Complex division, real part

Percentage Accurate: 61.1% → 85.4%

Time: 9.7s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((a * c) + (b * 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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + \frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+298)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (/ (+ b (/ 1.0 (/ (/ d c) a))) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+298) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b + (1.0 / ((d / c) / a))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+298)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b + Float64(1.0 / Float64(Float64(d / c) / a))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+298], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(1.0 / N[(N[(d / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.0000000000000003e298

   1. Initial program 78.9%

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

      1. fma-define 78.9%

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

      2. fma-define 78.9%

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

   3. Simplified 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. fma-define 78.9%

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

      3. add-sqr-sqrt 78.9%

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

      4. times-frac 78.9%

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

      5. fma-define 78.9%

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

      6. hypot-define 78.9%

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

      7. fma-define 78.9%

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

      8. fma-define 78.9%

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

      9. hypot-define 97.4%

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

   6. Applied egg-rr 97.4%

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

   if 5.0000000000000003e298 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 10.4%

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

      1. fma-define 10.4%

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

      2. fma-define 10.4%

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

   3. Simplified 10.4%

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

   5. Taylor expanded in d around inf 44.2%

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

      1. clear-num 44.2%

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

      2. inv-pow 44.2%

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

      3. *-commutative 44.2%

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

   7. Applied egg-rr 44.2%

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

      1. unpow-1 44.2%

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

      2. associate-/r* 57.9%

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

   9. Simplified 57.9%

      \[\leadsto \frac{b + \color{blue}{\frac{1}{\frac{\frac{d}{c}}{a}}}}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{b + \frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.15e-33)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 6.5e-162)
     (/ (+ b (/ 1.0 (/ (/ d c) a))) d)
     (if (<= c 2.3e+104)
       (/ (fma a c (* b d)) (fma c c (* d d)))
       (+ (/ a c) (/ d (* c (/ c b))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.15e-33) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 6.5e-162) {
		tmp = (b + (1.0 / ((d / c) / a))) / d;
	} else if (c <= 2.3e+104) {
		tmp = fma(a, c, (b * d)) / fma(c, c, (d * d));
	} else {
		tmp = (a / c) + (d / (c * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.15e-33)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 6.5e-162)
		tmp = Float64(Float64(b + Float64(1.0 / Float64(Float64(d / c) / a))) / d);
	elseif (c <= 2.3e+104)
		tmp = Float64(fma(a, c, Float64(b * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.15e-33], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 6.5e-162], N[(N[(b + N[(1.0 / N[(N[(d / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.3e+104], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.14999999999999993e-33

   1. Initial program 50.3%

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

      1. fma-define 50.3%

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

      2. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in d around 0 74.6%

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

      1. associate-*r/ 76.3%

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

   10. Simplified 76.3%

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

   if -1.14999999999999993e-33 < c < 6.49999999999999989e-162

   1. Initial program 73.2%

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

      1. fma-define 73.2%

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

      2. fma-define 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in d around inf 88.7%

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

      1. clear-num 88.7%

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

      2. inv-pow 88.7%

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

      3. *-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. unpow-1 88.7%

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

      2. associate-/r* 91.6%

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

   9. Simplified 91.6%

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

   if 6.49999999999999989e-162 < c < 2.29999999999999985e104

   1. Initial program 81.2%

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

      1. fma-define 81.2%

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

      2. fma-define 81.2%

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

   3. Simplified 81.2%

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

   if 2.29999999999999985e104 < c

   1. Initial program 35.2%

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

      1. fma-define 35.2%

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

      2. fma-define 35.2%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in d around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. times-frac 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. *-commutative 77.6%

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

      2. clear-num 77.6%

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

      3. frac-times 80.9%

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

      4. *-un-lft-identity 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.5 \cdot 10^{-162}:\\ \;\;\;\;\frac{b + \frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{+104}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{b + \frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.05e-32)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 2.9e-162)
     (/ (+ b (/ 1.0 (/ (/ d c) a))) d)
     (if (<= c 1.5e+104)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (+ (/ a c) (/ d (* c (/ c b))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.05e-32) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 2.9e-162) {
		tmp = (b + (1.0 / ((d / c) / a))) / d;
	} else if (c <= 1.5e+104) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + (d / (c * (c / b)));
	}
	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 <= (-1.05d-32)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 2.9d-162) then
        tmp = (b + (1.0d0 / ((d / c) / a))) / d
    else if (c <= 1.5d+104) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
    else
        tmp = (a / c) + (d / (c * (c / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.05e-32) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 2.9e-162) {
		tmp = (b + (1.0 / ((d / c) / a))) / d;
	} else if (c <= 1.5e+104) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (a / c) + (d / (c * (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.05e-32:
		tmp = (a + (b * (d / c))) / c
	elif c <= 2.9e-162:
		tmp = (b + (1.0 / ((d / c) / a))) / d
	elif c <= 1.5e+104:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (a / c) + (d / (c * (c / b)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.05e-32)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 2.9e-162)
		tmp = Float64(Float64(b + Float64(1.0 / Float64(Float64(d / c) / a))) / d);
	elseif (c <= 1.5e+104)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.05e-32)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 2.9e-162)
		tmp = (b + (1.0 / ((d / c) / a))) / d;
	elseif (c <= 1.5e+104)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (a / c) + (d / (c * (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.05e-32], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 2.9e-162], N[(N[(b + N[(1.0 / N[(N[(d / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.5e+104], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.05e-32

   1. Initial program 50.3%

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

      1. fma-define 50.3%

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

      2. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in d around 0 74.6%

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

      1. associate-*r/ 76.3%

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

   10. Simplified 76.3%

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

   if -1.05e-32 < c < 2.9000000000000001e-162

   1. Initial program 73.2%

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

      1. fma-define 73.2%

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

      2. fma-define 73.2%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in d around inf 88.7%

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

      1. clear-num 88.7%

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

      2. inv-pow 88.7%

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

      3. *-commutative 88.7%

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

   7. Applied egg-rr 88.7%

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

      1. unpow-1 88.7%

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

      2. associate-/r* 91.6%

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

   9. Simplified 91.6%

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

   if 2.9000000000000001e-162 < c < 1.49999999999999984e104

   1. Initial program 81.2%

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

   if 1.49999999999999984e104 < c

   1. Initial program 35.2%

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

      1. fma-define 35.2%

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

      2. fma-define 35.2%

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

   3. Simplified 35.2%

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

   5. Taylor expanded in d around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. unpow2 65.5%

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

      3. times-frac 77.6%

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

   7. Applied egg-rr 77.6%

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

      1. *-commutative 77.6%

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

      2. clear-num 77.6%

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

      3. frac-times 80.9%

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

      4. *-un-lft-identity 80.9%

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

   9. Applied egg-rr 80.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-162}:\\ \;\;\;\;\frac{b + \frac{1}{\frac{\frac{d}{c}}{a}}}{d}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 86:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.2e-32)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 86.0) (/ (+ b (* a (/ c d))) d) (+ (/ a c) (/ d (* c (/ c b)))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e-32) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 86.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a / c) + (d / (c * (c / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.2d-32)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 86.0d0) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a / c) + (d / (c * (c / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.2e-32) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 86.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a / c) + (d / (c * (c / b)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.2e-32:
		tmp = (a + (b * (d / c))) / c
	elif c <= 86.0:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a / c) + (d / (c * (c / b)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.2e-32)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 86.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a / c) + Float64(d / Float64(c * Float64(c / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.2e-32)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 86.0)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a / c) + (d / (c * (c / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.2e-32], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 86.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(d / N[(c * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.2000000000000001e-32

   1. Initial program 50.3%

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

      1. fma-define 50.3%

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

      2. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in d around 0 74.6%

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

      1. associate-*r/ 76.3%

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

   10. Simplified 76.3%

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

   if -1.2000000000000001e-32 < c < 86

   1. Initial program 73.7%

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

      1. fma-define 73.8%

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

      2. fma-define 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in d around inf 84.9%

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

      1. associate-/l* 86.4%

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

   7. Simplified 86.4%

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

   if 86 < c

   1. Initial program 52.2%

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

      1. fma-define 52.2%

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

      2. fma-define 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in d around 0 65.4%

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

      1. *-commutative 65.4%

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

      2. unpow2 65.4%

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

      3. times-frac 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. *-commutative 73.5%

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

      2. clear-num 73.5%

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

      3. frac-times 75.8%

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

      4. *-un-lft-identity 75.8%

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

   9. Applied egg-rr 75.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 86:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c \cdot \frac{c}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{-58} \lor \neg \left(c \leq 7 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.4e-58) (not (<= c 7e-68)))
   (/ (+ a (* b (/ d c))) c)
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.4e-58) || !(c <= 7e-68)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / 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 <= (-5.4d-58)) .or. (.not. (c <= 7d-68))) then
        tmp = (a + (b * (d / c))) / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.4e-58) || !(c <= 7e-68)) {
		tmp = (a + (b * (d / c))) / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.4e-58) or not (c <= 7e-68):
		tmp = (a + (b * (d / c))) / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.4e-58) || !(c <= 7e-68))
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.4e-58) || ~((c <= 7e-68)))
		tmp = (a + (b * (d / c))) / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.4e-58], N[Not[LessEqual[c, 7e-68]], $MachinePrecision]], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.3999999999999998e-58 or 7.00000000000000026e-68 < c

   1. Initial program 52.6%

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

      1. fma-define 52.6%

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

      2. fma-define 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in c around inf 69.3%

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

      1. *-commutative 69.3%

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

   7. Simplified 69.3%

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

   8. Taylor expanded in d around 0 69.3%

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

      1. associate-*r/ 72.9%

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

   10. Simplified 72.9%

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

   if -5.3999999999999998e-58 < c < 7.00000000000000026e-68

   1. Initial program 74.5%

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

      1. fma-define 74.5%

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

      2. fma-define 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in c around 0 69.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.4 \cdot 10^{-58} \lor \neg \left(c \leq 7 \cdot 10^{-68}\right):\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-33}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 125:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-33)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 125.0) (/ (+ b (* a (/ c d))) d) (/ (+ a (* d (/ b c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-33) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 125.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3d-33)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 125.0d0) then
        tmp = (b + (a * (c / d))) / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-33) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 125.0) {
		tmp = (b + (a * (c / d))) / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3e-33:
		tmp = (a + (b * (d / c))) / c
	elif c <= 125.0:
		tmp = (b + (a * (c / d))) / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-33)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 125.0)
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3e-33)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 125.0)
		tmp = (b + (a * (c / d))) / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-33], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 125.0], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.0000000000000002e-33

   1. Initial program 50.3%

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

      1. fma-define 50.3%

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

      2. fma-define 50.3%

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

   3. Simplified 50.3%

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

   5. Taylor expanded in c around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

   8. Taylor expanded in d around 0 74.6%

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

      1. associate-*r/ 76.3%

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

   10. Simplified 76.3%

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

   if -3.0000000000000002e-33 < c < 125

   1. Initial program 73.7%

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

      1. fma-define 73.8%

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

      2. fma-define 73.8%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in d around inf 84.9%

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

      1. associate-/l* 86.4%

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

   7. Simplified 86.4%

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

   if 125 < c

   1. Initial program 52.2%

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

      1. fma-define 52.2%

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

      2. fma-define 52.2%

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

   3. Simplified 52.2%

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

   5. Taylor expanded in c around inf 67.8%

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

      1. *-commutative 67.8%

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

   7. Simplified 67.8%

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

      1. associate-/l* 74.4%

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

   9. Applied egg-rr 74.4%

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

Alternative 7: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -9.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{a + b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 1.65 \cdot 10^{-66}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a + d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -9.8e-51)
   (/ (+ a (* b (/ d c))) c)
   (if (<= c 1.65e-66) (/ b d) (/ (+ a (* d (/ b c))) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.8e-51) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 1.65e-66) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-9.8d-51)) then
        tmp = (a + (b * (d / c))) / c
    else if (c <= 1.65d-66) then
        tmp = b / d
    else
        tmp = (a + (d * (b / c))) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -9.8e-51) {
		tmp = (a + (b * (d / c))) / c;
	} else if (c <= 1.65e-66) {
		tmp = b / d;
	} else {
		tmp = (a + (d * (b / c))) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -9.8e-51:
		tmp = (a + (b * (d / c))) / c
	elif c <= 1.65e-66:
		tmp = b / d
	else:
		tmp = (a + (d * (b / c))) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -9.8e-51)
		tmp = Float64(Float64(a + Float64(b * Float64(d / c))) / c);
	elseif (c <= 1.65e-66)
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a + Float64(d * Float64(b / c))) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -9.8e-51)
		tmp = (a + (b * (d / c))) / c;
	elseif (c <= 1.65e-66)
		tmp = b / d;
	else
		tmp = (a + (d * (b / c))) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -9.8e-51], N[(N[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 1.65e-66], N[(b / d), $MachinePrecision], N[(N[(a + N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.79999999999999948e-51

   1. Initial program 50.4%

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

      1. fma-define 50.4%

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

      2. fma-define 50.4%

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

   3. Simplified 50.4%

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

   5. Taylor expanded in c around inf 73.9%

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

      1. *-commutative 73.9%

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

   7. Simplified 73.9%

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

   8. Taylor expanded in d around 0 73.9%

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

      1. associate-*r/ 75.5%

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

   10. Simplified 75.5%

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

   if -9.79999999999999948e-51 < c < 1.6499999999999999e-66

   1. Initial program 74.5%

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

      1. fma-define 74.5%

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

      2. fma-define 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in c around 0 69.6%

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

   if 1.6499999999999999e-66 < c

   1. Initial program 54.3%

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

      1. fma-define 54.3%

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

      2. fma-define 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in c around inf 65.8%

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

      1. *-commutative 65.8%

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

   7. Simplified 65.8%

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

      1. associate-/l* 71.7%

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

   9. Applied egg-rr 71.7%

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

Alternative 8: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{-48} \lor \neg \left(c \leq 1.9 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.5e-48) (not (<= c 1.9e-66))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.5e-48) || !(c <= 1.9e-66)) {
		tmp = a / c;
	} else {
		tmp = b / 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 <= (-1.5d-48)) .or. (.not. (c <= 1.9d-66))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.5e-48) || !(c <= 1.9e-66)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.5e-48) or not (c <= 1.9e-66):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.5e-48) || !(c <= 1.9e-66))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.5e-48) || ~((c <= 1.9e-66)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.5e-48], N[Not[LessEqual[c, 1.9e-66]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.5e-48 or 1.8999999999999999e-66 < c

   1. Initial program 52.6%

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

      1. fma-define 52.6%

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

      2. fma-define 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in c around inf 62.6%

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

   if -1.5e-48 < c < 1.8999999999999999e-66

   1. Initial program 74.5%

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

      1. fma-define 74.5%

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

      2. fma-define 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in c around 0 69.6%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.5 \cdot 10^{-48} \lor \neg \left(c \leq 1.9 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.2% 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 61.7%

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

   1. fma-define 61.8%

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

   2. fma-define 61.8%

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

3. Simplified 61.8%

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

5. Taylor expanded in c around inf 43.0%

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

Developer Target 1: 99.4% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (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(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (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[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024160 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))