Complex division, imag part

Percentage Accurate: 61.3% → 98.4%

Time: 13.1s

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: 61.3% 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: 98.4% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.4%

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

   4. hypot-def 65.4%

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

   5. hypot-def 79.7%

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

3. Applied egg-rr 79.7%

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

   1. div-sub 79.7%

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

   2. sub-neg 79.7%

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

   3. *-commutative 79.7%

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

   4. associate-/l* 89.7%

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

   5. associate-/l* 98.0%

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

5. Applied egg-rr 98.0%

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

   1. sub-neg 98.0%

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

   2. associate-/r/ 99.6%

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

   3. associate-/r/ 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 87.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := c \cdot b - d \cdot a\\ \mathbf{if}\;\frac{t_1}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot b - a\right)\\ \end{array} \end{array} \]

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+298)
     (* t_0 (/ t_1 (hypot c d)))
     (* t_0 (- (* (/ c (hypot c d)) b) a)))))

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+298) {
		tmp = t_0 * (t_1 / hypot(c, d));
	} else {
		tmp = t_0 * (((c / hypot(c, d)) * b) - a);
	}
	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+298) {
		tmp = t_0 * (t_1 / Math.hypot(c, d));
	} else {
		tmp = t_0 * (((c / Math.hypot(c, d)) * b) - a);
	}
	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+298:
		tmp = t_0 * (t_1 / math.hypot(c, d))
	else:
		tmp = t_0 * (((c / math.hypot(c, d)) * b) - a)
	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+298)
		tmp = Float64(t_0 * Float64(t_1 / hypot(c, d)));
	else
		tmp = Float64(t_0 * Float64(Float64(Float64(c / hypot(c, d)) * b) - a));
	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+298)
		tmp = t_0 * (t_1 / hypot(c, d));
	else
		tmp = t_0 * (((c / hypot(c, d)) * b) - a);
	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+298], N[(t$95$0 * N[(t$95$1 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] - a), $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))) < 1.9999999999999999e298

   1. Initial program 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.8%

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

      4. hypot-def 78.8%

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

      5. hypot-def 95.0%

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

   3. Applied egg-rr 95.0%

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

   if 1.9999999999999999e298 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 14.0%

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

      1. *-un-lft-identity 14.0%

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

      2. add-sqr-sqrt 14.0%

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

      3. times-frac 14.0%

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

      4. hypot-def 14.0%

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

      5. hypot-def 21.2%

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

   3. Applied egg-rr 21.2%

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

      1. div-sub 21.2%

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

      2. sub-neg 21.2%

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

      3. *-commutative 21.2%

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

      4. associate-/l* 61.3%

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

      5. associate-/l* 95.9%

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

   5. Applied egg-rr 95.9%

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

      1. sub-neg 95.9%

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

      2. associate-/r/ 99.4%

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

      3. associate-/r/ 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in c around 0 69.7%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+298}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot b - a\right)\\ \end{array} \]

Alternative 3: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -8.5e+80)
     (/ b (* (hypot c d) (/ (hypot c d) c)))
     (if (<= c -2.05e-160)
       t_0
       (if (<= c 1.2e-116)
         (* (/ 1.0 d) (- (/ b (/ d c)) a))
         (if (<= c 1.75e-62)
           t_0
           (if (<= c 1.15e+18)
             (* (/ 1.0 d) (- (/ (* c b) d) a))
             (* (/ 1.0 (hypot c d)) (- b (* d (/ a (hypot c d))))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -8.5e+80) {
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	} else if (c <= -2.05e-160) {
		tmp = t_0;
	} else if (c <= 1.2e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.75e-62) {
		tmp = t_0;
	} else if (c <= 1.15e+18) {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (d * (a / hypot(c, d))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -8.5e+80) {
		tmp = b / (Math.hypot(c, d) * (Math.hypot(c, d) / c));
	} else if (c <= -2.05e-160) {
		tmp = t_0;
	} else if (c <= 1.2e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.75e-62) {
		tmp = t_0;
	} else if (c <= 1.15e+18) {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b - (d * (a / Math.hypot(c, d))));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -8.5e+80:
		tmp = b / (math.hypot(c, d) * (math.hypot(c, d) / c))
	elif c <= -2.05e-160:
		tmp = t_0
	elif c <= 1.2e-116:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 1.75e-62:
		tmp = t_0
	elif c <= 1.15e+18:
		tmp = (1.0 / d) * (((c * b) / d) - a)
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b - (d * (a / math.hypot(c, d))))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -8.5e+80)
		tmp = Float64(b / Float64(hypot(c, d) * Float64(hypot(c, d) / c)));
	elseif (c <= -2.05e-160)
		tmp = t_0;
	elseif (c <= 1.2e-116)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 1.75e-62)
		tmp = t_0;
	elseif (c <= 1.15e+18)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(Float64(c * b) / d) - a));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(d * Float64(a / hypot(c, d)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -8.5e+80)
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	elseif (c <= -2.05e-160)
		tmp = t_0;
	elseif (c <= 1.2e-116)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 1.75e-62)
		tmp = t_0;
	elseif (c <= 1.15e+18)
		tmp = (1.0 / d) * (((c * b) / d) - a);
	else
		tmp = (1.0 / hypot(c, d)) * (b - (d * (a / hypot(c, d))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.5e+80], N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.05e-160], t$95$0, If[LessEqual[c, 1.2e-116], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.75e-62], t$95$0, If[LessEqual[c, 1.15e+18], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b - N[(d * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -8.50000000000000007e80

   1. Initial program 45.6%

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

   2. Taylor expanded in b around inf 41.9%

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

      1. associate-/l* 48.8%

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

      2. +-commutative 48.8%

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

      3. unpow2 48.8%

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

      4. fma-udef 48.8%

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

   4. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. +-commutative 48.8%

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

      3. add-sqr-sqrt 48.8%

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

      4. pow2 48.8%

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

      5. hypot-udef 48.8%

         \[\leadsto \frac{b}{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{{c}^{2} + d \cdot d}}{c}} \]

      6. pow2 48.8%

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

      7. hypot-udef 48.8%

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

      8. *-un-lft-identity 48.8%

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

      9. times-frac 92.1%

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

      10. /-rgt-identity 92.1%

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

   6. Applied egg-rr 92.1%

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

   if -8.50000000000000007e80 < c < -2.05000000000000001e-160 or 1.19999999999999996e-116 < c < 1.7500000000000001e-62

   1. Initial program 85.2%

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

   if -2.05000000000000001e-160 < c < 1.19999999999999996e-116

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.2%

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

      4. hypot-def 74.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 45.0%

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

   5. Taylor expanded in c around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. +-commutative 95.0%

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

      3. unsub-neg 95.0%

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

      4. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

   if 1.7500000000000001e-62 < c < 1.15e18

   1. Initial program 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. times-frac 70.3%

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

      4. hypot-def 70.4%

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

      5. hypot-def 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in c around 0 42.5%

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

   5. Taylor expanded in c around 0 80.4%

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

   if 1.15e18 < c

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.4%

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

      4. hypot-def 49.4%

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

      5. hypot-def 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. div-sub 63.6%

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

      2. sub-neg 63.6%

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

      3. *-commutative 63.6%

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

      4. associate-/l* 87.9%

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

      5. associate-/l* 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. sub-neg 99.3%

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

      2. associate-/r/ 99.7%

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

      3. associate-/r/ 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in c around inf 93.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{b} - \frac{a}{\mathsf{hypot}\left(c, d\right)} \cdot d\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - d \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\right)\\ \end{array} \]

Alternative 4: 82.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -2.15e+80)
     (/ b (* (hypot c d) (/ (hypot c d) c)))
     (if (<= c -2.05e-160)
       t_0
       (if (<= c 3e-116)
         (* (/ 1.0 d) (- (/ b (/ d c)) a))
         (if (<= c 1.6e-61)
           t_0
           (if (<= c 1.15e+18)
             (* (/ 1.0 d) (- (/ (* c b) d) a))
             (* (/ 1.0 (hypot c d)) (- b (/ a (/ (hypot c d) d)))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.15e+80) {
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	} else if (c <= -2.05e-160) {
		tmp = t_0;
	} else if (c <= 3e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.6e-61) {
		tmp = t_0;
	} else if (c <= 1.15e+18) {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (a / (hypot(c, d) / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -2.15e+80) {
		tmp = b / (Math.hypot(c, d) * (Math.hypot(c, d) / c));
	} else if (c <= -2.05e-160) {
		tmp = t_0;
	} else if (c <= 3e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 1.6e-61) {
		tmp = t_0;
	} else if (c <= 1.15e+18) {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b - (a / (Math.hypot(c, d) / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -2.15e+80:
		tmp = b / (math.hypot(c, d) * (math.hypot(c, d) / c))
	elif c <= -2.05e-160:
		tmp = t_0
	elif c <= 3e-116:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 1.6e-61:
		tmp = t_0
	elif c <= 1.15e+18:
		tmp = (1.0 / d) * (((c * b) / d) - a)
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b - (a / (math.hypot(c, d) / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -2.15e+80)
		tmp = Float64(b / Float64(hypot(c, d) * Float64(hypot(c, d) / c)));
	elseif (c <= -2.05e-160)
		tmp = t_0;
	elseif (c <= 3e-116)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 1.6e-61)
		tmp = t_0;
	elseif (c <= 1.15e+18)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(Float64(c * b) / d) - a));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(a / Float64(hypot(c, d) / d))));
	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 <= -2.15e+80)
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	elseif (c <= -2.05e-160)
		tmp = t_0;
	elseif (c <= 3e-116)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 1.6e-61)
		tmp = t_0;
	elseif (c <= 1.15e+18)
		tmp = (1.0 / d) * (((c * b) / d) - a);
	else
		tmp = (1.0 / hypot(c, d)) * (b - (a / (hypot(c, d) / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.15e+80], N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.05e-160], t$95$0, If[LessEqual[c, 3e-116], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.6e-61], t$95$0, If[LessEqual[c, 1.15e+18], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision]), $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 5 regimes

2. if c < -2.15000000000000002e80

   1. Initial program 45.6%

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

   2. Taylor expanded in b around inf 41.9%

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

      1. associate-/l* 48.8%

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

      2. +-commutative 48.8%

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

      3. unpow2 48.8%

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

      4. fma-udef 48.8%

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

   4. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. +-commutative 48.8%

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

      3. add-sqr-sqrt 48.8%

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

      4. pow2 48.8%

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

      5. hypot-udef 48.8%

         \[\leadsto \frac{b}{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{{c}^{2} + d \cdot d}}{c}} \]

      6. pow2 48.8%

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

      7. hypot-udef 48.8%

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

      8. *-un-lft-identity 48.8%

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

      9. times-frac 92.1%

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

      10. /-rgt-identity 92.1%

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

   6. Applied egg-rr 92.1%

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

   if -2.15000000000000002e80 < c < -2.05000000000000001e-160 or 3.00000000000000026e-116 < c < 1.6000000000000001e-61

   1. Initial program 85.2%

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

   if -2.05000000000000001e-160 < c < 3.00000000000000026e-116

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.2%

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

      4. hypot-def 74.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 45.0%

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

   5. Taylor expanded in c around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. +-commutative 95.0%

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

      3. unsub-neg 95.0%

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

      4. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

   if 1.6000000000000001e-61 < c < 1.15e18

   1. Initial program 70.3%

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

      1. *-un-lft-identity 70.3%

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

      2. add-sqr-sqrt 70.3%

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

      3. times-frac 70.3%

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

      4. hypot-def 70.4%

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

      5. hypot-def 87.0%

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

   3. Applied egg-rr 87.0%

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

   4. Taylor expanded in c around 0 42.5%

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

   5. Taylor expanded in c around 0 80.4%

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

   if 1.15e18 < c

   1. Initial program 49.4%

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

      1. *-un-lft-identity 49.4%

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

      2. add-sqr-sqrt 49.4%

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

      3. times-frac 49.4%

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

      4. hypot-def 49.4%

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

      5. hypot-def 63.6%

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

   3. Applied egg-rr 63.6%

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

      1. div-sub 63.6%

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

      2. *-commutative 63.6%

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

      3. associate-/l* 87.9%

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

      4. associate-/l* 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in c around inf 93.4%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(\color{blue}{b} - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -2.05 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{\mathsf{hypot}\left(c, d\right)}{d}}\right)\\ \end{array} \]

Alternative 5: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -1.95 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{c}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -1e+81)
     (/ b (* (hypot c d) (/ (hypot c d) c)))
     (if (<= c -1.95e-160)
       t_0
       (if (<= c 3e-116)
         (* (/ 1.0 d) (- (/ b (/ d c)) a))
         (if (<= c 4.3e+117)
           t_0
           (* (/ 1.0 (hypot c d)) (- b (/ a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+81) {
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	} else if (c <= -1.95e-160) {
		tmp = t_0;
	} else if (c <= 3e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 4.3e+117) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (a / (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1e+81) {
		tmp = b / (Math.hypot(c, d) * (Math.hypot(c, d) / c));
	} else if (c <= -1.95e-160) {
		tmp = t_0;
	} else if (c <= 3e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 4.3e+117) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b - (a / (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -1e+81:
		tmp = b / (math.hypot(c, d) * (math.hypot(c, d) / c))
	elif c <= -1.95e-160:
		tmp = t_0
	elif c <= 3e-116:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 4.3e+117:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b - (a / (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1e+81)
		tmp = Float64(b / Float64(hypot(c, d) * Float64(hypot(c, d) / c)));
	elseif (c <= -1.95e-160)
		tmp = t_0;
	elseif (c <= 3e-116)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 4.3e+117)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(a / Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -1e+81)
		tmp = b / (hypot(c, d) * (hypot(c, d) / c));
	elseif (c <= -1.95e-160)
		tmp = t_0;
	elseif (c <= 3e-116)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 4.3e+117)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b - (a / (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1e+81], N[(b / N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] * N[(N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.95e-160], t$95$0, If[LessEqual[c, 3e-116], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.3e+117], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.99999999999999921e80

   1. Initial program 45.6%

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

   2. Taylor expanded in b around inf 41.9%

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

      1. associate-/l* 48.8%

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

      2. +-commutative 48.8%

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

      3. unpow2 48.8%

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

      4. fma-udef 48.8%

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

   4. Simplified 48.8%

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

      1. fma-udef 48.8%

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

      2. +-commutative 48.8%

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

      3. add-sqr-sqrt 48.8%

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

      4. pow2 48.8%

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

      5. hypot-udef 48.8%

         \[\leadsto \frac{b}{\frac{\color{blue}{\mathsf{hypot}\left(c, d\right)} \cdot \sqrt{{c}^{2} + d \cdot d}}{c}} \]

      6. pow2 48.8%

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

      7. hypot-udef 48.8%

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

      8. *-un-lft-identity 48.8%

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

      9. times-frac 92.1%

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

      10. /-rgt-identity 92.1%

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

   6. Applied egg-rr 92.1%

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

   if -9.99999999999999921e80 < c < -1.94999999999999995e-160 or 3.00000000000000026e-116 < c < 4.29999999999999998e117

   1. Initial program 80.8%

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

   if -1.94999999999999995e-160 < c < 3.00000000000000026e-116

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.2%

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

      4. hypot-def 74.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 45.0%

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

   5. Taylor expanded in c around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. +-commutative 95.0%

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

      3. unsub-neg 95.0%

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

      4. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

   if 4.29999999999999998e117 < c

   1. Initial program 29.1%

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

      1. *-un-lft-identity 29.1%

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

      2. add-sqr-sqrt 29.1%

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

      3. times-frac 29.1%

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

      4. hypot-def 29.1%

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

      5. hypot-def 50.1%

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

   3. Applied egg-rr 50.1%

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

   4. Taylor expanded in c around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. associate-/l* 90.8%

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

   6. Simplified 90.8%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\frac{b}{\mathsf{hypot}\left(c, d\right) \cdot \frac{\mathsf{hypot}\left(c, d\right)}{c}}\\ \mathbf{elif}\;c \leq -1.95 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{c}{d}}\right)\\ \end{array} \]

Alternative 6: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -7 \cdot 10^{+133}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{c}{d}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -7e+133)
     (- (/ b c) (* d (/ (/ a c) c)))
     (if (<= c -2.75e-160)
       t_0
       (if (<= c 1.18e-116)
         (* (/ 1.0 d) (- (/ b (/ d c)) a))
         (if (<= c 2.6e+120)
           t_0
           (* (/ 1.0 (hypot c d)) (- b (/ a (/ c d))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7e+133) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= -2.75e-160) {
		tmp = t_0;
	} else if (c <= 1.18e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 2.6e+120) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(c, d)) * (b - (a / (c / d)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -7e+133) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= -2.75e-160) {
		tmp = t_0;
	} else if (c <= 1.18e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 2.6e+120) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b - (a / (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -7e+133:
		tmp = (b / c) - (d * ((a / c) / c))
	elif c <= -2.75e-160:
		tmp = t_0
	elif c <= 1.18e-116:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 2.6e+120:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b - (a / (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -7e+133)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	elseif (c <= -2.75e-160)
		tmp = t_0;
	elseif (c <= 1.18e-116)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 2.6e+120)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b - Float64(a / Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -7e+133)
		tmp = (b / c) - (d * ((a / c) / c));
	elseif (c <= -2.75e-160)
		tmp = t_0;
	elseif (c <= 1.18e-116)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 2.6e+120)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(c, d)) * (b - (a / (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7e+133], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.75e-160], t$95$0, If[LessEqual[c, 1.18e-116], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+120], t$95$0, N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b - N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.9999999999999997e133

   1. Initial program 35.9%

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

   2. Taylor expanded in c around inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

      4. associate-/l* 96.1%

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

      5. associate-/r/ 96.1%

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

   4. Simplified 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. pow2 96.1%

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

      3. times-frac 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.3%

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

      2. *-lft-identity 96.3%

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

   8. Simplified 96.3%

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

   if -6.9999999999999997e133 < c < -2.75e-160 or 1.1800000000000001e-116 < c < 2.5999999999999999e120

   1. Initial program 80.3%

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

   if -2.75e-160 < c < 1.1800000000000001e-116

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.2%

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

      4. hypot-def 74.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 45.0%

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

   5. Taylor expanded in c around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. +-commutative 95.0%

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

      3. unsub-neg 95.0%

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

      4. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

   if 2.5999999999999999e120 < c

   1. Initial program 29.1%

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

      1. *-un-lft-identity 29.1%

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

      2. add-sqr-sqrt 29.1%

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

      3. times-frac 29.1%

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

      4. hypot-def 29.1%

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

      5. hypot-def 50.1%

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

   3. Applied egg-rr 50.1%

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

   4. Taylor expanded in c around inf 88.5%

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

      1. mul-1-neg 88.5%

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

      2. unsub-neg 88.5%

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

      3. associate-/l* 90.8%

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

   6. Simplified 90.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7 \cdot 10^{+133}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -2.75 \cdot 10^{-160}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+120}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b - \frac{a}{\frac{c}{d}}\right)\\ \end{array} \]

Alternative 7: 81.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -6.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -6.5e+133)
     (- (/ b c) (* d (/ (/ a c) c)))
     (if (<= c -6.2e-161)
       t_0
       (if (<= c 3.2e-116)
         (* (/ 1.0 d) (- (/ b (/ d c)) a))
         (if (<= c 4.8e+117) t_0 (- (/ b c) (/ (/ (* d a) c) 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 <= -6.5e+133) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= -6.2e-161) {
		tmp = t_0;
	} else if (c <= 3.2e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 4.8e+117) {
		tmp = t_0;
	} else {
		tmp = (b / c) - (((d * a) / 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) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    if (c <= (-6.5d+133)) then
        tmp = (b / c) - (d * ((a / c) / c))
    else if (c <= (-6.2d-161)) then
        tmp = t_0
    else if (c <= 3.2d-116) then
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    else if (c <= 4.8d+117) then
        tmp = t_0
    else
        tmp = (b / c) - (((d * a) / c) / 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 <= -6.5e+133) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else if (c <= -6.2e-161) {
		tmp = t_0;
	} else if (c <= 3.2e-116) {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	} else if (c <= 4.8e+117) {
		tmp = t_0;
	} else {
		tmp = (b / c) - (((d * a) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -6.5e+133:
		tmp = (b / c) - (d * ((a / c) / c))
	elif c <= -6.2e-161:
		tmp = t_0
	elif c <= 3.2e-116:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	elif c <= 4.8e+117:
		tmp = t_0
	else:
		tmp = (b / c) - (((d * a) / c) / 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 <= -6.5e+133)
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	elseif (c <= -6.2e-161)
		tmp = t_0;
	elseif (c <= 3.2e-116)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	elseif (c <= 4.8e+117)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(Float64(d * a) / c) / 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 <= -6.5e+133)
		tmp = (b / c) - (d * ((a / c) / c));
	elseif (c <= -6.2e-161)
		tmp = t_0;
	elseif (c <= 3.2e-116)
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	elseif (c <= 4.8e+117)
		tmp = t_0;
	else
		tmp = (b / c) - (((d * a) / c) / 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, -6.5e+133], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -6.2e-161], t$95$0, If[LessEqual[c, 3.2e-116], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.8e+117], t$95$0, N[(N[(b / c), $MachinePrecision] - N[(N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.5000000000000004e133

   1. Initial program 35.9%

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

   2. Taylor expanded in c around inf 93.5%

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

      1. +-commutative 93.5%

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

      2. mul-1-neg 93.5%

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

      3. unsub-neg 93.5%

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

      4. associate-/l* 96.1%

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

      5. associate-/r/ 96.1%

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

   4. Simplified 96.1%

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

      1. *-un-lft-identity 96.1%

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

      2. pow2 96.1%

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

      3. times-frac 96.3%

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

   6. Applied egg-rr 96.3%

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

      1. associate-*l/ 96.3%

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

      2. *-lft-identity 96.3%

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

   8. Simplified 96.3%

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

   if -6.5000000000000004e133 < c < -6.1999999999999997e-161 or 3.20000000000000009e-116 < c < 4.7999999999999998e117

   1. Initial program 80.3%

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

   if -6.1999999999999997e-161 < c < 3.20000000000000009e-116

   1. Initial program 74.2%

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

      1. *-un-lft-identity 74.2%

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

      2. add-sqr-sqrt 74.2%

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

      3. times-frac 74.2%

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

      4. hypot-def 74.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 45.0%

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

   5. Taylor expanded in c around 0 95.0%

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

      1. neg-mul-1 95.0%

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

      2. +-commutative 95.0%

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

      3. unsub-neg 95.0%

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

      4. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

   if 4.7999999999999998e117 < c

   1. Initial program 29.1%

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

   2. Taylor expanded in c around inf 81.6%

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

      1. +-commutative 81.6%

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

      2. mul-1-neg 81.6%

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

      3. unsub-neg 81.6%

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

      4. associate-/l* 83.5%

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

      5. associate-/r/ 83.5%

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

   4. Simplified 83.5%

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

      1. associate-*l/ 81.6%

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

      2. pow2 81.6%

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

      3. associate-/r* 88.8%

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

   6. Applied egg-rr 88.8%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+133}:\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -6.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{d \cdot a}{c}}{c}\\ \end{array} \]

Alternative 8: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-14} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.9e-14) (not (<= c 2.7e+18)))
   (/ b c)
   (* (/ 1.0 d) (- (* c (/ b d)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-14) || !(c <= 2.7e+18)) {
		tmp = b / c;
	} else {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.9d-14)) .or. (.not. (c <= 2.7d+18))) then
        tmp = b / c
    else
        tmp = (1.0d0 / d) * ((c * (b / d)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-14) || !(c <= 2.7e+18)) {
		tmp = b / c;
	} else {
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.9e-14) or not (c <= 2.7e+18):
		tmp = b / c
	else:
		tmp = (1.0 / d) * ((c * (b / d)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.9e-14) || !(c <= 2.7e+18))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c * Float64(b / d)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.9e-14) || ~((c <= 2.7e+18)))
		tmp = b / c;
	else
		tmp = (1.0 / d) * ((c * (b / d)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.9e-14], N[Not[LessEqual[c, 2.7e+18]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.8999999999999998e-14 or 2.7e18 < c

   1. Initial program 52.8%

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

   2. Taylor expanded in c around inf 76.8%

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

   if -3.8999999999999998e-14 < c < 2.7e18

   1. Initial program 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 40.8%

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

   5. Taylor expanded in c around 0 79.2%

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

      1. +-commutative 79.2%

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

      2. mul-1-neg 79.2%

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

      3. unsub-neg 79.2%

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

      4. div-inv 79.1%

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

      5. *-commutative 79.1%

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

      6. associate-*l* 77.1%

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

      7. div-inv 77.2%

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

   7. Applied egg-rr 77.2%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-14} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(c \cdot \frac{b}{d} - a\right)\\ \end{array} \]

Alternative 9: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-14} \lor \neg \left(c \leq 5.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.9e-14) (not (<= c 5.5e+18)))
   (/ b c)
   (* (/ 1.0 d) (- (/ b (/ d c)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-14) || !(c <= 5.5e+18)) {
		tmp = b / c;
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.9d-14)) .or. (.not. (c <= 5.5d+18))) then
        tmp = b / c
    else
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.9e-14) || !(c <= 5.5e+18)) {
		tmp = b / c;
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.9e-14) or not (c <= 5.5e+18):
		tmp = b / c
	else:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.9e-14) || !(c <= 5.5e+18))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.9e-14) || ~((c <= 5.5e+18)))
		tmp = b / c;
	else
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.9e-14], N[Not[LessEqual[c, 5.5e+18]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.8999999999999998e-14 or 5.5e18 < c

   1. Initial program 52.8%

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

   2. Taylor expanded in c around inf 76.8%

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

   if -3.8999999999999998e-14 < c < 5.5e18

   1. Initial program 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 40.8%

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

   5. Taylor expanded in c around 0 79.2%

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

      1. neg-mul-1 79.2%

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

      2. +-commutative 79.2%

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

      3. unsub-neg 79.2%

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

      4. associate-/l* 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.9 \cdot 10^{-14} \lor \neg \left(c \leq 5.5 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \]

Alternative 10: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.62 \cdot 10^{-15} \lor \neg \left(c \leq 1.2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.62e-15) (not (<= c 1.2e+18)))
   (- (/ b c) (* d (/ (/ a c) c)))
   (* (/ 1.0 d) (- (/ b (/ d c)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.62e-15) || !(c <= 1.2e+18)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.62d-15)) .or. (.not. (c <= 1.2d+18))) then
        tmp = (b / c) - (d * ((a / c) / c))
    else
        tmp = (1.0d0 / d) * ((b / (d / c)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.62e-15) || !(c <= 1.2e+18)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.62e-15) or not (c <= 1.2e+18):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = (1.0 / d) * ((b / (d / c)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.62e-15) || !(c <= 1.2e+18))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(b / Float64(d / c)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.62e-15) || ~((c <= 1.2e+18)))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = (1.0 / d) * ((b / (d / c)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.62e-15], N[Not[LessEqual[c, 1.2e+18]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.62000000000000009e-15 or 1.2e18 < c

   1. Initial program 52.8%

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

   2. Taylor expanded in c around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. mul-1-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. associate-/l* 79.6%

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

      5. associate-/r/ 80.4%

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

   4. Simplified 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. pow2 80.4%

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

      3. times-frac 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-*l/ 81.9%

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

      2. *-lft-identity 81.9%

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

   8. Simplified 81.9%

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

   if -1.62000000000000009e-15 < c < 1.2e18

   1. Initial program 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 40.8%

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

   5. Taylor expanded in c around 0 79.2%

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

      1. neg-mul-1 79.2%

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

      2. +-commutative 79.2%

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

      3. unsub-neg 79.2%

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

      4. associate-/l* 79.2%

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

   7. Simplified 79.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.62 \cdot 10^{-15} \lor \neg \left(c \leq 1.2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{b}{\frac{d}{c}} - a\right)\\ \end{array} \]

Alternative 11: 78.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.36 \cdot 10^{-14} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.36e-14) (not (<= c 2.7e+18)))
   (- (/ b c) (* d (/ (/ a c) c)))
   (* (/ 1.0 d) (- (/ (* c b) d) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.36e-14) || !(c <= 2.7e+18)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.36d-14)) .or. (.not. (c <= 2.7d+18))) then
        tmp = (b / c) - (d * ((a / c) / c))
    else
        tmp = (1.0d0 / d) * (((c * b) / d) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.36e-14) || !(c <= 2.7e+18)) {
		tmp = (b / c) - (d * ((a / c) / c));
	} else {
		tmp = (1.0 / d) * (((c * b) / d) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.36e-14) or not (c <= 2.7e+18):
		tmp = (b / c) - (d * ((a / c) / c))
	else:
		tmp = (1.0 / d) * (((c * b) / d) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.36e-14) || !(c <= 2.7e+18))
		tmp = Float64(Float64(b / c) - Float64(d * Float64(Float64(a / c) / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(Float64(c * b) / d) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.36e-14) || ~((c <= 2.7e+18)))
		tmp = (b / c) - (d * ((a / c) / c));
	else
		tmp = (1.0 / d) * (((c * b) / d) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.36e-14], N[Not[LessEqual[c, 2.7e+18]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(d * N[(N[(a / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.36e-14 or 2.7e18 < c

   1. Initial program 52.8%

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

   2. Taylor expanded in c around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. mul-1-neg 78.2%

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

      3. unsub-neg 78.2%

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

      4. associate-/l* 79.6%

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

      5. associate-/r/ 80.4%

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

   4. Simplified 80.4%

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

      1. *-un-lft-identity 80.4%

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

      2. pow2 80.4%

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

      3. times-frac 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-*l/ 81.9%

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

      2. *-lft-identity 81.9%

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

   8. Simplified 81.9%

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

   if -1.36e-14 < c < 2.7e18

   1. Initial program 77.3%

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

      1. *-un-lft-identity 77.3%

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

      2. add-sqr-sqrt 77.3%

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

      3. times-frac 77.2%

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

      4. hypot-def 77.2%

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

      5. hypot-def 91.1%

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

   3. Applied egg-rr 91.1%

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

   4. Taylor expanded in c around 0 40.8%

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

   5. Taylor expanded in c around 0 79.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.36 \cdot 10^{-14} \lor \neg \left(c \leq 2.7 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{b}{c} - d \cdot \frac{\frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c \cdot b}{d} - a\right)\\ \end{array} \]

Alternative 12: 61.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+66} \lor \neg \left(d \leq 3.6 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.2e+66) (not (<= d 3.6e+138))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.2e+66) || !(d <= 3.6e+138)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-3.2d+66)) .or. (.not. (d <= 3.6d+138))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.2e+66) || !(d <= 3.6e+138)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.2e+66) or not (d <= 3.6e+138):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.2e+66) || !(d <= 3.6e+138))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -3.2e+66) || ~((d <= 3.6e+138)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.2e+66], N[Not[LessEqual[d, 3.6e+138]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.2e66 or 3.6000000000000001e138 < d

   1. Initial program 47.5%

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

   2. Taylor expanded in c around 0 77.8%

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

      1. associate-*r/ 77.8%

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

      2. neg-mul-1 77.8%

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

   4. Simplified 77.8%

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

   if -3.2e66 < d < 3.6000000000000001e138

   1. Initial program 73.5%

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

   2. Taylor expanded in c around inf 61.8%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{+66} \lor \neg \left(d \leq 3.6 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 13: 46.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+199} \lor \neg \left(d \leq 8 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2e+199) (not (<= d 8e+154))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2e+199) || !(d <= 8e+154)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2d+199)) .or. (.not. (d <= 8d+154))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2e+199) || !(d <= 8e+154)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2e+199) or not (d <= 8e+154):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2e+199) || !(d <= 8e+154))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2e+199) || ~((d <= 8e+154)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2e+199], N[Not[LessEqual[d, 8e+154]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.00000000000000019e199 or 8.0000000000000003e154 < d

   1. Initial program 47.3%

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

      1. *-un-lft-identity 47.3%

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

      2. add-sqr-sqrt 47.3%

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

      3. times-frac 47.3%

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

      4. hypot-def 47.3%

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

      5. hypot-def 67.5%

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

   3. Applied egg-rr 67.5%

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

   4. Taylor expanded in c around 0 58.9%

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

   5. Taylor expanded in d around -inf 47.9%

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

   if -2.00000000000000019e199 < d < 8.0000000000000003e154

   1. Initial program 69.0%

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

   2. Taylor expanded in c around inf 54.7%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{+199} \lor \neg \left(d \leq 8 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 14: 10.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. *-un-lft-identity 65.5%

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

   2. add-sqr-sqrt 65.5%

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

   3. times-frac 65.4%

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

   4. hypot-def 65.4%

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

   5. hypot-def 79.7%

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

3. Applied egg-rr 79.7%

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

4. Taylor expanded in c around 0 25.6%

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

5. Taylor expanded in d around -inf 11.2%

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

6. Final simplification 11.2%

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

Developer target: 99.5% 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 2023307 
(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))))