Complex division, imag part

Percentage Accurate: 61.1% → 97.8%

Time: 17.3s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. add-sqr-sqrt 63.0%

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

   3. times-frac 63.0%

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

   4. hypot-def 63.0%

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

   5. hypot-def 77.4%

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

3. Applied egg-rr 77.4%

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

   1. div-sub 77.4%

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

   2. sub-neg 77.4%

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

5. Applied egg-rr 77.4%

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

   1. sub-neg 77.4%

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

   2. associate-/l* 88.9%

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

   3. associate-/r/ 87.4%

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

   4. *-commutative 87.4%

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

   5. associate-*r/ 96.7%

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

7. Simplified 96.7%

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

   1. associate-*l/ 96.9%

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

   2. *-un-lft-identity 96.9%

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

   3. div-sub 96.9%

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

   4. *-commutative 96.9%

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

9. Applied egg-rr 96.9%

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

10. Final simplification 96.9%

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

Alternative 2: 97.5% accurate, 0.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, c, d)
	tmp = (1.0 / hypot(c, d)) * ((c * (b / hypot(c, d))) - (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[(c * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(d * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. add-sqr-sqrt 63.0%

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

   3. times-frac 63.0%

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

   4. hypot-def 63.0%

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

   5. hypot-def 77.4%

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

3. Applied egg-rr 77.4%

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

   1. div-sub 77.4%

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

   2. sub-neg 77.4%

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

5. Applied egg-rr 77.4%

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

   1. sub-neg 77.4%

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

   2. associate-/l* 88.9%

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

   3. associate-/r/ 87.4%

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

   4. *-commutative 87.4%

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

   5. associate-*r/ 96.7%

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

7. Simplified 96.7%

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

8. Final simplification 96.7%

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

Alternative 3: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := t_0 \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -2.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.68 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(c \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)} - a \cdot \frac{d}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d)))
        (t_1 (* t_0 (/ (- (* c b) (* d a)) (hypot c d)))))
   (if (<= c -2.15e+71)
     (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
     (if (<= c -5.5e-101)
       t_1
       (if (<= c 1.1e-227)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 1.68e+80)
           t_1
           (* t_0 (- (* c (/ b (hypot c d))) (* a (/ d c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = t_0 * (((c * b) - (d * a)) / hypot(c, d));
	double tmp;
	if (c <= -2.15e+71) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -5.5e-101) {
		tmp = t_1;
	} else if (c <= 1.1e-227) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.68e+80) {
		tmp = t_1;
	} else {
		tmp = t_0 * ((c * (b / hypot(c, d))) - (a * (d / c)));
	}
	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 = t_0 * (((c * b) - (d * a)) / Math.hypot(c, d));
	double tmp;
	if (c <= -2.15e+71) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -5.5e-101) {
		tmp = t_1;
	} else if (c <= 1.1e-227) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.68e+80) {
		tmp = t_1;
	} else {
		tmp = t_0 * ((c * (b / Math.hypot(c, d))) - (a * (d / c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = 1.0 / math.hypot(c, d)
	t_1 = t_0 * (((c * b) - (d * a)) / math.hypot(c, d))
	tmp = 0
	if c <= -2.15e+71:
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))))
	elif c <= -5.5e-101:
		tmp = t_1
	elif c <= 1.1e-227:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 1.68e+80:
		tmp = t_1
	else:
		tmp = t_0 * ((c * (b / math.hypot(c, d))) - (a * (d / c)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(t_0 * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)))
	tmp = 0.0
	if (c <= -2.15e+71)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= -5.5e-101)
		tmp = t_1;
	elseif (c <= 1.1e-227)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 1.68e+80)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(Float64(c * Float64(b / hypot(c, d))) - Float64(a * Float64(d / c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = 1.0 / hypot(c, d);
	t_1 = t_0 * (((c * b) - (d * a)) / hypot(c, d));
	tmp = 0.0;
	if (c <= -2.15e+71)
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	elseif (c <= -5.5e-101)
		tmp = t_1;
	elseif (c <= 1.1e-227)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 1.68e+80)
		tmp = t_1;
	else
		tmp = t_0 * ((c * (b / hypot(c, d))) - (a * (d / c)));
	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[(t$95$0 * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.15e+71], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.5e-101], t$95$1, If[LessEqual[c, 1.1e-227], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.68e+80], t$95$1, N[(t$95$0 * N[(N[(c * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.14999999999999992e71

   1. Initial program 40.2%

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

   2. Taylor expanded in c around inf 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

      4. unpow2 71.9%

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

      5. times-frac 82.2%

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

   4. Simplified 82.2%

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

      1. associate-*r/ 82.2%

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

      2. clear-num 82.3%

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

   6. Applied egg-rr 82.3%

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

   if -2.14999999999999992e71 < c < -5.49999999999999973e-101 or 1.0999999999999999e-227 < c < 1.68000000000000006e80

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 85.7%

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

      4. hypot-def 85.7%

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

      5. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

   if -5.49999999999999973e-101 < c < 1.0999999999999999e-227

   1. Initial program 66.9%

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

   2. Taylor expanded in c around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. *-commutative 87.8%

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

      5. unpow2 87.8%

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

      6. times-frac 91.2%

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

   4. Simplified 91.2%

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

      1. associate-*r/ 95.6%

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

      2. sub-div 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 1.68000000000000006e80 < c

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 58.3%

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

   3. Applied egg-rr 58.3%

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

      1. div-sub 58.3%

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

      2. sub-neg 58.3%

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

   5. Applied egg-rr 58.3%

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

      1. sub-neg 58.3%

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

      2. associate-/l* 91.7%

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

      3. associate-/r/ 91.7%

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

      4. *-commutative 91.7%

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

      5. associate-*r/ 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in d around 0 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-/l* 94.0%

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

      3. associate-/r/ 94.0%

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

   10. Simplified 94.0%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.68 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(c \cdot \frac{b}{\mathsf{hypot}\left(c, d\right)} - a \cdot \frac{d}{c}\right)\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (hypot c d)) (/ (- (* c b) (* d a)) (hypot c d)))))
   (if (<= c -1.65e+69)
     (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
     (if (<= c -5.6e-107)
       t_0
       (if (<= c 1.05e-227)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 1.75e+81) t_0 (/ (- b (* a (/ d c))) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	double tmp;
	if (c <= -1.65e+69) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -5.6e-107) {
		tmp = t_0;
	} else if (c <= 1.05e-227) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.75e+81) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / Math.hypot(c, d)) * (((c * b) - (d * a)) / Math.hypot(c, d));
	double tmp;
	if (c <= -1.65e+69) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -5.6e-107) {
		tmp = t_0;
	} else if (c <= 1.05e-227) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.75e+81) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / math.hypot(c, d)) * (((c * b) - (d * a)) / math.hypot(c, d))
	tmp = 0
	if c <= -1.65e+69:
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))))
	elif c <= -5.6e-107:
		tmp = t_0
	elif c <= 1.05e-227:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 1.75e+81:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / hypot(c, d)) * Float64(Float64(Float64(c * b) - Float64(d * a)) / hypot(c, d)))
	tmp = 0.0
	if (c <= -1.65e+69)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= -5.6e-107)
		tmp = t_0;
	elseif (c <= 1.05e-227)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 1.75e+81)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / hypot(c, d)) * (((c * b) - (d * a)) / hypot(c, d));
	tmp = 0.0;
	if (c <= -1.65e+69)
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	elseif (c <= -5.6e-107)
		tmp = t_0;
	elseif (c <= 1.05e-227)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 1.75e+81)
		tmp = t_0;
	else
		tmp = (b - (a * (d / c))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+69], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.6e-107], t$95$0, If[LessEqual[c, 1.05e-227], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.75e+81], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.6499999999999999e69

   1. Initial program 40.2%

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

   2. Taylor expanded in c around inf 71.9%

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

      1. +-commutative 71.9%

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

      2. mul-1-neg 71.9%

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

      3. unsub-neg 71.9%

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

      4. unpow2 71.9%

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

      5. times-frac 82.2%

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

   4. Simplified 82.2%

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

      1. associate-*r/ 82.2%

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

      2. clear-num 82.3%

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

   6. Applied egg-rr 82.3%

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

   if -1.6499999999999999e69 < c < -5.5999999999999998e-107 or 1.05e-227 < c < 1.75e81

   1. Initial program 85.8%

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

      1. *-un-lft-identity 85.8%

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

      2. add-sqr-sqrt 85.7%

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

      3. times-frac 85.7%

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

      4. hypot-def 85.7%

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

      5. hypot-def 95.4%

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

   3. Applied egg-rr 95.4%

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

   if -5.5999999999999998e-107 < c < 1.05e-227

   1. Initial program 66.9%

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

   2. Taylor expanded in c around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. mul-1-neg 87.8%

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

      3. unsub-neg 87.8%

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

      4. *-commutative 87.8%

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

      5. unpow2 87.8%

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

      6. times-frac 91.2%

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

   4. Simplified 91.2%

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

      1. associate-*r/ 95.6%

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

      2. sub-div 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 1.75e81 < c

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 58.3%

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

   3. Applied egg-rr 58.3%

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

   4. Taylor expanded in c around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. associate-*l/ 92.1%

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

      3. sub-neg 92.1%

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

      4. *-commutative 92.1%

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

   6. Simplified 92.1%

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

      1. associate-*l/ 92.4%

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

      2. *-un-lft-identity 92.4%

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

      3. div-sub 92.4%

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

      4. associate-*r/ 90.1%

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

   8. Applied egg-rr 90.1%

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

      1. div-sub 90.1%

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

      2. associate-/l* 92.4%

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

      3. associate-/r/ 92.4%

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

   10. Simplified 92.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+69}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{c \cdot b - d \cdot a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 5: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-161}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* d d) (* c c)))))
   (if (<= c -6.5e+56)
     (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
     (if (<= c -3.8e-107)
       t_0
       (if (<= c 9e-161)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 4e+79) t_0 (/ (- b (* a (/ d c))) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -6.5e+56) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -3.8e-107) {
		tmp = t_0;
	} else if (c <= 9e-161) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 4e+79) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -6.5e+56) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -3.8e-107) {
		tmp = t_0;
	} else if (c <= 9e-161) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 4e+79) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c))
	tmp = 0
	if c <= -6.5e+56:
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))))
	elif c <= -3.8e-107:
		tmp = t_0
	elif c <= 9e-161:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 4e+79:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / 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(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -6.5e+56)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= -3.8e-107)
		tmp = t_0;
	elseif (c <= 9e-161)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 4e+79)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	tmp = 0.0;
	if (c <= -6.5e+56)
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	elseif (c <= -3.8e-107)
		tmp = t_0;
	elseif (c <= 9e-161)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 4e+79)
		tmp = t_0;
	else
		tmp = (b - (a * (d / c))) / 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[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.5e+56], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.8e-107], t$95$0, If[LessEqual[c, 9e-161], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4e+79], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.5000000000000001e56

   1. Initial program 43.6%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. associate-*r/ 83.2%

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

      2. clear-num 83.3%

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

   6. Applied egg-rr 83.3%

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

   if -6.5000000000000001e56 < c < -3.8000000000000002e-107 or 8.9999999999999993e-161 < c < 3.99999999999999987e79

   1. Initial program 87.5%

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

   if -3.8000000000000002e-107 < c < 8.9999999999999993e-161

   1. Initial program 68.4%

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

   2. Taylor expanded in c around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 3.99999999999999987e79 < c

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 58.3%

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

   3. Applied egg-rr 58.3%

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

   4. Taylor expanded in c around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. associate-*l/ 92.1%

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

      3. sub-neg 92.1%

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

      4. *-commutative 92.1%

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

   6. Simplified 92.1%

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

      1. associate-*l/ 92.4%

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

      2. *-un-lft-identity 92.4%

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

      3. div-sub 92.4%

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

      4. associate-*r/ 90.1%

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

   8. Applied egg-rr 90.1%

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

      1. div-sub 90.1%

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

      2. associate-/l* 92.4%

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

      3. associate-/r/ 92.4%

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

   10. Simplified 92.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -3.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-161}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+79}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 6: 82.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot b - d \cdot a\\ \mathbf{if}\;c \leq -1.45 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{t_0}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* c b) (* d a))))
   (if (<= c -1.45e+57)
     (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
     (if (<= c -1.5e-108)
       (/ t_0 (fma c c (* d d)))
       (if (<= c 2.45e-160)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 4.4e+80)
           (/ t_0 (+ (* d d) (* c c)))
           (/ (- b (* a (/ d c))) (hypot c d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * b) - (d * a);
	double tmp;
	if (c <= -1.45e+57) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -1.5e-108) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (c <= 2.45e-160) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 4.4e+80) {
		tmp = t_0 / ((d * d) + (c * c));
	} else {
		tmp = (b - (a * (d / c))) / hypot(c, d);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * b) - Float64(d * a))
	tmp = 0.0
	if (c <= -1.45e+57)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= -1.5e-108)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (c <= 2.45e-160)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 4.4e+80)
		tmp = Float64(t_0 / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / hypot(c, d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.45e+57], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -1.5e-108], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.45e-160], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.4e+80], N[(t$95$0 / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -1.4500000000000001e57

   1. Initial program 43.6%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. associate-*r/ 83.2%

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

      2. clear-num 83.3%

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

   6. Applied egg-rr 83.3%

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

   if -1.4500000000000001e57 < c < -1.49999999999999996e-108

   1. Initial program 89.0%

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

      1. fma-def 89.0%

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

   3. Simplified 89.0%

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

   if -1.49999999999999996e-108 < c < 2.4499999999999999e-160

   1. Initial program 68.4%

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

   2. Taylor expanded in c around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 2.4499999999999999e-160 < c < 4.40000000000000005e80

   1. Initial program 86.6%

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

   if 4.40000000000000005e80 < c

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 58.3%

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

   3. Applied egg-rr 58.3%

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

   4. Taylor expanded in c around inf 89.9%

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

      1. mul-1-neg 89.9%

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

      2. associate-*l/ 92.1%

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

      3. sub-neg 92.1%

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

      4. *-commutative 92.1%

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

   6. Simplified 92.1%

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

      1. associate-*l/ 92.4%

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

      2. *-un-lft-identity 92.4%

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

      3. div-sub 92.4%

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

      4. associate-*r/ 90.1%

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

   8. Applied egg-rr 90.1%

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

      1. div-sub 90.1%

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

      2. associate-/l* 92.4%

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

      3. associate-/r/ 92.4%

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

   10. Simplified 92.4%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+80}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 7: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{if}\;c \leq -4.3 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* d d) (* c c)))))
   (if (<= c -4.3e+57)
     (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
     (if (<= c -3.3e-106)
       t_0
       (if (<= c 2.2e-160)
         (/ (- (* b (/ c d)) a) d)
         (if (<= c 1.16e+77) t_0 (/ (- b (* a (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -4.3e+57) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -3.3e-106) {
		tmp = t_0;
	} else if (c <= 2.2e-160) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.16e+77) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c))
    if (c <= (-4.3d+57)) then
        tmp = (b / c) + ((-1.0d0) / (c / (d * (a / c))))
    else if (c <= (-3.3d-106)) then
        tmp = t_0
    else if (c <= 2.2d-160) then
        tmp = ((b * (c / d)) - a) / d
    else if (c <= 1.16d+77) then
        tmp = t_0
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	double tmp;
	if (c <= -4.3e+57) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= -3.3e-106) {
		tmp = t_0;
	} else if (c <= 2.2e-160) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (c <= 1.16e+77) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c))
	tmp = 0
	if c <= -4.3e+57:
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))))
	elif c <= -3.3e-106:
		tmp = t_0
	elif c <= 2.2e-160:
		tmp = ((b * (c / d)) - a) / d
	elif c <= 1.16e+77:
		tmp = t_0
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (c <= -4.3e+57)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= -3.3e-106)
		tmp = t_0;
	elseif (c <= 2.2e-160)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (c <= 1.16e+77)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((d * d) + (c * c));
	tmp = 0.0;
	if (c <= -4.3e+57)
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	elseif (c <= -3.3e-106)
		tmp = t_0;
	elseif (c <= 2.2e-160)
		tmp = ((b * (c / d)) - a) / d;
	elseif (c <= 1.16e+77)
		tmp = t_0;
	else
		tmp = (b - (a * (d / 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[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4.3e+57], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.3e-106], t$95$0, If[LessEqual[c, 2.2e-160], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.16e+77], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.30000000000000033e57

   1. Initial program 43.6%

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

   2. Taylor expanded in c around inf 73.5%

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

      1. +-commutative 73.5%

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

      2. mul-1-neg 73.5%

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

      3. unsub-neg 73.5%

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

      4. unpow2 73.5%

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

      5. times-frac 83.2%

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

   4. Simplified 83.2%

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

      1. associate-*r/ 83.2%

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

      2. clear-num 83.3%

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

   6. Applied egg-rr 83.3%

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

   if -4.30000000000000033e57 < c < -3.30000000000000016e-106 or 2.2e-160 < c < 1.1600000000000001e77

   1. Initial program 87.5%

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

   if -3.30000000000000016e-106 < c < 2.2e-160

   1. Initial program 68.4%

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

   2. Taylor expanded in c around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. *-commutative 86.4%

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

      5. unpow2 86.4%

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

      6. times-frac 89.4%

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

   4. Simplified 89.4%

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

      1. associate-*r/ 93.0%

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

      2. sub-div 94.2%

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

   6. Applied egg-rr 94.2%

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

   if 1.1600000000000001e77 < c

   1. Initial program 36.4%

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

      1. *-un-lft-identity 36.4%

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

      2. add-sqr-sqrt 36.4%

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

      3. times-frac 36.4%

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

      4. hypot-def 36.4%

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

      5. hypot-def 58.3%

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

   3. Applied egg-rr 58.3%

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

   4. Taylor expanded in c around inf 81.7%

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

      1. +-commutative 81.7%

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

      2. metadata-eval 81.7%

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

      3. unpow2 81.7%

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

      4. cancel-sign-sub-inv 81.7%

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

      5. *-lft-identity 81.7%

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

      6. associate-/r* 90.0%

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

      7. associate-*r/ 92.1%

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

      8. div-sub 92.1%

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

   6. Simplified 92.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.3 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq -3.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{-160}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.16 \cdot 10^{+77}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 8: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.8e-31)
   (+ (/ b c) (/ -1.0 (/ c (* d (/ a c)))))
   (if (<= c 1.6e-11) (/ (- (* b (/ c d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e-31) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= 1.6e-11) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.8d-31)) then
        tmp = (b / c) + ((-1.0d0) / (c / (d * (a / c))))
    else if (c <= 1.6d-11) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.8e-31) {
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	} else if (c <= 1.6e-11) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.8e-31:
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))))
	elif c <= 1.6e-11:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.8e-31)
		tmp = Float64(Float64(b / c) + Float64(-1.0 / Float64(c / Float64(d * Float64(a / c)))));
	elseif (c <= 1.6e-11)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.8e-31)
		tmp = (b / c) + (-1.0 / (c / (d * (a / c))));
	elseif (c <= 1.6e-11)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.8e-31], N[(N[(b / c), $MachinePrecision] + N[(-1.0 / N[(c / N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.6e-11], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.8e-31

   1. Initial program 53.6%

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

   2. Taylor expanded in c around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. clear-num 83.7%

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

   6. Applied egg-rr 83.7%

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

   if -3.8e-31 < c < 1.59999999999999997e-11

   1. Initial program 75.2%

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

   2. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

      6. times-frac 76.9%

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

   4. Simplified 76.9%

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

      1. associate-*r/ 80.5%

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

      2. sub-div 82.8%

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

   6. Applied egg-rr 82.8%

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

   if 1.59999999999999997e-11 < c

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.5%

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

      4. hypot-def 46.5%

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

      5. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in c around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. metadata-eval 80.9%

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

      3. unpow2 80.9%

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

      4. cancel-sign-sub-inv 80.9%

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

      5. *-lft-identity 80.9%

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

      6. associate-/r* 87.4%

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

      7. associate-*r/ 89.1%

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

      8. div-sub 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{b}{c} + \frac{-1}{\frac{c}{d \cdot \frac{a}{c}}}\\ \mathbf{elif}\;c \leq 1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 9: 67.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-26} \lor \neg \left(c \leq 8.5 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.1e-26) (not (<= c 8.5e-151)))
   (/ (- b (* a (/ d c))) c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.1e-26) || !(c <= 8.5e-151)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.1d-26)) .or. (.not. (c <= 8.5d-151))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.1e-26) || !(c <= 8.5e-151)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.1e-26) or not (c <= 8.5e-151):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.1e-26) || !(c <= 8.5e-151))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.1e-26) || ~((c <= 8.5e-151)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.1e-26], N[Not[LessEqual[c, 8.5e-151]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.0999999999999999e-26 or 8.49999999999999999e-151 < c

   1. Initial program 58.0%

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

      1. *-un-lft-identity 58.0%

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

      2. add-sqr-sqrt 58.0%

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

      3. times-frac 58.0%

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

      4. hypot-def 58.0%

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

      5. hypot-def 72.6%

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

   3. Applied egg-rr 72.6%

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

   4. Taylor expanded in c around inf 71.4%

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

      1. +-commutative 71.4%

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

      2. metadata-eval 71.4%

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

      3. unpow2 71.4%

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

      4. cancel-sign-sub-inv 71.4%

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

      5. *-lft-identity 71.4%

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

      6. associate-/r* 75.2%

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

      7. associate-*r/ 77.2%

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

      8. div-sub 77.2%

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

   6. Simplified 77.2%

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

   if -4.0999999999999999e-26 < c < 8.49999999999999999e-151

   1. Initial program 70.9%

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

   2. Taylor expanded in c around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.1 \cdot 10^{-26} \lor \neg \left(c \leq 8.5 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 10: 67.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.75e-32)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 8.5e-151) (/ (- a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.75e-32) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 8.5e-151) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3.75d-32)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 8.5d-151) then
        tmp = -a / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.75e-32) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 8.5e-151) {
		tmp = -a / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.75e-32:
		tmp = (b - (d * (a / c))) / c
	elif c <= 8.5e-151:
		tmp = -a / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.75e-32)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 8.5e-151)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3.75e-32)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 8.5e-151)
		tmp = -a / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.75e-32], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 8.5e-151], N[((-a) / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.74999999999999977e-32

   1. Initial program 53.6%

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

   2. Taylor expanded in c around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   6. Applied egg-rr 83.7%

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

   if -3.74999999999999977e-32 < c < 8.49999999999999999e-151

   1. Initial program 70.9%

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

   2. Taylor expanded in c around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

   if 8.49999999999999999e-151 < c

   1. Initial program 60.9%

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

      1. *-un-lft-identity 60.9%

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

      2. add-sqr-sqrt 60.9%

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

      3. times-frac 60.9%

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

      4. hypot-def 60.9%

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

      5. hypot-def 75.2%

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

   3. Applied egg-rr 75.2%

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

   4. Taylor expanded in c around inf 69.4%

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

      1. +-commutative 69.4%

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

      2. metadata-eval 69.4%

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

      3. unpow2 69.4%

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

      4. cancel-sign-sub-inv 69.4%

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

      5. *-lft-identity 69.4%

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

      6. associate-/r* 73.7%

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

      7. associate-*r/ 74.8%

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

      8. div-sub 74.8%

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

   6. Simplified 74.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.75 \cdot 10^{-32}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 11: 76.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.1e-29)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 1.42e-11) (/ (- (* c (/ b d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e-29) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 1.42e-11) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-2.1d-29)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 1.42d-11) then
        tmp = ((c * (b / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.1e-29) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 1.42e-11) {
		tmp = ((c * (b / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.1e-29:
		tmp = (b - (d * (a / c))) / c
	elif c <= 1.42e-11:
		tmp = ((c * (b / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.1e-29)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 1.42e-11)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -2.1e-29)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 1.42e-11)
		tmp = ((c * (b / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.1e-29], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.42e-11], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.09999999999999989e-29

   1. Initial program 53.6%

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

   2. Taylor expanded in c around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   6. Applied egg-rr 83.7%

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

   if -2.09999999999999989e-29 < c < 1.42e-11

   1. Initial program 75.2%

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

   2. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

      6. times-frac 76.9%

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

   4. Simplified 76.9%

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

   5. Taylor expanded in c around 0 75.1%

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

      1. neg-mul-1 75.1%

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

      2. unpow2 75.1%

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

      3. times-frac 76.9%

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

      4. *-commutative 76.9%

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

      5. +-commutative 76.9%

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

      6. unsub-neg 76.9%

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

      7. associate-*l/ 77.6%

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

      8. div-sub 79.9%

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

   7. Simplified 79.9%

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

   if 1.42e-11 < c

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.5%

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

      4. hypot-def 46.5%

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

      5. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in c around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. metadata-eval 80.9%

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

      3. unpow2 80.9%

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

      4. cancel-sign-sub-inv 80.9%

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

      5. *-lft-identity 80.9%

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

      6. associate-/r* 87.4%

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

      7. associate-*r/ 89.1%

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

      8. div-sub 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.1 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.42 \cdot 10^{-11}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 12: 77.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.2e-32)
   (/ (- b (* d (/ a c))) c)
   (if (<= c 1.18e-11) (/ (- (* b (/ c d)) a) d) (/ (- b (* a (/ d c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e-32) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 1.18e-11) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-8.2d-32)) then
        tmp = (b - (d * (a / c))) / c
    else if (c <= 1.18d-11) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.2e-32) {
		tmp = (b - (d * (a / c))) / c;
	} else if (c <= 1.18e-11) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8.2e-32:
		tmp = (b - (d * (a / c))) / c
	elif c <= 1.18e-11:
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.2e-32)
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	elseif (c <= 1.18e-11)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -8.2e-32)
		tmp = (b - (d * (a / c))) / c;
	elseif (c <= 1.18e-11)
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.2e-32], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.18e-11], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.1999999999999995e-32

   1. Initial program 53.6%

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

   2. Taylor expanded in c around inf 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. times-frac 82.4%

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

   4. Simplified 82.4%

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

      1. associate-*r/ 83.7%

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

      2. sub-div 83.7%

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

   6. Applied egg-rr 83.7%

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

   if -8.1999999999999995e-32 < c < 1.18e-11

   1. Initial program 75.2%

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

   2. Taylor expanded in c around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. mul-1-neg 75.1%

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

      3. unsub-neg 75.1%

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

      4. *-commutative 75.1%

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

      5. unpow2 75.1%

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

      6. times-frac 76.9%

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

   4. Simplified 76.9%

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

      1. associate-*r/ 80.5%

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

      2. sub-div 82.8%

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

   6. Applied egg-rr 82.8%

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

   if 1.18e-11 < c

   1. Initial program 46.5%

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

      1. *-un-lft-identity 46.5%

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

      2. add-sqr-sqrt 46.5%

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

      3. times-frac 46.5%

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

      4. hypot-def 46.5%

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

      5. hypot-def 65.4%

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

   3. Applied egg-rr 65.4%

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

   4. Taylor expanded in c around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. metadata-eval 80.9%

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

      3. unpow2 80.9%

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

      4. cancel-sign-sub-inv 80.9%

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

      5. *-lft-identity 80.9%

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

      6. associate-/r* 87.4%

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

      7. associate-*r/ 89.1%

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

      8. div-sub 89.1%

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

   6. Simplified 89.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.18 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 13: 60.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.2e-26) (/ b c) (if (<= c 8.2e-151) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e-26) {
		tmp = b / c;
	} else if (c <= 8.2e-151) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.2d-26)) then
        tmp = b / c
    else if (c <= 8.2d-151) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e-26) {
		tmp = b / c;
	} else if (c <= 8.2e-151) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.2e-26:
		tmp = b / c
	elif c <= 8.2e-151:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.2e-26)
		tmp = Float64(b / c);
	elseif (c <= 8.2e-151)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.2e-26)
		tmp = b / c;
	elseif (c <= 8.2e-151)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.2e-26], N[(b / c), $MachinePrecision], If[LessEqual[c, 8.2e-151], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.2e-26 or 8.2000000000000002e-151 < c

   1. Initial program 58.0%

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

   2. Taylor expanded in c around inf 64.0%

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

   if -1.2e-26 < c < 8.2000000000000002e-151

   1. Initial program 70.9%

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

   2. Taylor expanded in c around 0 65.5%

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

      1. associate-*r/ 65.5%

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

      2. neg-mul-1 65.5%

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

   4. Simplified 65.5%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 14: 9.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

   1. *-un-lft-identity 63.0%

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

   2. add-sqr-sqrt 63.0%

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

   3. times-frac 63.0%

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

   4. hypot-def 63.0%

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

   5. hypot-def 77.4%

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

3. Applied egg-rr 77.4%

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

4. Taylor expanded in c around inf 37.4%

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

   1. mul-1-neg 37.4%

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

   2. associate-*l/ 37.6%

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

   3. sub-neg 37.6%

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

   4. *-commutative 37.6%

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

6. Simplified 37.6%

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

7. Taylor expanded in d around -inf 9.9%

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

8. Final simplification 9.9%

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

Alternative 15: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in c around inf 45.5%

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

3. Final simplification 45.5%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023279 
(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))))