Complex division, imag part

Percentage Accurate: 60.9% → 99.0%

Time: 9.9s

Alternatives: 14

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: 60.9% 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: 99.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{b}{\frac{\mathsf{hypot}\left(c, d\right)}{c}} - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* (/ 1.0 (hypot c d)) (- (/ b (/ (hypot c d) c)) (/ a (/ (hypot c d) d)))))

C

double code(double a, double b, double c, double d) {
	return (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / c)) - (a / (hypot(c, d) / d)));
}

Java

public static double code(double a, double b, double c, double d) {
	return (1.0 / Math.hypot(c, d)) * ((b / (Math.hypot(c, d) / c)) - (a / (Math.hypot(c, d) / d)));
}

Python

def code(a, b, c, d):
	return (1.0 / math.hypot(c, d)) * ((b / (math.hypot(c, d) / c)) - (a / (math.hypot(c, d) / d)))

Julia

function code(a, b, c, d)
	return Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(b / Float64(hypot(c, d) / c)) - Float64(a / Float64(hypot(c, d) / d))))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (1.0 / hypot(c, d)) * ((b / (hypot(c, d) / c)) - (a / (hypot(c, d) / d)));
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] - N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.1%

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

   1. *-un-lft-identity 61.1%

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

   2. add-sqr-sqrt 61.1%

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

   3. times-frac 61.1%

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

   4. hypot-def 61.1%

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

   5. hypot-def 75.8%

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

3. Applied egg-rr 75.8%

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

   1. div-sub 75.8%

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

5. Applied egg-rr 75.8%

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

   1. associate-/l* 85.3%

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

   2. associate-/l* 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 87.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_1}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d))) (t_1 (- (* c b) (* d a))))
   (if (<= (/ t_1 (+ (* c c) (* d d))) 2e+307)
     (* t_0 (/ t_1 (hypot c d)))
     (* t_0 (- b (/ a (/ (hypot c d) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 2e+307) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = t_0 * (b - (a / (hypot(c, d) / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / Math.hypot(c, d);
	double t_1 = (c * b) - (d * a);
	double tmp;
	if ((t_1 / ((c * c) + (d * d))) <= 2e+307) {
		tmp = t_0 * (t_1 / Math.hypot(c, d));
	} else {
		tmp = t_0 * (b - (a / (Math.hypot(c, d) / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 1.0 / math.hypot(c, d)
	t_1 = (c * b) - (d * a)
	tmp = 0
	if (t_1 / ((c * c) + (d * d))) <= 2e+307:
		tmp = t_0 * (t_1 / math.hypot(c, d))
	else:
		tmp = t_0 * (b - (a / (math.hypot(c, d) / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (Float64(t_1 / Float64(Float64(c * c) + Float64(d * d))) <= 2e+307)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = Float64(t_0 * Float64(b - Float64(a / Float64(hypot(c, d) / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / hypot(c, d);
	t_1 = (c * b) - (d * a);
	tmp = 0.0;
	if ((t_1 / ((c * c) + (d * d))) <= 2e+307)
		tmp = t_0 * (t_1 / hypot(c, d));
	else
		tmp = t_0 * (b - (a / (hypot(c, d) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+307], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(b - N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))) < 1.99999999999999997e307

   1. Initial program 77.4%

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

      1. *-un-lft-identity 77.4%

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

      2. add-sqr-sqrt 77.4%

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

      3. times-frac 77.4%

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

      4. hypot-def 77.4%

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

      5. hypot-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 1.99999999999999997e307 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 10.1%

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

      1. *-un-lft-identity 10.1%

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

      2. add-sqr-sqrt 10.1%

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

      3. times-frac 10.1%

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

      4. hypot-def 10.1%

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

      5. hypot-def 13.9%

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

   3. Applied egg-rr 13.9%

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

      1. div-sub 13.9%

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

   5. Applied egg-rr 13.9%

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

      1. associate-/l* 48.3%

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

      2. associate-/l* 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in c around inf 75.1%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{b} - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \]

Alternative 3: 83.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.4e+83)
   (/ (- b (* d (/ a c))) c)
   (if (<= c -1.15e-151)
     (/ (fma c b (* d (- a))) (+ (* c c) (* d d)))
     (if (<= c 1.6e-144)
       (/ (- (/ (* c b) d) a) d)
       (* (/ 1.0 (hypot c d)) (- b (/ a (/ (hypot c d) d))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.4e+83) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -1.15e-151) {
		tmp = fma(c, b, (d * -a)) / ((c * c) + (d * d));
	} else if (c <= 1.6e-144) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (a / (hypot(c, d) / d)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.4e+83)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= -1.15e-151)
		tmp = Float64(fma(c, b, Float64(d * Float64(-a))) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (c <= 1.6e-144)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(a / Float64(hypot(c, d) / d))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.4e+83], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.15e-151], N[(N[(c * b + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.6e-144], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b - N[(a / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.39999999999999991e83

   1. Initial program 44.7%

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

   2. Taylor expanded in c around inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. times-frac 90.5%

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

   4. Simplified 90.5%

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

      1. associate-*r/ 91.2%

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

      2. sub-div 91.2%

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

   6. Applied egg-rr 91.2%

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

   if -2.39999999999999991e83 < c < -1.14999999999999998e-151

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. fma-neg 81.6%

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

      3. distribute-rgt-neg-in 81.6%

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

   3. Applied egg-rr 81.6%

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

   if -1.14999999999999998e-151 < c < 1.59999999999999986e-144

   1. Initial program 64.4%

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

   2. Taylor expanded in c around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. mul-1-neg 83.3%

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

      3. unsub-neg 83.3%

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

      4. unpow2 83.3%

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

      5. times-frac 85.3%

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

   4. Simplified 85.3%

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

      1. associate-*l/ 85.9%

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

   6. Applied egg-rr 85.9%

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

      1. *-un-lft-identity 85.9%

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

      2. sub-div 87.4%

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

   8. Applied egg-rr 87.4%

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

      1. *-lft-identity 87.4%

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

      2. associate-*r/ 93.0%

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

   10. Simplified 93.0%

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

   if 1.59999999999999986e-144 < c

   1. Initial program 55.5%

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

      1. *-un-lft-identity 55.5%

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

      2. add-sqr-sqrt 55.5%

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

      3. times-frac 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-def 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-def 71.1%

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

   3. Applied egg-rr 71.1%

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

      1. div-sub 71.1%

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

   5. Applied egg-rr 71.1%

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

      1. associate-/l* 84.4%

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

      2. associate-/l* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in c around inf 89.7%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{b} - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.15 \cdot 10^{-151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \]

Alternative 4: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -2.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -7.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{t_0}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))))
   (if (<= c -2.2e+83)
     (/ (- b (* d (/ a c))) c)
     (if (<= c -7.6e-152)
       (/ (fma c b (* d (- a))) t_0)
       (if (<= c 1.6e-55)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 2.6e+92)
           (/ (- (* c b) (* d a)) t_0)
           (- (/ b c) (* (/ a c) (/ d c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double tmp;
	if (c <= -2.2e+83) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -7.6e-152) {
		tmp = fma(c, b, (d * -a)) / t_0;
	} else if (c <= 1.6e-55) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 2.6e+92) {
		tmp = ((c * b) - (d * a)) / t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (c <= -2.2e+83)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= -7.6e-152)
		tmp = Float64(fma(c, b, Float64(d * Float64(-a))) / t_0);
	elseif (c <= 1.6e-55)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 2.6e+92)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / t_0);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.2e+83], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -7.6e-152], N[(N[(c * b + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[c, 1.6e-55], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.6e+92], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.19999999999999999e83

   1. Initial program 44.7%

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

   2. Taylor expanded in c around inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. times-frac 90.5%

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

   4. Simplified 90.5%

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

      1. associate-*r/ 91.2%

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

      2. sub-div 91.2%

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

   6. Applied egg-rr 91.2%

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

   if -2.19999999999999999e83 < c < -7.60000000000000024e-152

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. fma-neg 81.6%

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

      3. distribute-rgt-neg-in 81.6%

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

   3. Applied egg-rr 81.6%

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

   if -7.60000000000000024e-152 < c < 1.6000000000000001e-55

   1. Initial program 63.9%

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

   2. Taylor expanded in c around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. unpow2 84.5%

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

      5. times-frac 86.1%

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

   4. Simplified 86.1%

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

      1. associate-*l/ 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. sub-div 87.8%

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

   8. Applied egg-rr 87.8%

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

      1. *-lft-identity 87.8%

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

      2. associate-*r/ 92.2%

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

   10. Simplified 92.2%

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

   if 1.6000000000000001e-55 < c < 2.5999999999999999e92

   1. Initial program 80.4%

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

   if 2.5999999999999999e92 < c

   1. Initial program 35.3%

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

   2. Taylor expanded in c around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. unpow2 80.1%

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

      5. times-frac 91.7%

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

   4. Simplified 91.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -7.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, d \cdot \left(-a\right)\right)}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 5: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.45 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.32 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -2.45e+83)
     (/ (- b (* d (/ a c))) c)
     (if (<= c -1.32e-152)
       t_0
       (if (<= c 1.65e-55)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 8.2e+80) t_0 (- (/ b c) (* (/ a c) (/ d c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.45e+83) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -1.32e-152) {
		tmp = t_0;
	} else if (c <= 1.65e-55) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 8.2e+80) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (c <= (-2.45d+83)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= (-1.32d-152)) then
        tmp = t_0
    else if (c <= 1.65d-55) then
        tmp = (((c * b) / d) - a) / d
    else if (c <= 8.2d+80) then
        tmp = t_0
    else
        tmp = (b / c) - ((a / c) * (d / c))
    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 tmp;
	if (c <= -2.45e+83) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -1.32e-152) {
		tmp = t_0;
	} else if (c <= 1.65e-55) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 8.2e+80) {
		tmp = t_0;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.45e+83:
		tmp = (b - (d * (a / c))) / c
	elif c <= -1.32e-152:
		tmp = t_0
	elif c <= 1.65e-55:
		tmp = (((c * b) / d) - a) / d
	elif c <= 8.2e+80:
		tmp = t_0
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.45e+83)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= -1.32e-152)
		tmp = t_0;
	elseif (c <= 1.65e-55)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 8.2e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -2.45e+83)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= -1.32e-152)
		tmp = t_0;
	elseif (c <= 1.65e-55)
		tmp = (((c * b) / d) - a) / d;
	elseif (c <= 8.2e+80)
		tmp = t_0;
	else
		tmp = (b / c) - ((a / c) * (d / c));
	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]}, If[LessEqual[c, -2.45e+83], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.32e-152], t$95$0, If[LessEqual[c, 1.65e-55], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8.2e+80], t$95$0, N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.44999999999999989e83

   1. Initial program 44.7%

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

   2. Taylor expanded in c around inf 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. unsub-neg 77.9%

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

      4. unpow2 77.9%

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

      5. times-frac 90.5%

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

   4. Simplified 90.5%

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

      1. associate-*r/ 91.2%

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

      2. sub-div 91.2%

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

   6. Applied egg-rr 91.2%

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

   if -2.44999999999999989e83 < c < -1.31999999999999995e-152 or 1.65e-55 < c < 8.20000000000000003e80

   1. Initial program 81.1%

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

   if -1.31999999999999995e-152 < c < 1.65e-55

   1. Initial program 63.9%

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

   2. Taylor expanded in c around 0 84.5%

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

      1. +-commutative 84.5%

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

      2. mul-1-neg 84.5%

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

      3. unsub-neg 84.5%

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

      4. unpow2 84.5%

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

      5. times-frac 86.1%

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

   4. Simplified 86.1%

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

      1. associate-*l/ 86.6%

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

   6. Applied egg-rr 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. sub-div 87.8%

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

   8. Applied egg-rr 87.8%

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

      1. *-lft-identity 87.8%

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

      2. associate-*r/ 92.2%

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

   10. Simplified 92.2%

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

   if 8.20000000000000003e80 < c

   1. Initial program 35.3%

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

   2. Taylor expanded in c around inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. unpow2 80.1%

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

      5. times-frac 91.7%

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

   4. Simplified 91.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.45 \cdot 10^{+83}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.32 \cdot 10^{-152}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 6: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{c}{\frac{c \cdot c + d \cdot d}{b}}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.3e+80)
   (/ (- b (* d (/ a c))) c)
   (if (<= c -1.1e-21)
     (/ c (/ (+ (* c c) (* d d)) b))
     (if (<= c 2.7e+44)
       (/ (- (/ (* c b) d) a) d)
       (- (/ b c) (* (/ a c) (/ d c)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.3e+80) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -1.1e-21) {
		tmp = c / (((c * c) + (d * d)) / b);
	} else if (c <= 2.7e+44) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b / c) - ((a / c) * (d / 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 <= (-4.3d+80)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= (-1.1d-21)) then
        tmp = c / (((c * c) + (d * d)) / b)
    else if (c <= 2.7d+44) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = (b / c) - ((a / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.3e+80) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= -1.1e-21) {
		tmp = c / (((c * c) + (d * d)) / b);
	} else if (c <= 2.7e+44) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.3e+80:
		tmp = (b - (d * (a / c))) / c
	elif c <= -1.1e-21:
		tmp = c / (((c * c) + (d * d)) / b)
	elif c <= 2.7e+44:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.3e+80)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= -1.1e-21)
		tmp = Float64(c / Float64(Float64(Float64(c * c) + Float64(d * d)) / b));
	elseif (c <= 2.7e+44)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -4.3e+80)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= -1.1e-21)
		tmp = c / (((c * c) + (d * d)) / b);
	elseif (c <= 2.7e+44)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.3e+80], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -1.1e-21], N[(c / N[(N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.7e+44], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.30000000000000004e80

   1. Initial program 45.9%

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

   2. Taylor expanded in c around inf 76.3%

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

      1. +-commutative 76.3%

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

      2. mul-1-neg 76.3%

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

      3. unsub-neg 76.3%

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

      4. unpow2 76.3%

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

      5. times-frac 88.6%

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

   4. Simplified 88.6%

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

      1. associate-*r/ 89.2%

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

      2. sub-div 89.2%

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

   6. Applied egg-rr 89.2%

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

   if -4.30000000000000004e80 < c < -1.1e-21

   1. Initial program 83.9%

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

      1. *-un-lft-identity 83.9%

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

      2. add-sqr-sqrt 83.9%

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

      3. times-frac 83.9%

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

      4. hypot-def 83.9%

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

      5. hypot-def 84.2%

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

   3. Applied egg-rr 84.2%

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

   4. Taylor expanded in b around inf 81.2%

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

      1. associate-/l* 81.4%

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

      2. unpow2 81.4%

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

      3. unpow2 81.4%

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

   6. Simplified 81.4%

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

   if -1.1e-21 < c < 2.7e44

   1. Initial program 69.1%

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

   2. Taylor expanded in c around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. mul-1-neg 73.7%

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

      3. unsub-neg 73.7%

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

      4. unpow2 73.7%

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

      5. times-frac 76.6%

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

   4. Simplified 76.6%

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

      1. associate-*l/ 76.9%

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

   6. Applied egg-rr 76.9%

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

      1. *-un-lft-identity 76.9%

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

      2. sub-div 77.7%

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

   8. Applied egg-rr 77.7%

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

      1. *-lft-identity 77.7%

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

      2. associate-*r/ 80.6%

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

   10. Simplified 80.6%

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

   if 2.7e44 < c

   1. Initial program 44.5%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 84.9%

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

   4. Simplified 84.9%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -1.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{c}{\frac{c \cdot c + d \cdot d}{b}}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 7: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1e+52)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 3.1e+44)
     (/ (- (/ (* c b) d) a) d)
     (- (/ b c) (* (/ a c) (/ d c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+52) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3.1e+44) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b / c) - ((a / c) * (d / 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 <= (-1d+52)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 3.1d+44) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = (b / c) - ((a / c) * (d / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1e+52) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 3.1e+44) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = (b / c) - ((a / c) * (d / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1e+52:
		tmp = (b - (d * (a / c))) / c
	elif c <= 3.1e+44:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = (b / c) - ((a / c) * (d / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1e+52)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 3.1e+44)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1e+52)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 3.1e+44)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = (b / c) - ((a / c) * (d / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1e+52], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 3.1e+44], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.9999999999999999e51

   1. Initial program 50.2%

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

   2. Taylor expanded in c around inf 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. unpow2 75.3%

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

      5. times-frac 86.2%

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

   4. Simplified 86.2%

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

      1. associate-*r/ 86.8%

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

      2. sub-div 86.8%

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

   6. Applied egg-rr 86.8%

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

   if -9.9999999999999999e51 < c < 3.09999999999999996e44

   1. Initial program 71.0%

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

   2. Taylor expanded in c around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. unpow2 70.9%

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

      5. times-frac 73.5%

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

   4. Simplified 73.5%

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

      1. associate-*l/ 73.8%

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

   6. Applied egg-rr 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. sub-div 74.4%

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

   8. Applied egg-rr 74.4%

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

      1. *-lft-identity 74.4%

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

      2. associate-*r/ 76.9%

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

   10. Simplified 76.9%

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

   if 3.09999999999999996e44 < c

   1. Initial program 44.5%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. +-commutative 75.5%

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

      2. mul-1-neg 75.5%

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

      3. unsub-neg 75.5%

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

      4. unpow2 75.5%

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

      5. times-frac 84.9%

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

   4. Simplified 84.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 3.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 8: 73.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1300000000 \lor \neg \left(d \leq 1.05 \cdot 10^{+32}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1300000000.0) (not (<= d 1.05e+32)))
   (- (/ a d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1300000000.0) || !(d <= 1.05e+32)) {
		tmp = -(a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1300000000.0d0)) .or. (.not. (d <= 1.05d+32))) then
        tmp = -(a / d)
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1300000000.0) || !(d <= 1.05e+32)) {
		tmp = -(a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1300000000.0) or not (d <= 1.05e+32):
		tmp = -(a / d)
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1300000000.0) || !(d <= 1.05e+32))
		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 ((d <= -1300000000.0) || ~((d <= 1.05e+32)))
		tmp = -(a / d);
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1300000000.0], N[Not[LessEqual[d, 1.05e+32]], $MachinePrecision]], (-N[(a / d), $MachinePrecision]), N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.3e9 or 1.05e32 < d

   1. Initial program 46.3%

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

   2. Taylor expanded in c around 0 68.5%

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

      1. associate-*r/ 68.5%

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

      2. neg-mul-1 68.5%

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

   4. Simplified 68.5%

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

   if -1.3e9 < d < 1.05e32

   1. Initial program 72.7%

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

      1. *-un-lft-identity 72.7%

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

      2. add-sqr-sqrt 72.7%

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

      3. times-frac 72.7%

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

      4. hypot-def 72.7%

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

      5. hypot-def 84.6%

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

   3. Applied egg-rr 84.6%

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

   4. Taylor expanded in c around inf 71.6%

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

      1. +-commutative 71.6%

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

      2. metadata-eval 71.6%

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

      3. unpow2 71.6%

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

      4. cancel-sign-sub-inv 71.6%

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

      5. *-lft-identity 71.6%

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

      6. associate-/r* 76.0%

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

      7. div-sub 77.5%

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

      8. associate-*r/ 78.3%

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

   6. Simplified 78.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1300000000 \lor \neg \left(d \leq 1.05 \cdot 10^{+32}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 9: 77.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-73} \lor \neg \left(d \leq 3.6 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.7e-73) (not (<= d 3.6e+30)))
   (/ (- (* 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 <= -3.7e-73) || !(d <= 3.6e+30)) {
		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 <= (-3.7d-73)) .or. (.not. (d <= 3.6d+30))) 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 <= -3.7e-73) || !(d <= 3.6e+30)) {
		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 <= -3.7e-73) or not (d <= 3.6e+30):
		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 <= -3.7e-73) || !(d <= 3.6e+30))
		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 <= -3.7e-73) || ~((d <= 3.6e+30)))
		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, -3.7e-73], N[Not[LessEqual[d, 3.6e+30]], $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 < -3.7000000000000001e-73 or 3.6000000000000002e30 < d

   1. Initial program 53.6%

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

   2. Taylor expanded in c around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. mul-1-neg 71.1%

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

      3. unsub-neg 71.1%

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

      4. unpow2 71.1%

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

      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%

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

   6. Applied egg-rr 75.7%

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

      1. sub-div 75.7%

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

   8. Applied egg-rr 75.7%

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

   if -3.7000000000000001e-73 < d < 3.6000000000000002e30

   1. Initial program 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. add-sqr-sqrt 69.2%

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

      3. times-frac 69.2%

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

      4. hypot-def 69.2%

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

      5. hypot-def 82.9%

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

   3. Applied egg-rr 82.9%

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

   4. Taylor expanded in c around inf 77.3%

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

      1. +-commutative 77.3%

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

      2. metadata-eval 77.3%

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

      3. unpow2 77.3%

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

      4. cancel-sign-sub-inv 77.3%

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

      5. *-lft-identity 77.3%

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

      6. associate-/r* 82.4%

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

      7. div-sub 84.1%

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

      8. associate-*r/ 85.0%

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

   6. Simplified 85.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.7 \cdot 10^{-73} \lor \neg \left(d \leq 3.6 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 10: 77.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.45 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.45e+52)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 2.7e+44) (/ (- (/ (* 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 <= -3.45e+52) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 2.7e+44) {
		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 <= (-3.45d+52)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 2.7d+44) 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 <= -3.45e+52) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 2.7e+44) {
		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 <= -3.45e+52:
		tmp = (b - (d * (a / c))) / c
	elif c <= 2.7e+44:
		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 <= -3.45e+52)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 2.7e+44)
		tmp = Float64(Float64(Float64(Float64(c * 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 <= -3.45e+52)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 2.7e+44)
		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, -3.45e+52], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.7e+44], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $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 < -3.44999999999999998e52

   1. Initial program 50.2%

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

   2. Taylor expanded in c around inf 75.3%

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

      1. +-commutative 75.3%

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

      2. mul-1-neg 75.3%

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

      3. unsub-neg 75.3%

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

      4. unpow2 75.3%

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

      5. times-frac 86.2%

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

   4. Simplified 86.2%

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

      1. associate-*r/ 86.8%

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

      2. sub-div 86.8%

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

   6. Applied egg-rr 86.8%

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

   if -3.44999999999999998e52 < c < 2.7e44

   1. Initial program 71.0%

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

   2. Taylor expanded in c around 0 70.9%

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

      1. +-commutative 70.9%

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

      2. mul-1-neg 70.9%

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

      3. unsub-neg 70.9%

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

      4. unpow2 70.9%

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

      5. times-frac 73.5%

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

   4. Simplified 73.5%

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

      1. associate-*l/ 73.8%

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

   6. Applied egg-rr 73.8%

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

      1. *-un-lft-identity 73.8%

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

      2. sub-div 74.4%

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

   8. Applied egg-rr 74.4%

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

      1. *-lft-identity 74.4%

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

      2. associate-*r/ 76.9%

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

   10. Simplified 76.9%

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

   if 2.7e44 < c

   1. Initial program 44.5%

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

      1. *-un-lft-identity 44.5%

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

      2. add-sqr-sqrt 44.5%

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

      3. times-frac 44.5%

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

      4. hypot-def 44.5%

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

      5. hypot-def 63.3%

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

   3. Applied egg-rr 63.3%

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

   4. Taylor expanded in c around inf 75.5%

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

      1. +-commutative 75.5%

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

      2. metadata-eval 75.5%

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

      3. unpow2 75.5%

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

      4. cancel-sign-sub-inv 75.5%

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

      5. *-lft-identity 75.5%

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

      6. associate-/r* 77.7%

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

      7. div-sub 77.7%

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

      8. associate-*r/ 84.9%

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

   6. Simplified 84.9%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.45 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 11: 63.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-21} \lor \neg \left(c \leq 2.5 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9.5e-21) (not (<= c 2.5e+48))) (/ b c) (- (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.5e-21) || !(c <= 2.5e+48)) {
		tmp = b / c;
	} else {
		tmp = -(a / d);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-9.5d-21)) .or. (.not. (c <= 2.5d+48))) then
        tmp = b / c
    else
        tmp = -(a / d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.5e-21) || !(c <= 2.5e+48)) {
		tmp = b / c;
	} else {
		tmp = -(a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -9.5e-21) or not (c <= 2.5e+48):
		tmp = b / c
	else:
		tmp = -(a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9.5e-21) || !(c <= 2.5e+48))
		tmp = Float64(b / c);
	else
		tmp = Float64(-Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -9.5e-21) || ~((c <= 2.5e+48)))
		tmp = b / c;
	else
		tmp = -(a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9.5e-21], N[Not[LessEqual[c, 2.5e+48]], $MachinePrecision]], N[(b / c), $MachinePrecision], (-N[(a / d), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if c < -9.4999999999999994e-21 or 2.49999999999999987e48 < c

   1. Initial program 52.5%

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

   2. Taylor expanded in c around inf 68.4%

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

   if -9.4999999999999994e-21 < c < 2.49999999999999987e48

   1. Initial program 69.0%

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

   2. Taylor expanded in c around 0 63.1%

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

      1. associate-*r/ 63.1%

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

      2. neg-mul-1 63.1%

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

   4. Simplified 63.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.5 \cdot 10^{-21} \lor \neg \left(c \leq 2.5 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]

Alternative 12: 14.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.2e+58) (/ a d) (if (<= d 1.9e+145) (/ a c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e+58) {
		tmp = a / d;
	} else if (d <= 1.9e+145) {
		tmp = a / 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 (d <= (-5.2d+58)) then
        tmp = a / d
    else if (d <= 1.9d+145) then
        tmp = a / 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 (d <= -5.2e+58) {
		tmp = a / d;
	} else if (d <= 1.9e+145) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.2e+58:
		tmp = a / d
	elif d <= 1.9e+145:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.2e+58)
		tmp = Float64(a / d);
	elseif (d <= 1.9e+145)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -5.2e+58)
		tmp = a / d;
	elseif (d <= 1.9e+145)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.2e+58], N[(a / d), $MachinePrecision], If[LessEqual[d, 1.9e+145], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.19999999999999976e58 or 1.90000000000000006e145 < d

   1. Initial program 36.8%

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

      1. *-un-lft-identity 36.8%

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

      2. add-sqr-sqrt 36.8%

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

      3. times-frac 36.8%

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

      4. hypot-def 36.8%

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

      5. hypot-def 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in c around 0 46.1%

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

      1. mul-1-neg 46.1%

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

   6. Simplified 46.1%

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

   7. Taylor expanded in d around -inf 26.3%

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

   if -5.19999999999999976e58 < d < 1.90000000000000006e145

   1. Initial program 71.9%

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

      1. *-un-lft-identity 71.9%

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

      2. add-sqr-sqrt 71.9%

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

      3. times-frac 71.9%

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

      4. hypot-def 71.9%

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

      5. hypot-def 83.1%

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

   3. Applied egg-rr 83.1%

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

   4. Taylor expanded in c around 0 19.7%

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

      1. mul-1-neg 19.7%

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

   6. Simplified 19.7%

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

   7. Taylor expanded in c around -inf 10.0%

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

4. Final simplification 15.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+145}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 13: 46.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+245}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.5e+188) (/ a d) (if (<= d 2.05e+245) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.5e+188) {
		tmp = a / d;
	} else if (d <= 2.05e+245) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-5.5d+188)) then
        tmp = a / d
    else if (d <= 2.05d+245) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.5e+188) {
		tmp = a / d;
	} else if (d <= 2.05e+245) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.5e+188:
		tmp = a / d
	elif d <= 2.05e+245:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.5e+188)
		tmp = Float64(a / d);
	elseif (d <= 2.05e+245)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -5.5e+188)
		tmp = a / d;
	elseif (d <= 2.05e+245)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.5e+188], N[(a / d), $MachinePrecision], If[LessEqual[d, 2.05e+245], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.50000000000000013e188 or 2.05000000000000002e245 < d

   1. Initial program 44.6%

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

      1. *-un-lft-identity 44.6%

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

      2. add-sqr-sqrt 44.6%

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

      3. times-frac 44.6%

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

      4. hypot-def 44.6%

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

      5. hypot-def 57.5%

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

   3. Applied egg-rr 57.5%

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

   4. Taylor expanded in c around 0 56.8%

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

      1. mul-1-neg 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in d around -inf 42.6%

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

   if -5.50000000000000013e188 < d < 2.05000000000000002e245

   1. Initial program 63.4%

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

   2. Taylor expanded in c around inf 45.9%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.05 \cdot 10^{+245}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 14: 9.2% 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 61.1%

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

   1. *-un-lft-identity 61.1%

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

   2. add-sqr-sqrt 61.1%

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

   3. times-frac 61.1%

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

   4. hypot-def 61.1%

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

   5. hypot-def 75.8%

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

3. Applied egg-rr 75.8%

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

4. Taylor expanded in c around 0 27.8%

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

   1. mul-1-neg 27.8%

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

6. Simplified 27.8%

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

7. Taylor expanded in c around -inf 9.5%

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

8. Final simplification 9.5%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023216 
(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))))