Complex division, imag part

Percentage Accurate: 61.4% → 85.6%

Time: 9.2s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{if}\;c \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0
         (fma
          (/ c (hypot c d))
          (/ b (hypot c d))
          (* a (/ (- d) (pow (hypot c d) 2.0))))))
   (if (<= c -1.1e-76)
     t_0
     (if (<= c 2.25e-80)
       (/ (- (/ b (/ d c)) a) d)
       (if (<= c 9.2e+138) t_0 (/ (- b (* a (/ d c))) c))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((c / hypot(c, d)), (b / hypot(c, d)), (a * (-d / pow(hypot(c, d), 2.0))));
	double tmp;
	if (c <= -1.1e-76) {
		tmp = t_0;
	} else if (c <= 2.25e-80) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 9.2e+138) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(Float64(c / hypot(c, d)), Float64(b / hypot(c, d)), Float64(a * Float64(Float64(-d) / (hypot(c, d) ^ 2.0))))
	tmp = 0.0
	if (c <= -1.1e-76)
		tmp = t_0;
	elseif (c <= 2.25e-80)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 9.2e+138)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] + N[(a * N[((-d) / N[Power[N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.1e-76], t$95$0, If[LessEqual[c, 2.25e-80], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.2e+138], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.1e-76 or 2.2500000000000001e-80 < c < 9.2000000000000003e138

   1. Initial program 69.7%

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

      1. div-sub 69.7%

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

      2. *-commutative 69.7%

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

      3. add-sqr-sqrt 69.7%

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

      4. times-frac 74.9%

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

      5. fma-neg 74.9%

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

      6. hypot-define 74.9%

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

      7. hypot-define 86.7%

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

      8. associate-/l* 89.3%

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

      9. add-sqr-sqrt 89.3%

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

      10. pow2 89.3%

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

      11. hypot-define 89.3%

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

   4. Applied egg-rr 89.3%

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

   if -1.1e-76 < c < 2.2500000000000001e-80

   1. Initial program 68.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. unpow2 87.5%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.5%

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

      7. associate-/l* 93.5%

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

   5. Simplified 93.5%

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

      1. clear-num 93.5%

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

      2. un-div-inv 93.5%

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

   7. Applied egg-rr 93.5%

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

   if 9.2000000000000003e138 < c

   1. Initial program 37.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 89.0%

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

      1. mul-1-neg 89.0%

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

      2. unsub-neg 89.0%

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

      3. associate-/l* 94.4%

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

   5. Simplified 94.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.1 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{elif}\;c \leq 2.25 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.2 \cdot 10^{+138}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{hypot}\left(c, d\right)}, \frac{b}{\mathsf{hypot}\left(c, d\right)}, a \cdot \frac{-d}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(c, d\right)}\\ t_1 := t\_0 \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+100}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(a, \frac{d}{c}, -b\right)\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ 1.0 (hypot c d)))
        (t_1 (* t_0 (/ (fma b c (* d (- a))) (hypot c d)))))
   (if (<= c -2.7e+100)
     (* t_0 (fma a (/ d c) (- b)))
     (if (<= c -5.6e-80)
       t_1
       (if (<= c 3.4e-81)
         (/ (- (/ b (/ d c)) a) d)
         (if (<= c 9.8e+58) t_1 (/ (- b (* a (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = 1.0 / hypot(c, d);
	double t_1 = t_0 * (fma(b, c, (d * -a)) / hypot(c, d));
	double tmp;
	if (c <= -2.7e+100) {
		tmp = t_0 * fma(a, (d / c), -b);
	} else if (c <= -5.6e-80) {
		tmp = t_1;
	} else if (c <= 3.4e-81) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 9.8e+58) {
		tmp = t_1;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(1.0 / hypot(c, d))
	t_1 = Float64(t_0 * Float64(fma(b, c, Float64(d * Float64(-a))) / hypot(c, d)))
	tmp = 0.0
	if (c <= -2.7e+100)
		tmp = Float64(t_0 * fma(a, Float64(d / c), Float64(-b)));
	elseif (c <= -5.6e-80)
		tmp = t_1;
	elseif (c <= 3.4e-81)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 9.8e+58)
		tmp = t_1;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(b * c + N[(d * (-a)), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.7e+100], N[(t$95$0 * N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -5.6e-80], t$95$1, If[LessEqual[c, 3.4e-81], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.8e+58], t$95$1, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.69999999999999998e100

   1. Initial program 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.4%

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

      4. hypot-define 44.4%

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

      5. fma-neg 44.4%

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

      6. distribute-rgt-neg-in 44.4%

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

      7. hypot-define 56.2%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in c around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-/l* 86.1%

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

      3. fma-define 86.1%

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

      4. mul-1-neg 86.1%

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

   7. Simplified 86.1%

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

   if -2.69999999999999998e100 < c < -5.59999999999999978e-80 or 3.3999999999999999e-81 < c < 9.80000000000000037e58

   1. Initial program 88.2%

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

      1. *-un-lft-identity 88.2%

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

      2. add-sqr-sqrt 88.2%

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

      3. times-frac 88.0%

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

      4. hypot-define 88.0%

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

      5. fma-neg 88.0%

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

      6. distribute-rgt-neg-in 88.0%

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

      7. hypot-define 93.8%

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

   4. Applied egg-rr 93.8%

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

   if -5.59999999999999978e-80 < c < 3.3999999999999999e-81

   1. Initial program 68.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. unpow2 87.5%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.5%

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

      7. associate-/l* 93.5%

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

   5. Simplified 93.5%

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

      1. clear-num 93.5%

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

      2. un-div-inv 93.5%

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

   7. Applied egg-rr 93.5%

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

   if 9.80000000000000037e58 < c

   1. Initial program 43.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

      3. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.7 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, \frac{d}{c}, -b\right)\\ \mathbf{elif}\;c \leq -5.6 \cdot 10^{-80}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(b, c, d \cdot \left(-a\right)\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, \frac{d}{c}, -b\right)\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d)))))
   (if (<= c -1.2e+100)
     (* (/ 1.0 (hypot c d)) (fma a (/ d c) (- b)))
     (if (<= c -8e-81)
       t_0
       (if (<= c 2.65e-76)
         (/ (- (/ b (/ d c)) a) d)
         (if (<= c 9.8e+58) t_0 (/ (- b (* a (/ d c))) c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.2e+100) {
		tmp = (1.0 / hypot(c, d)) * fma(a, (d / c), -b);
	} else if (c <= -8e-81) {
		tmp = t_0;
	} else if (c <= 2.65e-76) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 9.8e+58) {
		tmp = t_0;
	} else {
		tmp = (b - (a * (d / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.2e+100)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * fma(a, Float64(d / c), Float64(-b)));
	elseif (c <= -8e-81)
		tmp = t_0;
	elseif (c <= 2.65e-76)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 9.8e+58)
		tmp = t_0;
	else
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.2e+100], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -8e-81], t$95$0, If[LessEqual[c, 2.65e-76], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 9.8e+58], t$95$0, N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.20000000000000006e100

   1. Initial program 44.3%

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

      1. *-un-lft-identity 44.3%

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

      2. add-sqr-sqrt 44.3%

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

      3. times-frac 44.4%

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

      4. hypot-define 44.4%

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

      5. fma-neg 44.4%

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

      6. distribute-rgt-neg-in 44.4%

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

      7. hypot-define 56.2%

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

   4. Applied egg-rr 56.2%

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

   5. Taylor expanded in c around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. associate-/l* 86.1%

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

      3. fma-define 86.1%

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

      4. mul-1-neg 86.1%

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

   7. Simplified 86.1%

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

   if -1.20000000000000006e100 < c < -7.9999999999999997e-81 or 2.65e-76 < c < 9.80000000000000037e58

   1. Initial program 88.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -7.9999999999999997e-81 < c < 2.65e-76

   1. Initial program 68.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. unpow2 87.5%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.5%

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

      7. associate-/l* 93.5%

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

   5. Simplified 93.5%

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

      1. clear-num 93.5%

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

      2. un-div-inv 93.5%

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

   7. Applied egg-rr 93.5%

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

   if 9.80000000000000037e58 < c

   1. Initial program 43.0%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 82.6%

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

      1. mul-1-neg 82.6%

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

      2. unsub-neg 82.6%

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

      3. associate-/l* 86.7%

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

   5. Simplified 86.7%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \mathsf{fma}\left(a, \frac{d}{c}, -b\right)\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-81}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.65 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ t_1 := \frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{if}\;c \leq -9 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-78}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (+ (* c c) (* d d))))
        (t_1 (/ (- b (* a (/ d c))) c)))
   (if (<= c -9e+99)
     t_1
     (if (<= c -8.5e-78)
       t_0
       (if (<= c 4.8e-81)
         (/ (- (/ b (/ d c)) a) d)
         (if (<= c 2e+58) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -9e+99) {
		tmp = t_1;
	} else if (c <= -8.5e-78) {
		tmp = t_0;
	} else if (c <= 4.8e-81) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 2e+58) {
		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 = ((c * b) - (d * a)) / ((c * c) + (d * d))
    t_1 = (b - (a * (d / c))) / c
    if (c <= (-9d+99)) then
        tmp = t_1
    else if (c <= (-8.5d-78)) then
        tmp = t_0
    else if (c <= 4.8d-81) then
        tmp = ((b / (d / c)) - a) / d
    else if (c <= 2d+58) 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 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	double t_1 = (b - (a * (d / c))) / c;
	double tmp;
	if (c <= -9e+99) {
		tmp = t_1;
	} else if (c <= -8.5e-78) {
		tmp = t_0;
	} else if (c <= 4.8e-81) {
		tmp = ((b / (d / c)) - a) / d;
	} else if (c <= 2e+58) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d))
	t_1 = (b - (a * (d / c))) / c
	tmp = 0
	if c <= -9e+99:
		tmp = t_1
	elif c <= -8.5e-78:
		tmp = t_0
	elif c <= 4.8e-81:
		tmp = ((b / (d / c)) - a) / d
	elif c <= 2e+58:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b - Float64(a * Float64(d / c))) / c)
	tmp = 0.0
	if (c <= -9e+99)
		tmp = t_1;
	elseif (c <= -8.5e-78)
		tmp = t_0;
	elseif (c <= 4.8e-81)
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	elseif (c <= 2e+58)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((c * b) - (d * a)) / ((c * c) + (d * d));
	t_1 = (b - (a * (d / c))) / c;
	tmp = 0.0;
	if (c <= -9e+99)
		tmp = t_1;
	elseif (c <= -8.5e-78)
		tmp = t_0;
	elseif (c <= 4.8e-81)
		tmp = ((b / (d / c)) - a) / d;
	elseif (c <= 2e+58)
		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[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -9e+99], t$95$1, If[LessEqual[c, -8.5e-78], t$95$0, If[LessEqual[c, 4.8e-81], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2e+58], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.9999999999999999e99 or 1.99999999999999989e58 < c

   1. Initial program 43.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 82.8%

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

      1. mul-1-neg 82.8%

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

      2. unsub-neg 82.8%

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

      3. associate-/l* 86.2%

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

   5. Simplified 86.2%

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

   if -8.9999999999999999e99 < c < -8.49999999999999957e-78 or 4.7999999999999998e-81 < c < 1.99999999999999989e58

   1. Initial program 88.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   if -8.49999999999999957e-78 < c < 4.7999999999999998e-81

   1. Initial program 68.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 87.5%

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

      1. +-commutative 87.5%

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

      2. mul-1-neg 87.5%

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

      3. unsub-neg 87.5%

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

      4. unpow2 87.5%

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

      5. associate-/r* 92.6%

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

      6. div-sub 92.5%

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

      7. associate-/l* 93.5%

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

   5. Simplified 93.5%

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

      1. clear-num 93.5%

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

      2. un-div-inv 93.5%

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

   7. Applied egg-rr 93.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9 \cdot 10^{+99}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \mathbf{elif}\;c \leq 2 \cdot 10^{+58}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{-48} \lor \neg \left(c \leq 1.15 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.3e-48) (not (<= c 1.15e+37)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ b (/ d c)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.3e-48) || !(c <= 1.15e+37)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b / (d / c)) - 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 <= (-1.3d-48)) .or. (.not. (c <= 1.15d+37))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b / (d / c)) - 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 <= -1.3e-48) || !(c <= 1.15e+37)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b / (d / c)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.3e-48) or not (c <= 1.15e+37):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b / (d / c)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.3e-48) || !(c <= 1.15e+37))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b / Float64(d / c)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.3e-48) || ~((c <= 1.15e+37)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b / (d / c)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.3e-48], N[Not[LessEqual[c, 1.15e+37]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b / N[(d / c), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.29999999999999994e-48 or 1.15000000000000001e37 < c

   1. Initial program 54.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. unsub-neg 79.5%

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

      3. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

   if -1.29999999999999994e-48 < c < 1.15000000000000001e37

   1. Initial program 73.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. mul-1-neg 83.3%

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

      3. unsub-neg 83.3%

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

      4. unpow2 83.3%

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

      5. associate-/r* 87.0%

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

      6. div-sub 87.1%

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

      7. associate-/l* 87.8%

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

   5. Simplified 87.8%

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

      1. clear-num 87.8%

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

      2. un-div-inv 87.8%

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

   7. Applied egg-rr 87.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{-48} \lor \neg \left(c \leq 1.15 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b}{\frac{d}{c}} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{-49} \lor \neg \left(c \leq 6.5 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e-49) (not (<= c 6.5e+33)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (* b (/ c d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e-49) || !(c <= 6.5e+33)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / 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 <= (-2.6d-49)) .or. (.not. (c <= 6.5d+33))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = ((b * (c / 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 <= -2.6e-49) || !(c <= 6.5e+33)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = ((b * (c / d)) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.6e-49) or not (c <= 6.5e+33):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = ((b * (c / d)) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e-49) || !(c <= 6.5e+33))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.6e-49) || ~((c <= 6.5e+33)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = ((b * (c / d)) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.6e-49], N[Not[LessEqual[c, 6.5e+33]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.59999999999999995e-49 or 6.49999999999999993e33 < c

   1. Initial program 54.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 79.5%

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

      1. mul-1-neg 79.5%

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

      2. unsub-neg 79.5%

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

      3. associate-/l* 82.1%

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

   5. Simplified 82.1%

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

   if -2.59999999999999995e-49 < c < 6.49999999999999993e33

   1. Initial program 73.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 83.3%

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

      1. +-commutative 83.3%

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

      2. mul-1-neg 83.3%

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

      3. unsub-neg 83.3%

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

      4. unpow2 83.3%

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

      5. associate-/r* 87.0%

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

      6. div-sub 87.1%

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

      7. associate-/l* 87.8%

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

   5. Simplified 87.8%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{-49} \lor \neg \left(c \leq 6.5 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-53} \lor \neg \left(c \leq 2.5 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.9e-53) (not (<= c 2.5e+33)))
   (/ (- b (* a (/ d c))) c)
   (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-53) || !(c <= 2.5e+33)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.9d-53)) .or. (.not. (c <= 2.5d+33))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.9e-53) || !(c <= 2.5e+33)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.9e-53) or not (c <= 2.5e+33):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.9e-53) || !(c <= 2.5e+33))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.9e-53) || ~((c <= 2.5e+33)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.9e-53], N[Not[LessEqual[c, 2.5e+33]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.8999999999999999e-53 or 2.49999999999999986e33 < c

   1. Initial program 54.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

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

      3. associate-/l* 81.2%

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

   5. Simplified 81.2%

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

   if -1.8999999999999999e-53 < c < 2.49999999999999986e33

   1. Initial program 73.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 71.8%

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

      1. associate-*r/ 71.8%

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

      2. neg-mul-1 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.9 \cdot 10^{-53} \lor \neg \left(c \leq 2.5 \cdot 10^{+33}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.4 \cdot 10^{+64} \lor \neg \left(c \leq 3.3 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -9.4e+64) (not (<= c 3.3e+46))) (/ b c) (/ a (- d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.4e+64) || !(c <= 3.3e+46)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-9.4d+64)) .or. (.not. (c <= 3.3d+46))) then
        tmp = b / c
    else
        tmp = a / -d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -9.4e+64) || !(c <= 3.3e+46)) {
		tmp = b / c;
	} else {
		tmp = a / -d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -9.4e+64) or not (c <= 3.3e+46):
		tmp = b / c
	else:
		tmp = a / -d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -9.4e+64) || !(c <= 3.3e+46))
		tmp = Float64(b / c);
	else
		tmp = Float64(a / Float64(-d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -9.4e+64) || ~((c <= 3.3e+46)))
		tmp = b / c;
	else
		tmp = a / -d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -9.4e+64], N[Not[LessEqual[c, 3.3e+46]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(a / (-d)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -9.40000000000000058e64 or 3.2999999999999998e46 < c

   1. Initial program 48.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 73.3%

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

   if -9.40000000000000058e64 < c < 3.2999999999999998e46

   1. Initial program 74.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 65.8%

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

      1. associate-*r/ 65.8%

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

      2. neg-mul-1 65.8%

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

   5. Simplified 65.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.4 \cdot 10^{+64} \lor \neg \left(c \leq 3.3 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 5.8 \cdot 10^{+187}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 5.8e+187) (/ b c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 5.8e+187) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 5.8d+187) then
        tmp = b / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 5.8e+187) {
		tmp = b / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 5.8e+187:
		tmp = b / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 5.8e+187)
		tmp = Float64(b / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 5.8e+187)
		tmp = b / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 5.8e+187], N[(b / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 5.8000000000000002e187

   1. Initial program 65.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around inf 41.1%

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

   if 5.8000000000000002e187 < d

   1. Initial program 54.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 83.6%

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

      1. +-commutative 83.6%

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

      2. mul-1-neg 83.6%

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

      3. unsub-neg 83.6%

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

      4. unpow2 83.6%

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

      5. associate-/r* 79.9%

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

      6. div-sub 79.9%

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

      7. *-commutative 79.9%

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

      8. associate-/l* 88.1%

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

      9. fma-neg 88.1%

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

   5. Simplified 88.1%

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

   6. Taylor expanded in d around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. +-commutative 55.6%

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

      3. sub-neg 55.6%

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

      4. *-commutative 55.6%

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

      5. *-commutative 55.6%

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

   8. Simplified 55.6%

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

      1. clear-num 55.6%

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

      2. inv-pow 55.6%

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

   10. Applied egg-rr 51.0%

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

       1. unpow-1 51.0%

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

       2. *-rgt-identity 51.0%

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

       3. associate-*r/ 51.0%

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

       4. *-commutative 51.0%

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

       5. fma-define 51.0%

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

       6. +-commutative 51.0%

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

       7. distribute-rgt-in 51.0%

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

       8. /-rgt-identity 51.0%

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

       9. times-frac 51.0%

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

       10. *-commutative 51.0%

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

       11. times-frac 51.0%

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

       12. metadata-eval 51.0%

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

       13. associate-/l* 51.4%

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

       14. *-inverses 51.4%

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

       15. *-rgt-identity 51.4%

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

       16. *-lft-identity 51.4%

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

       17. *-commutative 51.4%

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

       18. *-commutative 51.4%

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

       19. /-rgt-identity 51.4%

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

       20. associate-/r/ 51.4%

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

       21. associate-*l/ 51.4%

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

   12. Simplified 51.4%

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

   13. Taylor expanded in d around inf 55.3%

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

Alternative 10: 12.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq 3 \cdot 10^{+83}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d 3e+83) (/ a c) (/ a d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 3e+83) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= 3d+83) then
        tmp = a / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= 3e+83) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= 3e+83:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= 3e+83)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= 3e+83)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, 3e+83], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < 3e83

   1. Initial program 67.8%

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

      1. *-un-lft-identity 67.8%

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

      2. add-sqr-sqrt 67.8%

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

      3. times-frac 67.7%

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

      4. hypot-define 67.8%

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

      5. fma-neg 67.8%

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

      6. distribute-rgt-neg-in 67.8%

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

      7. hypot-define 77.7%

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

   4. Applied egg-rr 77.7%

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

   5. Taylor expanded in c around inf 33.1%

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

      1. mul-1-neg 33.1%

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

      2. unsub-neg 33.1%

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

      3. associate-/l* 34.0%

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

   7. Simplified 34.0%

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

   8. Taylor expanded in d around -inf 7.4%

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

   if 3e83 < d

   1. Initial program 51.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. mul-1-neg 74.3%

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

      3. unsub-neg 74.3%

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

      4. unpow2 74.3%

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

      5. associate-/r* 72.7%

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

      6. div-sub 72.7%

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

      7. *-commutative 72.7%

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

      8. associate-/l* 78.4%

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

      9. fma-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in d around 0 54.5%

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

      1. neg-mul-1 54.5%

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

      2. +-commutative 54.5%

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

      3. sub-neg 54.5%

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

      4. *-commutative 54.5%

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

      5. *-commutative 54.5%

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

   8. Simplified 54.5%

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

      1. clear-num 54.3%

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

      2. inv-pow 54.3%

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

   10. Applied egg-rr 29.9%

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

       1. unpow-1 29.9%

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

       2. *-rgt-identity 29.9%

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

       3. associate-*r/ 29.9%

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

       4. *-commutative 29.9%

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

       5. fma-define 29.9%

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

       6. +-commutative 29.9%

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

       7. distribute-rgt-in 29.9%

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

       8. /-rgt-identity 29.9%

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

       9. times-frac 29.9%

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

       10. *-commutative 29.9%

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

       11. times-frac 29.9%

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

       12. metadata-eval 29.9%

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

       13. associate-/l* 30.2%

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

       14. *-inverses 30.2%

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

       15. *-rgt-identity 30.2%

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

       16. *-lft-identity 30.2%

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

       17. *-commutative 30.2%

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

       18. *-commutative 30.2%

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

       19. /-rgt-identity 30.2%

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

       20. associate-/r/ 30.2%

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

       21. associate-*l/ 30.1%

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

   12. Simplified 32.1%

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

   13. Taylor expanded in d around inf 32.3%

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

Alternative 11: 9.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. *-un-lft-identity 64.5%

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

   2. add-sqr-sqrt 64.5%

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

   3. times-frac 64.4%

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

   4. hypot-define 64.4%

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

   5. fma-neg 64.4%

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

   6. distribute-rgt-neg-in 64.4%

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

   7. hypot-define 75.1%

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

4. Applied egg-rr 75.1%

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

5. Taylor expanded in c around inf 30.4%

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

   1. mul-1-neg 30.4%

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

   2. unsub-neg 30.4%

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

   3. associate-/l* 31.1%

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

7. Simplified 31.1%

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

8. Taylor expanded in d around -inf 8.2%

   \[\leadsto \color{blue}{\frac{a}{c}} \]
9. Add Preprocessing

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 2024106 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (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))))