Complex division, imag part

Percentage Accurate: 62.0% → 84.4%

Time: 7.3s

Alternatives: 9

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% 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: 84.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 76.5%

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

      1. *-un-lft-identity 76.5%

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

      2. add-sqr-sqrt 76.5%

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

      3. times-frac 76.5%

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

      4. hypot-def 76.6%

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

      5. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in b around 0 1.4%

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

      1. associate-*r/ 1.4%

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

      2. mul-1-neg 1.4%

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

      3. distribute-rgt-neg-out 1.4%

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

      4. associate-/l* 3.7%

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

      5. +-commutative 3.7%

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

      6. unpow2 3.7%

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

      7. fma-def 3.7%

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

      8. unpow2 3.7%

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

   4. Simplified 3.7%

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

   5. Taylor expanded in d around 0 54.1%

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

      1. neg-mul-1 54.1%

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

      2. +-commutative 54.1%

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

      3. unsub-neg 54.1%

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

      4. mul-1-neg 54.1%

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

      5. unpow2 54.1%

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

      6. associate-*l/ 58.5%

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

      7. *-commutative 58.5%

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

      8. distribute-rgt-neg-in 58.5%

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

   7. Simplified 58.5%

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

4. Final simplification 88.9%

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

Alternative 2: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(a - \frac{b \cdot c}{d}\right)\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (* (/ a c) (/ d c)))))
   (if (<= c -2.7e+97)
     t_1
     (if (<= c -2e-14)
       t_0
       (if (<= c 1.15e-122)
         (* (/ -1.0 d) (- a (/ (* b c) d)))
         (if (<= c 1.15e+99) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -2.7e+97) {
		tmp = t_1;
	} else if (c <= -2e-14) {
		tmp = t_0;
	} else if (c <= 1.15e-122) {
		tmp = (-1.0 / d) * (a - ((b * c) / d));
	} else if (c <= 1.15e+99) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a / c) * (d / c))
    if (c <= (-2.7d+97)) then
        tmp = t_1
    else if (c <= (-2d-14)) then
        tmp = t_0
    else if (c <= 1.15d-122) then
        tmp = ((-1.0d0) / d) * (a - ((b * c) / d))
    else if (c <= 1.15d+99) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -2.7e+97) {
		tmp = t_1;
	} else if (c <= -2e-14) {
		tmp = t_0;
	} else if (c <= 1.15e-122) {
		tmp = (-1.0 / d) * (a - ((b * c) / d));
	} else if (c <= 1.15e+99) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a / c) * (d / c))
	tmp = 0
	if c <= -2.7e+97:
		tmp = t_1
	elif c <= -2e-14:
		tmp = t_0
	elif c <= 1.15e-122:
		tmp = (-1.0 / d) * (a - ((b * c) / d))
	elif c <= 1.15e+99:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -2.7e+97)
		tmp = t_1;
	elseif (c <= -2e-14)
		tmp = t_0;
	elseif (c <= 1.15e-122)
		tmp = Float64(Float64(-1.0 / d) * Float64(a - Float64(Float64(b * c) / d)));
	elseif (c <= 1.15e+99)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a / c) * (d / c));
	tmp = 0.0;
	if (c <= -2.7e+97)
		tmp = t_1;
	elseif (c <= -2e-14)
		tmp = t_0;
	elseif (c <= 1.15e-122)
		tmp = (-1.0 / d) * (a - ((b * c) / d));
	elseif (c <= 1.15e+99)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+97], t$95$1, If[LessEqual[c, -2e-14], t$95$0, If[LessEqual[c, 1.15e-122], N[(N[(-1.0 / d), $MachinePrecision] * N[(a - N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.15e+99], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.69999999999999993e97 or 1.1500000000000001e99 < c

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. add-sqr-sqrt 43.1%

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

      3. times-frac 43.1%

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

      4. hypot-def 43.1%

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

      5. hypot-def 70.2%

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

   3. Applied egg-rr 70.2%

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

   4. Taylor expanded in c around inf 78.4%

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

      1. fma-def 78.4%

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

      2. unpow2 78.4%

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

      3. times-frac 92.3%

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

      4. fma-def 92.3%

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

      5. neg-mul-1 92.3%

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

      6. +-commutative 92.3%

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

      7. sub-neg 92.3%

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

   6. Simplified 92.3%

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

   if -2.69999999999999993e97 < c < -2e-14 or 1.15000000000000003e-122 < c < 1.1500000000000001e99

   1. Initial program 71.0%

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

   if -2e-14 < c < 1.15000000000000003e-122

   1. Initial program 74.4%

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

      1. *-un-lft-identity 74.4%

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

      2. add-sqr-sqrt 74.4%

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

      3. times-frac 74.4%

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

      4. hypot-def 74.5%

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

      5. hypot-def 88.1%

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

   3. Applied egg-rr 88.1%

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

   4. Taylor expanded in d around -inf 53.2%

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

      1. associate-*r/ 53.2%

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

      2. mul-1-neg 53.2%

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

      3. *-commutative 53.2%

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

   6. Simplified 53.2%

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

   7. Taylor expanded in d around -inf 96.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+97}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-122}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(a - \frac{b \cdot c}{d}\right)\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 3: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1250:\\ \;\;\;\;\frac{a}{\left(-d\right) - c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-9} \lor \neg \left(c \leq 1.3 \cdot 10^{+57}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(a - \frac{b \cdot c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ b c) (* (/ a c) (/ d c)))))
   (if (<= c -3.3e+91)
     t_0
     (if (<= c -1250.0)
       (/ a (- (- d) (* c (/ c d))))
       (if (or (<= c -1.7e-9) (not (<= c 1.3e+57)))
         t_0
         (* (/ -1.0 d) (- a (/ (* b c) d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -3.3e+91) {
		tmp = t_0;
	} else if (c <= -1250.0) {
		tmp = a / (-d - (c * (c / d)));
	} else if ((c <= -1.7e-9) || !(c <= 1.3e+57)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * (a - ((b * 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) :: t_0
    real(8) :: tmp
    t_0 = (b / c) - ((a / c) * (d / c))
    if (c <= (-3.3d+91)) then
        tmp = t_0
    else if (c <= (-1250.0d0)) then
        tmp = a / (-d - (c * (c / d)))
    else if ((c <= (-1.7d-9)) .or. (.not. (c <= 1.3d+57))) then
        tmp = t_0
    else
        tmp = ((-1.0d0) / d) * (a - ((b * c) / d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -3.3e+91) {
		tmp = t_0;
	} else if (c <= -1250.0) {
		tmp = a / (-d - (c * (c / d)));
	} else if ((c <= -1.7e-9) || !(c <= 1.3e+57)) {
		tmp = t_0;
	} else {
		tmp = (-1.0 / d) * (a - ((b * c) / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / c) - ((a / c) * (d / c))
	tmp = 0
	if c <= -3.3e+91:
		tmp = t_0
	elif c <= -1250.0:
		tmp = a / (-d - (c * (c / d)))
	elif (c <= -1.7e-9) or not (c <= 1.3e+57):
		tmp = t_0
	else:
		tmp = (-1.0 / d) * (a - ((b * c) / d))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -3.3e+91)
		tmp = t_0;
	elseif (c <= -1250.0)
		tmp = Float64(a / Float64(Float64(-d) - Float64(c * Float64(c / d))));
	elseif ((c <= -1.7e-9) || !(c <= 1.3e+57))
		tmp = t_0;
	else
		tmp = Float64(Float64(-1.0 / d) * Float64(a - Float64(Float64(b * c) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / c) - ((a / c) * (d / c));
	tmp = 0.0;
	if (c <= -3.3e+91)
		tmp = t_0;
	elseif (c <= -1250.0)
		tmp = a / (-d - (c * (c / d)));
	elseif ((c <= -1.7e-9) || ~((c <= 1.3e+57)))
		tmp = t_0;
	else
		tmp = (-1.0 / d) * (a - ((b * c) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.3e+91], t$95$0, If[LessEqual[c, -1250.0], N[(a / N[((-d) - N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[c, -1.7e-9], N[Not[LessEqual[c, 1.3e+57]], $MachinePrecision]], t$95$0, N[(N[(-1.0 / d), $MachinePrecision] * N[(a - N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.30000000000000017e91 or -1250 < c < -1.6999999999999999e-9 or 1.3e57 < c

   1. Initial program 48.9%

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

      1. *-un-lft-identity 48.9%

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

      2. add-sqr-sqrt 48.9%

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

      3. times-frac 48.9%

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

      4. hypot-def 48.9%

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

      5. hypot-def 73.3%

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

   3. Applied egg-rr 73.3%

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

   4. Taylor expanded in c around inf 77.2%

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

      1. fma-def 77.2%

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

      2. unpow2 77.2%

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

      3. times-frac 87.8%

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

      4. fma-def 87.8%

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

      5. neg-mul-1 87.8%

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

      6. +-commutative 87.8%

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

      7. sub-neg 87.8%

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

   6. Simplified 87.8%

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

   if -3.30000000000000017e91 < c < -1250

   1. Initial program 62.0%

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

   2. Taylor expanded in b around 0 35.1%

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

      1. associate-*r/ 35.1%

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

      2. mul-1-neg 35.1%

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

      3. distribute-rgt-neg-out 35.1%

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

      4. associate-/l* 40.4%

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

      5. +-commutative 40.4%

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

      6. unpow2 40.4%

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

      7. fma-def 40.4%

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

      8. unpow2 40.4%

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

   4. Simplified 40.4%

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

   5. Taylor expanded in d around 0 63.4%

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

      1. neg-mul-1 63.4%

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

      2. +-commutative 63.4%

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

      3. unsub-neg 63.4%

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

      4. mul-1-neg 63.4%

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

      5. unpow2 63.4%

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

      6. associate-*l/ 63.5%

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

      7. *-commutative 63.5%

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

      8. distribute-rgt-neg-in 63.5%

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

   7. Simplified 63.5%

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

   if -1.6999999999999999e-9 < c < 1.3e57

   1. Initial program 74.1%

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

      1. *-un-lft-identity 74.1%

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

      2. add-sqr-sqrt 74.1%

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

      3. times-frac 74.1%

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

      4. hypot-def 74.2%

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

      5. hypot-def 85.6%

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

   3. Applied egg-rr 85.6%

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

   4. Taylor expanded in d around -inf 51.0%

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

      1. associate-*r/ 51.0%

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

      2. mul-1-neg 51.0%

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

      3. *-commutative 51.0%

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

   6. Simplified 51.0%

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

   7. Taylor expanded in d around -inf 86.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -1250:\\ \;\;\;\;\frac{a}{\left(-d\right) - c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq -1.7 \cdot 10^{-9} \lor \neg \left(c \leq 1.3 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{d} \cdot \left(a - \frac{b \cdot c}{d}\right)\\ \end{array} \]

Alternative 4: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+87} \lor \neg \left(c \leq 3.2 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.8e+87) (not (<= c 3.2e+58)))
   (- (/ b c) (* (/ a c) (/ d c)))
   (/ (- (/ c (/ d b)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.8e+87) || !(c <= 3.2e+58)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = ((c / (d / b)) - 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 <= (-1.8d+87)) .or. (.not. (c <= 3.2d+58))) then
        tmp = (b / c) - ((a / c) * (d / c))
    else
        tmp = ((c / (d / b)) - 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 <= -1.8e+87) || !(c <= 3.2e+58)) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.8e+87) or not (c <= 3.2e+58):
		tmp = (b / c) - ((a / c) * (d / c))
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.8e+87) || !(c <= 3.2e+58))
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.8e+87) || ~((c <= 3.2e+58)))
		tmp = (b / c) - ((a / c) * (d / c));
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.8e+87], N[Not[LessEqual[c, 3.2e+58]], $MachinePrecision]], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.79999999999999997e87 or 3.20000000000000015e58 < c

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. add-sqr-sqrt 48.4%

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

      3. times-frac 48.4%

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

      4. hypot-def 48.4%

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

      5. hypot-def 73.0%

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

   3. Applied egg-rr 73.0%

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

   4. Taylor expanded in c around inf 76.0%

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

      1. fma-def 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 86.7%

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

      4. fma-def 86.7%

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

      5. neg-mul-1 86.7%

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

      6. +-commutative 86.7%

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

      7. sub-neg 86.7%

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

   6. Simplified 86.7%

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

   if -1.79999999999999997e87 < c < 3.20000000000000015e58

   1. Initial program 72.6%

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

      1. *-un-lft-identity 72.6%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

      4. hypot-def 72.7%

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

      5. hypot-def 83.2%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in d around -inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

      3. *-commutative 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in d around -inf 81.4%

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

   8. Taylor expanded in d around 0 74.2%

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

      1. associate-*r/ 74.2%

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

      2. associate-*l/ 74.0%

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

      3. +-commutative 74.0%

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

      4. *-commutative 74.0%

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

      5. unpow2 74.0%

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

      6. associate-*l/ 74.2%

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

      7. associate-*r/ 74.2%

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

      8. mul-1-neg 74.2%

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

      9. sub-neg 74.2%

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

      10. associate-/r* 80.8%

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

      11. div-sub 81.6%

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

      12. associate-/l* 80.5%

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

   10. Simplified 80.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+87} \lor \neg \left(c \leq 3.2 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]

Alternative 5: 72.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+24} \lor \neg \left(d \leq 2.3 \cdot 10^{+49}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8e+24) (not (<= d 2.3e+49)))
   (- (/ a d))
   (/ (- b (* a (/ d c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8e+24) || !(d <= 2.3e+49)) {
		tmp = -(a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8d+24)) .or. (.not. (d <= 2.3d+49))) then
        tmp = -(a / d)
    else
        tmp = (b - (a * (d / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8e+24) || !(d <= 2.3e+49)) {
		tmp = -(a / d);
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8e+24) or not (d <= 2.3e+49):
		tmp = -(a / d)
	else:
		tmp = (b - (a * (d / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8e+24) || !(d <= 2.3e+49))
		tmp = Float64(-Float64(a / d));
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8e+24) || ~((d <= 2.3e+49)))
		tmp = -(a / d);
	else
		tmp = (b - (a * (d / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8e+24], N[Not[LessEqual[d, 2.3e+49]], $MachinePrecision]], (-N[(a / d), $MachinePrecision]), N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.9999999999999999e24 or 2.30000000000000002e49 < d

   1. Initial program 51.1%

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

   2. Taylor expanded in c around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. neg-mul-1 75.7%

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

   4. Simplified 75.7%

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

   if -7.9999999999999999e24 < d < 2.30000000000000002e49

   1. Initial program 73.6%

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

      1. *-un-lft-identity 73.6%

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

      2. add-sqr-sqrt 73.6%

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

      3. times-frac 73.6%

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

      4. hypot-def 73.7%

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

      5. hypot-def 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in c around inf 69.1%

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

      1. fma-def 69.1%

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

      2. unpow2 69.1%

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

      3. times-frac 73.3%

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

      4. fma-def 73.3%

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

      5. neg-mul-1 73.3%

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

      6. +-commutative 73.3%

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

      7. sub-neg 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in b around 0 69.1%

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

      1. +-commutative 69.1%

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

      2. mul-1-neg 69.1%

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

      3. *-commutative 69.1%

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

      4. unpow2 69.1%

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

      5. times-frac 73.3%

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

      6. distribute-lft-neg-in 73.3%

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

      7. cancel-sign-sub-inv 73.3%

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

      8. *-commutative 73.3%

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

      9. associate-*l/ 73.3%

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

      10. div-sub 75.5%

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

   9. Simplified 75.5%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8 \cdot 10^{+24} \lor \neg \left(d \leq 2.3 \cdot 10^{+49}\right):\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]

Alternative 6: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+88} \lor \neg \left(c \leq 9.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.6e+88) (not (<= c 9.5e+57)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ c (/ d b)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.6e+88) || !(c <= 9.5e+57)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-3.6d+88)) .or. (.not. (c <= 9.5d+57))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((c / (d / b)) - a) / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.6e+88) || !(c <= 9.5e+57)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((c / (d / b)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.6e+88) or not (c <= 9.5e+57):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((c / (d / b)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.6e+88) || !(c <= 9.5e+57))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(c / Float64(d / b)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.6e+88) || ~((c <= 9.5e+57)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((c / (d / b)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.6e+88], N[Not[LessEqual[c, 9.5e+57]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.6000000000000002e88 or 9.4999999999999997e57 < c

   1. Initial program 48.4%

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

      1. *-un-lft-identity 48.4%

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

      2. add-sqr-sqrt 48.4%

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

      3. times-frac 48.4%

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

      4. hypot-def 48.4%

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

      5. hypot-def 73.0%

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

   3. Applied egg-rr 73.0%

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

   4. Taylor expanded in c around inf 76.0%

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

      1. fma-def 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 86.7%

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

      4. fma-def 86.7%

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

      5. neg-mul-1 86.7%

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

      6. +-commutative 86.7%

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

      7. sub-neg 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in b around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. mul-1-neg 76.0%

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

      3. *-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. times-frac 86.7%

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

      6. distribute-lft-neg-in 86.7%

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

      7. cancel-sign-sub-inv 86.7%

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

      8. *-commutative 86.7%

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

      9. associate-*l/ 86.7%

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

      10. div-sub 86.7%

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

   9. Simplified 86.7%

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

   if -3.6000000000000002e88 < c < 9.4999999999999997e57

   1. Initial program 72.6%

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

      1. *-un-lft-identity 72.6%

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

      2. add-sqr-sqrt 72.6%

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

      3. times-frac 72.7%

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

      4. hypot-def 72.7%

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

      5. hypot-def 83.2%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in d around -inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. mul-1-neg 47.9%

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

      3. *-commutative 47.9%

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

   6. Simplified 47.9%

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

   7. Taylor expanded in d around -inf 81.4%

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

   8. Taylor expanded in d around 0 74.2%

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

      1. associate-*r/ 74.2%

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

      2. associate-*l/ 74.0%

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

      3. +-commutative 74.0%

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

      4. *-commutative 74.0%

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

      5. unpow2 74.0%

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

      6. associate-*l/ 74.2%

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

      7. associate-*r/ 74.2%

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

      8. mul-1-neg 74.2%

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

      9. sub-neg 74.2%

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

      10. associate-/r* 80.8%

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

      11. div-sub 81.6%

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

      12. associate-/l* 80.5%

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

   10. Simplified 80.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+88} \lor \neg \left(c \leq 9.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{\frac{d}{b}} - a}{d}\\ \end{array} \]

Alternative 7: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+56}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.5e+91) (/ b c) (if (<= c 2.7e+56) (- (/ a d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.5e+91) {
		tmp = b / c;
	} else if (c <= 2.7e+56) {
		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 <= (-3.5d+91)) then
        tmp = b / c
    else if (c <= 2.7d+56) 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 <= -3.5e+91) {
		tmp = b / c;
	} else if (c <= 2.7e+56) {
		tmp = -(a / d);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.5e+91:
		tmp = b / c
	elif c <= 2.7e+56:
		tmp = -(a / d)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.5e+91)
		tmp = Float64(b / c);
	elseif (c <= 2.7e+56)
		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 <= -3.5e+91)
		tmp = b / c;
	elseif (c <= 2.7e+56)
		tmp = -(a / d);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.5e+91], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.7e+56], (-N[(a / d), $MachinePrecision]), N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.50000000000000001e91 or 2.7000000000000001e56 < c

   1. Initial program 47.3%

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

   2. Taylor expanded in c around inf 75.7%

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

   if -3.50000000000000001e91 < c < 2.7000000000000001e56

   1. Initial program 73.0%

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

   2. Taylor expanded in c around 0 67.5%

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

      1. associate-*r/ 67.5%

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

      2. neg-mul-1 67.5%

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

   4. Simplified 67.5%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.5 \cdot 10^{+91}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+56}:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 8: 47.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+170} \lor \neg \left(d \leq 4.6 \cdot 10^{+175}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.7e+170) (not (<= d 4.6e+175))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.7e+170) || !(d <= 4.6e+175)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-2.7d+170)) .or. (.not. (d <= 4.6d+175))) then
        tmp = a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.7e+170) || !(d <= 4.6e+175)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.7e+170) or not (d <= 4.6e+175):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.7e+170) || !(d <= 4.6e+175))
		tmp = Float64(a / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -2.7e+170) || ~((d <= 4.6e+175)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.7e+170], N[Not[LessEqual[d, 4.6e+175]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.7000000000000002e170 or 4.5999999999999999e175 < d

   1. Initial program 35.4%

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

      1. *-un-lft-identity 35.4%

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

      2. add-sqr-sqrt 35.4%

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

      3. times-frac 35.4%

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

      4. hypot-def 35.4%

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

      5. hypot-def 58.0%

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

   3. Applied egg-rr 58.0%

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

   4. Taylor expanded in d around -inf 65.5%

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

      1. associate-*r/ 65.5%

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

      2. mul-1-neg 65.5%

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

      3. *-commutative 65.5%

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

   6. Simplified 65.5%

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

   7. Taylor expanded in c around 0 34.7%

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

   if -2.7000000000000002e170 < d < 4.5999999999999999e175

   1. Initial program 71.5%

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

   2. Taylor expanded in c around inf 47.9%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.7 \cdot 10^{+170} \lor \neg \left(d \leq 4.6 \cdot 10^{+175}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9: 11.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 63.4%

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

   1. *-un-lft-identity 63.4%

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

   2. add-sqr-sqrt 63.3%

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

   3. times-frac 63.4%

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

   4. hypot-def 63.4%

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

   5. hypot-def 79.3%

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

3. Applied egg-rr 79.3%

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

4. Taylor expanded in d around -inf 35.5%

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

   1. associate-*r/ 35.5%

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

   2. mul-1-neg 35.5%

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

   3. *-commutative 35.5%

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

6. Simplified 35.5%

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

7. Taylor expanded in c around 0 12.2%

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

8. Final simplification 12.2%

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

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 2023297 
(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))))