Complex division, imag part

Percentage Accurate: 60.6% → 85.5%

Time: 8.5s

Alternatives: 12

Speedup: 1.8×

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: 60.6% 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: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+280}:\\ \;\;\;\;\frac{t_0 \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+280)
     (/ (* t_0 (/ 1.0 (hypot c d))) (hypot c d))
     (/ (- (* c (/ b d)) a) d))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+280) {
		tmp = (t_0 * (1.0 / hypot(c, d))) / hypot(c, d);
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+280) {
		tmp = (t_0 * (1.0 / Math.hypot(c, d))) / Math.hypot(c, d);
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+280:
		tmp = (t_0 * (1.0 / math.hypot(c, d))) / math.hypot(c, d)
	else:
		tmp = ((c * (b / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+280)
		tmp = Float64(Float64(t_0 * Float64(1.0 / hypot(c, d))) / hypot(c, d));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+280)
		tmp = (t_0 * (1.0 / hypot(c, d))) / hypot(c, d);
	else
		tmp = ((c * (b / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+280], N[(N[(t$95$0 * N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e280

   1. Initial program 72.9%

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

      1. *-un-lft-identity 72.9%

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

      2. add-sqr-sqrt 72.9%

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

      3. times-frac 72.8%

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

      4. hypot-def 72.8%

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

      5. hypot-def 97.0%

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

   3. Applied egg-rr 97.0%

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

      1. associate-*r/ 97.1%

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

   5. Applied egg-rr 97.1%

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

   if 1e280 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 13.2%

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

   2. Taylor expanded in c around 0 42.6%

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

      1. +-commutative 42.6%

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

      2. mul-1-neg 42.6%

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

      3. unsub-neg 42.6%

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

      4. unpow2 42.6%

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

      5. times-frac 57.0%

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

   4. Simplified 57.0%

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

      1. associate-*l/ 57.0%

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

      2. sub-div 57.5%

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

   6. Applied egg-rr 57.5%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+280}:\\ \;\;\;\;\frac{\left(b \cdot c - a \cdot d\right) \cdot \frac{1}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]

Alternative 2: 84.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, 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)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= Inf)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.7%

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

      1. *-un-lft-identity 70.7%

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

      2. add-sqr-sqrt 70.7%

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

      3. times-frac 70.6%

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

      4. hypot-def 70.6%

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

      5. hypot-def 95.0%

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

   3. Applied egg-rr 95.0%

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

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

   1. Initial program 0.0%

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

      1. *-un-lft-identity 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 0.0%

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

      4. hypot-def 0.0%

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

      5. hypot-def 2.6%

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

   3. Applied egg-rr 2.6%

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

      1. associate-*r/ 2.6%

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

   5. Applied egg-rr 2.6%

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

   6. Taylor expanded in c around inf 34.8%

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

      1. mul-1-neg 34.8%

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

      2. unpow2 34.8%

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

      3. associate-/l* 39.9%

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

      4. associate-*l/ 47.2%

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

      5. +-commutative 47.2%

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

      6. unsub-neg 47.2%

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

      7. associate-/r* 51.2%

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

      8. associate-/l* 35.5%

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

      9. associate-*r/ 51.2%

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

      10. div-sub 51.3%

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

   8. Simplified 51.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 3: 78.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 18000000000:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.7e+35)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 1.5e-60)
     (/ (- (* b (/ c d)) a) d)
     (if (<= c 18000000000.0)
       (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
       (if (<= c 2.05e+29)
         (/ (- (* c (/ b d)) a) d)
         (/ (- b (* a (/ d c))) (hypot c d)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.7e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 1.5e-60) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 18000000000.0) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 2.05e+29) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.7e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 1.5e-60) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 18000000000.0) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 2.05e+29) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.7e+35:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 1.5e-60:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 18000000000.0:
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
	elif c <= 2.05e+29:
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.7e+35)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 1.5e-60)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 18000000000.0)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.05e+29)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.7e+35)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 1.5e-60)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 18000000000.0)
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	elseif (c <= 2.05e+29)
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.7e+35], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.5e-60], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 18000000000.0], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.05e+29], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -4.70000000000000033e35

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. clear-num 81.4%

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

      2. frac-times 83.2%

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

      3. *-un-lft-identity 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -4.70000000000000033e35 < c < 1.50000000000000009e-60

   1. Initial program 65.5%

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

   2. Taylor expanded in c around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. mul-1-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. unpow2 78.2%

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

      5. times-frac 84.6%

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

   4. Simplified 84.6%

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

      1. associate-*r/ 86.6%

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

      2. sub-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 1.50000000000000009e-60 < c < 1.8e10

   1. Initial program 84.8%

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

   if 1.8e10 < c < 2.0500000000000002e29

   1. Initial program 48.4%

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

   2. Taylor expanded in c around 0 56.8%

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

      1. +-commutative 56.8%

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

      2. mul-1-neg 56.8%

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

      3. unsub-neg 56.8%

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

      4. unpow2 56.8%

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

      5. times-frac 75.4%

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

   4. Simplified 75.4%

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

      1. associate-*l/ 75.7%

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

      2. sub-div 76.4%

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

   6. Applied egg-rr 76.4%

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

   if 2.0500000000000002e29 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 81.5%

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

      1. +-commutative 81.5%

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

      2. associate-*r/ 87.6%

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

      3. neg-mul-1 87.6%

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

      4. sub-neg 87.6%

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

   8. Simplified 87.6%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 18000000000:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 4: 78.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 660000000:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4e+35)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 2.6e-61)
     (/ (- (* b (/ c d)) a) d)
     (if (<= c 660000000.0)
       (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
       (if (<= c 2.6e+29)
         (/ (- (* c (/ b d)) a) d)
         (/ (- b (* a (/ d c))) c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 2.6e-61) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 660000000.0) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 2.6e+29) {
		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 (c <= (-4d+35)) then
        tmp = (b / c) - (d / (c * (c / a)))
    else if (c <= 2.6d-61) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 660000000.0d0) then
        tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
    else if (c <= 2.6d+29) 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 (c <= -4e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 2.6e-61) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 660000000.0) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else if (c <= 2.6e+29) {
		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 c <= -4e+35:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 2.6e-61:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 660000000.0:
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d))
	elif c <= 2.6e+29:
		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 (c <= -4e+35)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 2.6e-61)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 660000000.0)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 2.6e+29)
		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 (c <= -4e+35)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 2.6e-61)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 660000000.0)
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	elseif (c <= 2.6e+29)
		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[LessEqual[c, -4e+35], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e-61], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 660000000.0], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+29], 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 5 regimes

2. if c < -3.9999999999999999e35

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. clear-num 81.4%

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

      2. frac-times 83.2%

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

      3. *-un-lft-identity 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -3.9999999999999999e35 < c < 2.6000000000000001e-61

   1. Initial program 65.5%

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

   2. Taylor expanded in c around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. mul-1-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. unpow2 78.2%

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

      5. times-frac 84.6%

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

   4. Simplified 84.6%

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

      1. associate-*r/ 86.6%

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

      2. sub-div 86.9%

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

   6. Applied egg-rr 86.9%

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

   if 2.6000000000000001e-61 < c < 6.6e8

   1. Initial program 84.8%

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

   if 6.6e8 < c < 2.6e29

   1. Initial program 48.4%

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

   2. Taylor expanded in c around 0 56.8%

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

      1. +-commutative 56.8%

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

      2. mul-1-neg 56.8%

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

      3. unsub-neg 56.8%

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

      4. unpow2 56.8%

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

      5. times-frac 75.4%

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

   4. Simplified 75.4%

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

      1. associate-*l/ 75.7%

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

      2. sub-div 76.4%

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

   6. Applied egg-rr 76.4%

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

   if 2.6e29 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unpow2 69.3%

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

      3. associate-/l* 67.6%

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

      4. associate-*l/ 73.3%

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

      5. +-commutative 73.3%

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

      6. unsub-neg 73.3%

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

      7. associate-/r* 87.4%

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

      8. associate-/l* 81.4%

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

      9. associate-*r/ 87.4%

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

      10. div-sub 87.4%

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

   8. Simplified 87.4%

      \[\leadsto \color{blue}{\frac{b - a \cdot \frac{d}{c}}{c}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 660000000:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 5: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{+15} \lor \neg \left(c \leq 4.2 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.9e+15) (not (<= c 4.2e+26)))
   (/ (- b (* a (/ d c))) c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e+15) || !(c <= 4.2e+26)) {
		tmp = (b - (a * (d / c))) / 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 <= (-3.9d+15)) .or. (.not. (c <= 4.2d+26))) then
        tmp = (b - (a * (d / c))) / 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 <= -3.9e+15) || !(c <= 4.2e+26)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.9e+15) or not (c <= 4.2e+26):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.9e+15) || !(c <= 4.2e+26))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / 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 <= -3.9e+15) || ~((c <= 4.2e+26)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.9e+15], N[Not[LessEqual[c, 4.2e+26]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.9e15 or 4.2000000000000002e26 < c

   1. Initial program 43.8%

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

      1. *-un-lft-identity 43.8%

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

      2. add-sqr-sqrt 43.8%

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

      3. times-frac 43.7%

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

      4. hypot-def 43.7%

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

      5. hypot-def 66.0%

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

   3. Applied egg-rr 66.0%

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

      1. associate-*r/ 66.0%

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

   5. Applied egg-rr 66.0%

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

   6. Taylor expanded in c around inf 70.8%

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

      1. mul-1-neg 70.8%

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

      2. unpow2 70.8%

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

      3. associate-/l* 70.3%

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

      4. associate-*l/ 72.7%

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

      5. +-commutative 72.7%

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

      6. unsub-neg 72.7%

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

      7. associate-/r* 80.5%

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

      8. associate-/l* 75.9%

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

      9. associate-*r/ 80.5%

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

      10. div-sub 80.5%

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

   8. Simplified 80.5%

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

   if -3.9e15 < c < 4.2000000000000002e26

   1. Initial program 66.8%

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

   2. Taylor expanded in c around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. neg-mul-1 70.0%

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

   4. Simplified 70.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{+15} \lor \neg \left(c \leq 4.2 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 6: 69.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -17000000000000:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -17000000000000.0)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 3.2e+29) (/ (- a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -17000000000000.0) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3.2e+29) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-17000000000000.0d0)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 3.2d+29) then
        tmp = -a / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -17000000000000.0) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3.2e+29) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -17000000000000.0:
		tmp = (b - (d * (a / c))) / c
	elif c <= 3.2e+29:
		tmp = -a / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -17000000000000.0)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 3.2e+29)
		tmp = Float64(Float64(-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 (c <= -17000000000000.0)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 3.2e+29)
		tmp = -a / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -17000000000000.0], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.2e+29], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.7e13

   1. Initial program 45.3%

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

   2. Taylor expanded in c around inf 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

      4. unpow2 71.9%

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

      5. times-frac 75.5%

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

   4. Simplified 75.5%

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

      1. associate-*r/ 77.1%

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

      2. sub-div 77.1%

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

   6. Applied egg-rr 77.1%

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

   if -1.7e13 < c < 3.19999999999999987e29

   1. Initial program 66.8%

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

   2. Taylor expanded in c around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. neg-mul-1 70.0%

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

   4. Simplified 70.0%

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

   if 3.19999999999999987e29 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unpow2 69.3%

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

      3. associate-/l* 67.6%

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

      4. associate-*l/ 73.3%

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

      5. +-commutative 73.3%

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

      6. unsub-neg 73.3%

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

      7. associate-/r* 87.4%

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

      8. associate-/l* 81.4%

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

      9. associate-*r/ 87.4%

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

      10. div-sub 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -17000000000000:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 7: 77.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.2e+35)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 3e+29) (/ (- (* c (/ b d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.2e+35) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3e+29) {
		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 (c <= (-3.2d+35)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 3d+29) 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 (c <= -3.2e+35) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3e+29) {
		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 c <= -3.2e+35:
		tmp = (b - (d * (a / c))) / c
	elif c <= 3e+29:
		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 (c <= -3.2e+35)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 3e+29)
		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 (c <= -3.2e+35)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 3e+29)
		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[LessEqual[c, -3.2e+35], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3e+29], 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 3 regimes

2. if c < -3.19999999999999983e35

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. associate-*r/ 83.2%

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

      2. sub-div 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -3.19999999999999983e35 < c < 2.9999999999999999e29

   1. Initial program 65.9%

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

   2. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. unpow2 74.5%

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

      5. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*l/ 81.5%

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

      2. sub-div 81.8%

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

   6. Applied egg-rr 81.8%

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

   if 2.9999999999999999e29 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unpow2 69.3%

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

      3. associate-/l* 67.6%

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

      4. associate-*l/ 73.3%

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

      5. +-commutative 73.3%

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

      6. unsub-neg 73.3%

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

      7. associate-/r* 87.4%

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

      8. associate-/l* 81.4%

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

      9. associate-*r/ 87.4%

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

      10. div-sub 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+29}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 8: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.62 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.62e+35)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 2.65e+26) (/ (- (* b (/ c d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.62e+35) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 2.65e+26) {
		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 (c <= (-1.62d+35)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 2.65d+26) 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 (c <= -1.62e+35) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 2.65e+26) {
		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 c <= -1.62e+35:
		tmp = (b - (d * (a / c))) / c
	elif c <= 2.65e+26:
		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 (c <= -1.62e+35)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 2.65e+26)
		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 (c <= -1.62e+35)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 2.65e+26)
		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[LessEqual[c, -1.62e+35], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.65e+26], 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 3 regimes

2. if c < -1.62e35

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. associate-*r/ 83.2%

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

      2. sub-div 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -1.62e35 < c < 2.64999999999999984e26

   1. Initial program 65.9%

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

   2. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. unpow2 74.5%

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

      5. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 82.9%

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

      2. sub-div 83.2%

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

   6. Applied egg-rr 83.2%

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

   if 2.64999999999999984e26 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unpow2 69.3%

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

      3. associate-/l* 67.6%

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

      4. associate-*l/ 73.3%

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

      5. +-commutative 73.3%

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

      6. unsub-neg 73.3%

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

      7. associate-/r* 87.4%

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

      8. associate-/l* 81.4%

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

      9. associate-*r/ 87.4%

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

      10. div-sub 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.62 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{+26}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 9: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.05e+35)
   (- (/ b c) (/ d (* c (/ c a))))
   (if (<= c 6.4e+29) (/ (- (* b (/ c d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.05e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 6.4e+29) {
		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 (c <= (-2.05d+35)) then
        tmp = (b / c) - (d / (c * (c / a)))
    else if (c <= 6.4d+29) 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 (c <= -2.05e+35) {
		tmp = (b / c) - (d / (c * (c / a)));
	} else if (c <= 6.4e+29) {
		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 c <= -2.05e+35:
		tmp = (b / c) - (d / (c * (c / a)))
	elif c <= 6.4e+29:
		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 (c <= -2.05e+35)
		tmp = Float64(Float64(b / c) - Float64(d / Float64(c * Float64(c / a))));
	elseif (c <= 6.4e+29)
		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 (c <= -2.05e+35)
		tmp = (b / c) - (d / (c * (c / a)));
	elseif (c <= 6.4e+29)
		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[LessEqual[c, -2.05e+35], N[(N[(b / c), $MachinePrecision] - N[(d / N[(c * N[(c / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.4e+29], 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 3 regimes

2. if c < -2.0499999999999999e35

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. mul-1-neg 77.3%

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

      3. unsub-neg 77.3%

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

      4. unpow2 77.3%

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

      5. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. clear-num 81.4%

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

      2. frac-times 83.2%

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

      3. *-un-lft-identity 83.2%

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

   6. Applied egg-rr 83.2%

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

   if -2.0499999999999999e35 < c < 6.39999999999999973e29

   1. Initial program 65.9%

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

   2. Taylor expanded in c around 0 74.5%

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

      1. +-commutative 74.5%

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

      2. mul-1-neg 74.5%

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

      3. unsub-neg 74.5%

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

      4. unpow2 74.5%

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

      5. times-frac 81.2%

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

   4. Simplified 81.2%

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

      1. associate-*r/ 82.9%

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

      2. sub-div 83.2%

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

   6. Applied egg-rr 83.2%

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

   if 6.39999999999999973e29 < c

   1. Initial program 41.6%

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

      1. *-un-lft-identity 41.6%

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

      2. add-sqr-sqrt 41.6%

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

      3. times-frac 41.6%

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

      4. hypot-def 41.6%

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

      5. hypot-def 67.2%

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

   3. Applied egg-rr 67.2%

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

      1. associate-*r/ 67.2%

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

   5. Applied egg-rr 67.2%

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

   6. Taylor expanded in c around inf 69.3%

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

      1. mul-1-neg 69.3%

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

      2. unpow2 69.3%

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

      3. associate-/l* 67.6%

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

      4. associate-*l/ 73.3%

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

      5. +-commutative 73.3%

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

      6. unsub-neg 73.3%

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

      7. associate-/r* 87.4%

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

      8. associate-/l* 81.4%

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

      9. associate-*r/ 87.4%

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

      10. div-sub 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c} - \frac{d}{c \cdot \frac{c}{a}}\\ \mathbf{elif}\;c \leq 6.4 \cdot 10^{+29}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 10: 64.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.6e+35) (/ b c) (if (<= c 1.5e+29) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e+35) {
		tmp = b / c;
	} else if (c <= 1.5e+29) {
		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 (c <= (-4.6d+35)) then
        tmp = b / c
    else if (c <= 1.5d+29) 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 (c <= -4.6e+35) {
		tmp = b / c;
	} else if (c <= 1.5e+29) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.6e+35:
		tmp = b / c
	elif c <= 1.5e+29:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.6e+35)
		tmp = Float64(b / c);
	elseif (c <= 1.5e+29)
		tmp = Float64(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 (c <= -4.6e+35)
		tmp = b / c;
	elseif (c <= 1.5e+29)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.6e+35], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.5e+29], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.5999999999999996e35 or 1.5e29 < c

   1. Initial program 43.6%

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

   2. Taylor expanded in c around inf 67.0%

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

   if -4.5999999999999996e35 < c < 1.5e29

   1. Initial program 65.9%

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

   2. Taylor expanded in c around 0 68.2%

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

      1. associate-*r/ 68.2%

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

      2. neg-mul-1 68.2%

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

   4. Simplified 68.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 11: 9.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

   1. *-un-lft-identity 57.7%

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

   2. add-sqr-sqrt 57.7%

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

   3. times-frac 57.7%

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

   4. hypot-def 57.7%

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

   5. hypot-def 78.0%

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

3. Applied egg-rr 78.0%

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

4. Taylor expanded in c around inf 27.0%

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

5. Taylor expanded in d around -inf 7.5%

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

6. Final simplification 7.5%

   \[\leadsto \frac{a}{c} \]

Alternative 12: 41.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in c around inf 34.4%

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

3. Final simplification 34.4%

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

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 2023199 
(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))))