hypot-define 78.0%

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

4. Applied egg-rr 78.0%

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

5. Taylor expanded in c around -inf 32.4%

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

   1. +-commutative 32.4%

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

   2. neg-mul-1 32.4%

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

   3. sub-neg 32.4%

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

   4. associate-/l* 33.9%

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

7. Simplified 33.9%

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

8. Taylor expanded in d around inf 7.5%

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

   1. +-commutative 7.5%

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

   2. neg-mul-1 7.5%

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

   3. sub-neg 7.5%

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

10. Simplified 7.5%

    \[\leadsto \color{blue}{\frac{a}{c} - \frac{b}{d}} \]
11. 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 2024179 
(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))))