Complex division, imag part

Percentage Accurate: 62.1% → 89.3%

Time: 8.5s

Alternatives: 10

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: 62.1% 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: 89.3% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 10^{+281}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e+281)))
     (fma (/ c (hypot c d)) (/ b (hypot c d)) (/ (- a) d))
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e+281)) {
		tmp = fma((c / hypot(c, d)), (b / hypot(c, d)), (-a / d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e+281))
		tmp = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(Float64(-a) / d));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e+281]], $MachinePrecision]], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision], 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]]]]

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 or 1e281 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 17.4%

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

      1. div-sub 7.3%

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

      2. *-commutative 7.3%

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

      3. add-sqr-sqrt 7.3%

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

      4. times-frac 15.9%

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

      5. fma-neg 15.9%

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

      6. hypot-def 15.9%

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

      7. hypot-def 45.1%

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

      8. associate-/l* 56.7%

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

      9. add-sqr-sqrt 56.7%

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

      10. pow2 56.7%

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

      11. hypot-def 56.7%

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

   3. Applied egg-rr 56.7%

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

   4. Taylor expanded in c around 0 80.0%

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

   if -inf.0 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e281

   1. Initial program 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. add-sqr-sqrt 83.7%

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

      3. times-frac 83.7%

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

      4. hypot-def 83.7%

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

      5. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq -\infty \lor \neg \left(\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+281}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, \frac{-a}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]

Alternative 2: 84.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot c - a \cdot d\\ \mathbf{if}\;\frac{t_0}{c \cdot c + d \cdot d} \leq 10^{+281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (* b c) (* a d))))
   (if (<= (/ t_0 (+ (* c c) (* d d))) 1e+281)
     (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
     (* (/ 1.0 d) (- (/ c (/ d b)) a)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+281) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) - (a * d);
	double tmp;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+281) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) - (a * d)
	tmp = 0
	if (t_0 / ((c * c) + (d * d))) <= 1e+281:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (1.0 / d) * ((c / (d / b)) - a)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) - Float64(a * d))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(c * c) + Float64(d * d))) <= 1e+281)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c / Float64(d / b)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) - (a * d);
	tmp = 0.0;
	if ((t_0 / ((c * c) + (d * d))) <= 1e+281)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+281], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 1e281

   1. Initial program 80.0%

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

      1. *-un-lft-identity 80.0%

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

      2. add-sqr-sqrt 80.0%

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

      3. times-frac 80.0%

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

      4. hypot-def 80.0%

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

      5. hypot-def 95.9%

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

   3. Applied egg-rr 95.9%

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

   if 1e281 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 16.6%

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

   2. Taylor expanded in c around 0 17.4%

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

      1. unpow2 17.4%

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

   4. Simplified 17.4%

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

      1. clear-num 17.4%

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

      2. inv-pow 17.4%

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

      3. *-commutative 17.4%

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

      4. *-commutative 17.4%

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

   6. Applied egg-rr 17.4%

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

      1. unpow-1 17.4%

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

      2. associate-/l* 22.2%

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

      3. associate-/r/ 22.2%

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

      4. *-commutative 22.2%

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

      5. div-sub 22.2%

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

      6. associate-/l* 26.3%

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

      7. associate-/l* 60.3%

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

      8. *-inverses 60.3%

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

   8. Simplified 60.3%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 10^{+281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{b \cdot c - a \cdot d}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \]

Alternative 3: 82.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot c + d \cdot d\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{t_0}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* c c) (* d d))))
   (if (<= c -5.5e+42)
     (- (/ b c) (* (/ a c) (/ d c)))
     (if (<= c -6.5e-97)
       (/ (- (* b c) (* a d)) t_0)
       (if (<= c 8.5e-137)
         (* (/ 1.0 d) (- (/ c (/ d b)) a))
         (if (<= c 1.5e+66)
           (/ (fma (- d) a (* b c)) t_0)
           (/ (- b (* d (/ a c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (c * c) + (d * d);
	double tmp;
	if (c <= -5.5e+42) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else if (c <= -6.5e-97) {
		tmp = ((b * c) - (a * d)) / t_0;
	} else if (c <= 8.5e-137) {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	} else if (c <= 1.5e+66) {
		tmp = fma(-d, a, (b * c)) / t_0;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(c * c) + Float64(d * d))
	tmp = 0.0
	if (c <= -5.5e+42)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	elseif (c <= -6.5e-97)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / t_0);
	elseif (c <= 8.5e-137)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c / Float64(d / b)) - a));
	elseif (c <= 1.5e+66)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / t_0);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.5e+42], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -6.5e-97], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[c, 8.5e-137], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.5e+66], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -5.50000000000000001e42

   1. Initial program 59.6%

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

   2. Taylor expanded in c around inf 90.3%

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

      1. +-commutative 90.3%

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

      2. mul-1-neg 90.3%

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

      3. unsub-neg 90.3%

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

      4. unpow2 90.3%

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

      5. times-frac 88.6%

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

   4. Simplified 88.6%

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

   if -5.50000000000000001e42 < c < -6.5000000000000004e-97

   1. Initial program 79.3%

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

   if -6.5000000000000004e-97 < c < 8.5000000000000001e-137

   1. Initial program 67.9%

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

   2. Taylor expanded in c around 0 65.8%

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

      1. unpow2 65.8%

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

   4. Simplified 65.8%

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

      1. clear-num 65.8%

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

      2. inv-pow 65.8%

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

      3. *-commutative 65.8%

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

      4. *-commutative 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. unpow-1 65.8%

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

      2. associate-/l* 79.2%

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

      3. associate-/r/ 79.4%

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

      4. *-commutative 79.4%

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

      5. div-sub 79.4%

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

      6. associate-/l* 79.3%

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

      7. associate-/l* 92.7%

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

      8. *-inverses 92.7%

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

   8. Simplified 92.7%

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

   if 8.5000000000000001e-137 < c < 1.50000000000000001e66

   1. Initial program 89.6%

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

      1. sub-neg 89.6%

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

      2. +-commutative 89.6%

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

      3. *-commutative 89.6%

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

      4. distribute-lft-neg-in 89.6%

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

      5. fma-def 89.7%

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

   3. Applied egg-rr 89.7%

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

   if 1.50000000000000001e66 < c

   1. Initial program 52.1%

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

   2. Taylor expanded in c around inf 81.5%

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

      1. +-commutative 81.5%

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

      2. mul-1-neg 81.5%

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

      3. unsub-neg 81.5%

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

      4. unpow2 81.5%

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

      5. times-frac 82.0%

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

   4. Simplified 82.0%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 85.5%

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

   6. Applied egg-rr 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+66}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 4: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -4.8e+42)
     (- (/ b c) (* (/ a c) (/ d c)))
     (if (<= c -2.8e-98)
       t_0
       (if (<= c 1.05e-136)
         (* (/ 1.0 d) (- (/ c (/ d b)) a))
         (if (<= c 5.8e+69) t_0 (/ (- b (* d (/ a c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.8e+42) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else if (c <= -2.8e-98) {
		tmp = t_0;
	} else if (c <= 1.05e-136) {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	} else if (c <= 5.8e+69) {
		tmp = t_0;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    if (c <= (-4.8d+42)) then
        tmp = (b / c) - ((a / c) * (d / c))
    else if (c <= (-2.8d-98)) then
        tmp = t_0
    else if (c <= 1.05d-136) then
        tmp = (1.0d0 / d) * ((c / (d / b)) - a)
    else if (c <= 5.8d+69) then
        tmp = t_0
    else
        tmp = (b - (d * (a / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -4.8e+42) {
		tmp = (b / c) - ((a / c) * (d / c));
	} else if (c <= -2.8e-98) {
		tmp = t_0;
	} else if (c <= 1.05e-136) {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	} else if (c <= 5.8e+69) {
		tmp = t_0;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -4.8e+42:
		tmp = (b / c) - ((a / c) * (d / c))
	elif c <= -2.8e-98:
		tmp = t_0
	elif c <= 1.05e-136:
		tmp = (1.0 / d) * ((c / (d / b)) - a)
	elif c <= 5.8e+69:
		tmp = t_0
	else:
		tmp = (b - (d * (a / c))) / c
	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)))
	tmp = 0.0
	if (c <= -4.8e+42)
		tmp = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)));
	elseif (c <= -2.8e-98)
		tmp = t_0;
	elseif (c <= 1.05e-136)
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c / Float64(d / b)) - a));
	elseif (c <= 5.8e+69)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -4.8e+42)
		tmp = (b / c) - ((a / c) * (d / c));
	elseif (c <= -2.8e-98)
		tmp = t_0;
	elseif (c <= 1.05e-136)
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	elseif (c <= 5.8e+69)
		tmp = t_0;
	else
		tmp = (b - (d * (a / c))) / c;
	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]}, If[LessEqual[c, -4.8e+42], N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -2.8e-98], t$95$0, If[LessEqual[c, 1.05e-136], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.8e+69], t$95$0, N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -4.7999999999999997e42

   1. Initial program 59.6%

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

   2. Taylor expanded in c around inf 90.3%

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

      1. +-commutative 90.3%

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

      2. mul-1-neg 90.3%

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

      3. unsub-neg 90.3%

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

      4. unpow2 90.3%

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

      5. times-frac 88.6%

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

   4. Simplified 88.6%

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

   if -4.7999999999999997e42 < c < -2.7999999999999999e-98 or 1.0499999999999999e-136 < c < 5.7999999999999997e69

   1. Initial program 85.7%

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

   if -2.7999999999999999e-98 < c < 1.0499999999999999e-136

   1. Initial program 67.9%

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

   2. Taylor expanded in c around 0 65.8%

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

      1. unpow2 65.8%

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

   4. Simplified 65.8%

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

      1. clear-num 65.8%

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

      2. inv-pow 65.8%

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

      3. *-commutative 65.8%

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

      4. *-commutative 65.8%

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

   6. Applied egg-rr 65.8%

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

      1. unpow-1 65.8%

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

      2. associate-/l* 79.2%

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

      3. associate-/r/ 79.4%

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

      4. *-commutative 79.4%

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

      5. div-sub 79.4%

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

      6. associate-/l* 79.3%

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

      7. associate-/l* 92.7%

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

      8. *-inverses 92.7%

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

   8. Simplified 92.7%

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

   if 5.7999999999999997e69 < c

   1. Initial program 52.1%

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

   2. Taylor expanded in c around inf 81.5%

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

      1. +-commutative 81.5%

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

      2. mul-1-neg 81.5%

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

      3. unsub-neg 81.5%

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

      4. unpow2 81.5%

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

      5. times-frac 82.0%

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

   4. Simplified 82.0%

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

      1. associate-*r/ 85.5%

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

      2. sub-div 85.5%

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

   6. Applied egg-rr 85.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-98}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -10500 \lor \neg \left(c \leq 9.7 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -10500.0) (not (<= c 9.7e-26)))
   (/ (- b (* d (/ a c))) c)
   (- (* (/ c d) (/ b d)) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -10500.0) || !(c <= 9.7e-26)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((c / d) * (b / d)) - (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 <= (-10500.0d0)) .or. (.not. (c <= 9.7d-26))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = ((c / d) * (b / d)) - (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 <= -10500.0) || !(c <= 9.7e-26)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = ((c / d) * (b / d)) - (a / d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -10500.0) or not (c <= 9.7e-26):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = ((c / d) * (b / d)) - (a / d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -10500.0) || !(c <= 9.7e-26))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(Float64(c / d) * Float64(b / d)) - Float64(a / d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -10500.0) || ~((c <= 9.7e-26)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = ((c / d) * (b / d)) - (a / d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -10500.0], N[Not[LessEqual[c, 9.7e-26]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision]), $MachinePrecision] - N[(a / d), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -10500 or 9.7000000000000001e-26 < c

   1. Initial program 61.0%

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

   2. Taylor expanded in c around inf 81.4%

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

      1. +-commutative 81.4%

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

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. unpow2 81.4%

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

      5. times-frac 80.9%

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

   4. Simplified 80.9%

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

      1. associate-*r/ 82.4%

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

      2. sub-div 82.4%

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

   6. Applied egg-rr 82.4%

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

   if -10500 < c < 9.7000000000000001e-26

   1. Initial program 73.2%

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

   2. Taylor expanded in c around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. mul-1-neg 74.2%

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

      3. unsub-neg 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -10500 \lor \neg \left(c \leq 9.7 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{d} \cdot \frac{b}{d} - \frac{a}{d}\\ \end{array} \]

Alternative 6: 77.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6200000000 \lor \neg \left(c \leq 2.75 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6200000000.0) (not (<= c 2.75e-21)))
   (/ (- b (* d (/ a c))) c)
   (* (/ 1.0 d) (- (/ c (/ d b)) a))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6200000000.0) || !(c <= 2.75e-21)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-6200000000.0d0)) .or. (.not. (c <= 2.75d-21))) then
        tmp = (b - (d * (a / c))) / c
    else
        tmp = (1.0d0 / d) * ((c / (d / b)) - a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6200000000.0) || !(c <= 2.75e-21)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6200000000.0) or not (c <= 2.75e-21):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = (1.0 / d) * ((c / (d / b)) - a)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6200000000.0) || !(c <= 2.75e-21))
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	else
		tmp = Float64(Float64(1.0 / d) * Float64(Float64(c / Float64(d / b)) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -6200000000.0) || ~((c <= 2.75e-21)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = (1.0 / d) * ((c / (d / b)) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -6200000000.0], N[Not[LessEqual[c, 2.75e-21]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(N[(c / N[(d / b), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -6.2e9 or 2.74999999999999989e-21 < c

   1. Initial program 61.0%

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

   2. Taylor expanded in c around inf 81.4%

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

      1. +-commutative 81.4%

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

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. unpow2 81.4%

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

      5. times-frac 80.9%

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

   4. Simplified 80.9%

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

      1. associate-*r/ 82.4%

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

      2. sub-div 82.4%

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

   6. Applied egg-rr 82.4%

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

   if -6.2e9 < c < 2.74999999999999989e-21

   1. Initial program 73.2%

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

   2. Taylor expanded in c around 0 59.8%

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

      1. unpow2 59.8%

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

   4. Simplified 59.8%

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

      1. clear-num 59.7%

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

      2. inv-pow 59.7%

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

      3. *-commutative 59.7%

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

      4. *-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

      1. unpow-1 59.7%

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

      2. associate-/l* 72.6%

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

      3. associate-/r/ 72.7%

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

      4. *-commutative 72.7%

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

      5. div-sub 72.7%

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

      6. associate-/l* 72.6%

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

      7. associate-/l* 82.7%

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

      8. *-inverses 82.7%

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

   8. Simplified 82.7%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6200000000 \lor \neg \left(c \leq 2.75 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(\frac{c}{\frac{d}{b}} - a\right)\\ \end{array} \]

Alternative 7: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-14} \lor \neg \left(c \leq 3.1 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -8.5e-14) (not (<= c 3.1e-68)))
   (/ (- b (* d (/ a c))) c)
   (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -8.5e-14) || !(c <= 3.1e-68)) {
		tmp = (b - (d * (a / 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 <= (-8.5d-14)) .or. (.not. (c <= 3.1d-68))) then
        tmp = (b - (d * (a / 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 <= -8.5e-14) || !(c <= 3.1e-68)) {
		tmp = (b - (d * (a / c))) / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -8.5e-14) or not (c <= 3.1e-68):
		tmp = (b - (d * (a / c))) / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -8.5e-14) || !(c <= 3.1e-68))
		tmp = Float64(Float64(b - Float64(d * Float64(a / 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 <= -8.5e-14) || ~((c <= 3.1e-68)))
		tmp = (b - (d * (a / c))) / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -8.5e-14], N[Not[LessEqual[c, 3.1e-68]], $MachinePrecision]], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -8.50000000000000038e-14 or 3.0999999999999999e-68 < c

   1. Initial program 63.7%

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

   2. Taylor expanded in c around inf 77.4%

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

      1. +-commutative 77.4%

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

      2. mul-1-neg 77.4%

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

      3. unsub-neg 77.4%

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

      4. unpow2 77.4%

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

      5. times-frac 77.0%

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

   4. Simplified 77.0%

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

      1. associate-*r/ 78.3%

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

      2. sub-div 78.3%

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

   6. Applied egg-rr 78.3%

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

   if -8.50000000000000038e-14 < c < 3.0999999999999999e-68

   1. Initial program 71.7%

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

   2. Taylor expanded in c around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.5 \cdot 10^{-14} \lor \neg \left(c \leq 3.1 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 8: 62.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-15) (/ b c) (if (<= c 5.4e-68) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-15) {
		tmp = b / c;
	} else if (c <= 5.4e-68) {
		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 <= (-3d-15)) then
        tmp = b / c
    else if (c <= 5.4d-68) 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 <= -3e-15) {
		tmp = b / c;
	} else if (c <= 5.4e-68) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3e-15:
		tmp = b / c
	elif c <= 5.4e-68:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-15)
		tmp = Float64(b / c);
	elseif (c <= 5.4e-68)
		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 <= -3e-15)
		tmp = b / c;
	elseif (c <= 5.4e-68)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-15], N[(b / c), $MachinePrecision], If[LessEqual[c, 5.4e-68], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3e-15 or 5.4000000000000003e-68 < c

   1. Initial program 63.7%

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

   2. Taylor expanded in c around inf 65.8%

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

   if -3e-15 < c < 5.4000000000000003e-68

   1. Initial program 71.7%

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

   2. Taylor expanded in c around 0 70.4%

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

      1. associate-*r/ 70.4%

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

      2. neg-mul-1 70.4%

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

   4. Simplified 70.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 5.4 \cdot 10^{-68}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 9: 46.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+191}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -3.6e+90) (/ a d) (if (<= d 2.6e+191) (/ b c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.6e+90) {
		tmp = a / d;
	} else if (d <= 2.6e+191) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-3.6d+90)) then
        tmp = a / d
    else if (d <= 2.6d+191) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -3.6e+90) {
		tmp = a / d;
	} else if (d <= 2.6e+191) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -3.6e+90:
		tmp = a / d
	elif d <= 2.6e+191:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -3.6e+90)
		tmp = Float64(a / d);
	elseif (d <= 2.6e+191)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -3.6e+90)
		tmp = a / d;
	elseif (d <= 2.6e+191)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -3.6e+90], N[(a / d), $MachinePrecision], If[LessEqual[d, 2.6e+191], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -3.6e90 or 2.6e191 < d

   1. Initial program 46.8%

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

      1. *-un-lft-identity 46.8%

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

      2. add-sqr-sqrt 46.8%

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

      3. times-frac 46.8%

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

      4. hypot-def 46.8%

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

      5. hypot-def 70.1%

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

   3. Applied egg-rr 70.1%

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

   4. Taylor expanded in c around 0 58.6%

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

      1. neg-mul-1 58.6%

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

      2. unsub-neg 58.6%

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

      3. associate-/l* 65.4%

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

   6. Simplified 65.4%

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

   7. Taylor expanded in d around -inf 42.4%

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

   if -3.6e90 < d < 2.6e191

   1. Initial program 73.5%

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

   2. Taylor expanded in c around inf 50.7%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.6 \cdot 10^{+90}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.6 \cdot 10^{+191}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 10: 11.2% 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 67.4%

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

   1. *-un-lft-identity 67.4%

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

   2. add-sqr-sqrt 67.4%

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

   3. times-frac 67.4%

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

   4. hypot-def 67.4%

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

   5. hypot-def 81.9%

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

3. Applied egg-rr 81.9%

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

4. Taylor expanded in c around 0 35.4%

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

   1. neg-mul-1 35.4%

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

   2. unsub-neg 35.4%

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

   3. associate-/l* 37.0%

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

6. Simplified 37.0%

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

7. Taylor expanded in d around -inf 12.0%

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

8. Final simplification 12.0%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023178 
(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))))