Complex division, real part

Percentage Accurate: 61.9% → 80.5%

Time: 13.1s

Alternatives: 12

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: 61.9% 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: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -9.2 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-97}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))) (t_1 (+ (/ a c) (* (/ d c) (/ b c)))))
   (if (<= c -9.2e+50)
     t_1
     (if (<= c -2e-97)
       (/ t_0 (fma c c (* d d)))
       (if (<= c 1.7e-160)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 1.12e-11) (/ t_0 (+ (* c c) (* d d))) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = (a / c) + ((d / c) * (b / c));
	double tmp;
	if (c <= -9.2e+50) {
		tmp = t_1;
	} else if (c <= -2e-97) {
		tmp = t_0 / fma(c, c, (d * d));
	} else if (c <= 1.7e-160) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.12e-11) {
		tmp = t_0 / ((c * c) + (d * d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	t_1 = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)))
	tmp = 0.0
	if (c <= -9.2e+50)
		tmp = t_1;
	elseif (c <= -2e-97)
		tmp = Float64(t_0 / fma(c, c, Float64(d * d)));
	elseif (c <= 1.7e-160)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 1.12e-11)
		tmp = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.2e+50], t$95$1, If[LessEqual[c, -2e-97], N[(t$95$0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e-160], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.12e-11], N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -9.19999999999999987e50 or 1.1200000000000001e-11 < c

   1. Initial program 44.4%

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

   2. Taylor expanded in c around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. times-frac 85.2%

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

   4. Simplified 85.2%

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

   if -9.19999999999999987e50 < c < -2.00000000000000007e-97

   1. Initial program 87.5%

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

      1. fma-def 87.5%

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

      2. fma-def 87.6%

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

   3. Simplified 87.6%

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

      1. fma-def 87.6%

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

      2. +-commutative 87.6%

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

   5. Applied egg-rr 87.6%

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

   if -2.00000000000000007e-97 < c < 1.70000000000000011e-160

   1. Initial program 73.3%

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

   2. Taylor expanded in c around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. times-frac 91.7%

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

   4. Simplified 91.7%

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

      1. +-commutative 91.7%

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

      2. frac-times 81.4%

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

      3. associate-/l/ 92.9%

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

      4. div-inv 92.9%

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

      5. div-inv 92.6%

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

      6. distribute-rgt-out 93.8%

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

      7. div-inv 93.9%

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

      8. *-commutative 93.9%

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

      9. associate-*l* 93.8%

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

      10. div-inv 93.9%

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

   6. Applied egg-rr 93.9%

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

   if 1.70000000000000011e-160 < c < 1.1200000000000001e-11

   1. Initial program 91.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -2 \cdot 10^{-97}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 2: 85.7% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 1e+305)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 1e+305) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	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))) <= 1e+305)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	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], 1e+305], 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[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $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))) < 9.9999999999999994e304

   1. Initial program 79.6%

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

      1. *-un-lft-identity 79.6%

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

      2. add-sqr-sqrt 79.6%

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

      3. times-frac 79.6%

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

      4. hypot-def 79.6%

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

      5. fma-def 79.6%

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

      6. hypot-def 95.6%

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

   3. Applied egg-rr 95.6%

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

   if 9.9999999999999994e304 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 17.6%

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

   2. Taylor expanded in c around inf 53.9%

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

      1. *-commutative 53.9%

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

      2. unpow2 53.9%

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

      3. times-frac 66.3%

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

   4. Simplified 66.3%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 3: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -2.95 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ a c) (* (/ d c) (/ b c)))))
   (if (<= c -2.95e+50)
     t_1
     (if (<= c -9e-109)
       t_0
       (if (<= c 1.15e-160)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 1.12e-11) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (a / c) + ((d / c) * (b / c));
	double tmp;
	if (c <= -2.95e+50) {
		tmp = t_1;
	} else if (c <= -9e-109) {
		tmp = t_0;
	} else if (c <= 1.15e-160) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.12e-11) {
		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 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    t_1 = (a / c) + ((d / c) * (b / c))
    if (c <= (-2.95d+50)) then
        tmp = t_1
    else if (c <= (-9d-109)) then
        tmp = t_0
    else if (c <= 1.15d-160) then
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    else if (c <= 1.12d-11) 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 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double t_1 = (a / c) + ((d / c) * (b / c));
	double tmp;
	if (c <= -2.95e+50) {
		tmp = t_1;
	} else if (c <= -9e-109) {
		tmp = t_0;
	} else if (c <= 1.15e-160) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 1.12e-11) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	t_1 = (a / c) + ((d / c) * (b / c))
	tmp = 0
	if c <= -2.95e+50:
		tmp = t_1
	elif c <= -9e-109:
		tmp = t_0
	elif c <= 1.15e-160:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 1.12e-11:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)))
	tmp = 0.0
	if (c <= -2.95e+50)
		tmp = t_1;
	elseif (c <= -9e-109)
		tmp = t_0;
	elseif (c <= 1.15e-160)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 1.12e-11)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	t_1 = (a / c) + ((d / c) * (b / c));
	tmp = 0.0;
	if (c <= -2.95e+50)
		tmp = t_1;
	elseif (c <= -9e-109)
		tmp = t_0;
	elseif (c <= 1.15e-160)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 1.12e-11)
		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[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.95e+50], t$95$1, If[LessEqual[c, -9e-109], t$95$0, If[LessEqual[c, 1.15e-160], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.12e-11], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.9499999999999999e50 or 1.1200000000000001e-11 < c

   1. Initial program 44.4%

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

   2. Taylor expanded in c around inf 79.1%

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

      1. *-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. times-frac 85.2%

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

   4. Simplified 85.2%

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

   if -2.9499999999999999e50 < c < -9.0000000000000002e-109 or 1.14999999999999992e-160 < c < 1.1200000000000001e-11

   1. Initial program 89.9%

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

   if -9.0000000000000002e-109 < c < 1.14999999999999992e-160

   1. Initial program 73.3%

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

   2. Taylor expanded in c around 0 81.4%

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

      1. *-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. times-frac 91.7%

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

   4. Simplified 91.7%

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

      1. +-commutative 91.7%

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

      2. frac-times 81.4%

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

      3. associate-/l/ 92.9%

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

      4. div-inv 92.9%

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

      5. div-inv 92.6%

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

      6. distribute-rgt-out 93.8%

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

      7. div-inv 93.9%

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

      8. *-commutative 93.9%

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

      9. associate-*l* 93.8%

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

      10. div-inv 93.9%

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

   6. Applied egg-rr 93.9%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.95 \cdot 10^{+50}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{-109}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 1.12 \cdot 10^{-11}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 4: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{-28} \lor \neg \left(c \leq 7.5 \cdot 10^{-109}\right) \land \left(c \leq 2.4 \cdot 10^{-31} \lor \neg \left(c \leq 1.45 \cdot 10^{-11}\right)\right):\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.85e-28)
         (and (not (<= c 7.5e-109)) (or (<= c 2.4e-31) (not (<= c 1.45e-11)))))
   (+ (/ a c) (* b (/ d (* c c))))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.85e-28) || (!(c <= 7.5e-109) && ((c <= 2.4e-31) || !(c <= 1.45e-11)))) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else {
		tmp = (1.0 / d) * (b + (a * (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 ((c <= (-1.85d-28)) .or. (.not. (c <= 7.5d-109)) .and. (c <= 2.4d-31) .or. (.not. (c <= 1.45d-11))) then
        tmp = (a / c) + (b * (d / (c * c)))
    else
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.85e-28) || (!(c <= 7.5e-109) && ((c <= 2.4e-31) || !(c <= 1.45e-11)))) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.85e-28) or (not (c <= 7.5e-109) and ((c <= 2.4e-31) or not (c <= 1.45e-11))):
		tmp = (a / c) + (b * (d / (c * c)))
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.85e-28) || (!(c <= 7.5e-109) && ((c <= 2.4e-31) || !(c <= 1.45e-11))))
		tmp = Float64(Float64(a / c) + Float64(b * Float64(d / Float64(c * c))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.85e-28) || (~((c <= 7.5e-109)) && ((c <= 2.4e-31) || ~((c <= 1.45e-11)))))
		tmp = (a / c) + (b * (d / (c * c)));
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.85e-28], And[N[Not[LessEqual[c, 7.5e-109]], $MachinePrecision], Or[LessEqual[c, 2.4e-31], N[Not[LessEqual[c, 1.45e-11]], $MachinePrecision]]]], N[(N[(a / c), $MachinePrecision] + N[(b * N[(d / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.8500000000000001e-28 or 7.49999999999999982e-109 < c < 2.4e-31 or 1.45e-11 < c

   1. Initial program 54.8%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. unpow2 77.3%

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

      2. associate-/l* 77.6%

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

   4. Simplified 77.6%

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

   5. Taylor expanded in b around 0 77.3%

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

      1. unpow2 77.3%

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

      2. *-lft-identity 77.3%

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

      3. times-frac 77.6%

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

      4. /-rgt-identity 77.6%

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

   7. Simplified 77.6%

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

   if -1.8500000000000001e-28 < c < 7.49999999999999982e-109 or 2.4e-31 < c < 1.45e-11

   1. Initial program 75.9%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. frac-times 79.0%

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

      3. associate-/l/ 87.4%

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

      4. div-inv 87.4%

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

      5. div-inv 87.2%

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

      6. distribute-rgt-out 88.0%

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

      7. div-inv 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l* 88.0%

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

      10. div-inv 88.0%

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

   6. Applied egg-rr 88.0%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.85 \cdot 10^{-28} \lor \neg \left(c \leq 7.5 \cdot 10^{-109}\right) \land \left(c \leq 2.4 \cdot 10^{-31} \lor \neg \left(c \leq 1.45 \cdot 10^{-11}\right)\right):\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 5: 72.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ t_1 := \frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{if}\;c \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* a (/ c d)))))
        (t_1 (+ (/ a c) (* b (/ d (* c c))))))
   (if (<= c -1.45e-31)
     t_1
     (if (<= c 3.4e-105)
       t_0
       (if (<= c 5.2e-38)
         t_1
         (if (<= c 1.3e-11) t_0 (+ (/ a c) (* d (/ b (* c c))))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = (a / c) + (b * (d / (c * c)));
	double tmp;
	if (c <= -1.45e-31) {
		tmp = t_1;
	} else if (c <= 3.4e-105) {
		tmp = t_0;
	} else if (c <= 5.2e-38) {
		tmp = t_1;
	} else if (c <= 1.3e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / d) * (b + (a * (c / d)))
    t_1 = (a / c) + (b * (d / (c * c)))
    if (c <= (-1.45d-31)) then
        tmp = t_1
    else if (c <= 3.4d-105) then
        tmp = t_0
    else if (c <= 5.2d-38) then
        tmp = t_1
    else if (c <= 1.3d-11) then
        tmp = t_0
    else
        tmp = (a / c) + (d * (b / (c * c)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = (a / c) + (b * (d / (c * c)));
	double tmp;
	if (c <= -1.45e-31) {
		tmp = t_1;
	} else if (c <= 3.4e-105) {
		tmp = t_0;
	} else if (c <= 5.2e-38) {
		tmp = t_1;
	} else if (c <= 1.3e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + (d * (b / (c * c)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a * (c / d)))
	t_1 = (a / c) + (b * (d / (c * c)))
	tmp = 0
	if c <= -1.45e-31:
		tmp = t_1
	elif c <= 3.4e-105:
		tmp = t_0
	elif c <= 5.2e-38:
		tmp = t_1
	elif c <= 1.3e-11:
		tmp = t_0
	else:
		tmp = (a / c) + (d * (b / (c * c)))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))))
	t_1 = Float64(Float64(a / c) + Float64(b * Float64(d / Float64(c * c))))
	tmp = 0.0
	if (c <= -1.45e-31)
		tmp = t_1;
	elseif (c <= 3.4e-105)
		tmp = t_0;
	elseif (c <= 5.2e-38)
		tmp = t_1;
	elseif (c <= 1.3e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(d * Float64(b / Float64(c * c))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a * (c / d)));
	t_1 = (a / c) + (b * (d / (c * c)));
	tmp = 0.0;
	if (c <= -1.45e-31)
		tmp = t_1;
	elseif (c <= 3.4e-105)
		tmp = t_0;
	elseif (c <= 5.2e-38)
		tmp = t_1;
	elseif (c <= 1.3e-11)
		tmp = t_0;
	else
		tmp = (a / c) + (d * (b / (c * c)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(b * N[(d / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.45e-31], t$95$1, If[LessEqual[c, 3.4e-105], t$95$0, If[LessEqual[c, 5.2e-38], t$95$1, If[LessEqual[c, 1.3e-11], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.45e-31 or 3.39999999999999992e-105 < c < 5.20000000000000022e-38

   1. Initial program 60.6%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. unpow2 75.5%

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

      2. associate-/l* 75.8%

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

   4. Simplified 75.8%

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

   5. Taylor expanded in b around 0 75.5%

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

      1. unpow2 75.5%

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

      2. *-lft-identity 75.5%

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

      3. times-frac 75.8%

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

      4. /-rgt-identity 75.8%

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

   7. Simplified 75.8%

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

   if -1.45e-31 < c < 3.39999999999999992e-105 or 5.20000000000000022e-38 < c < 1.3e-11

   1. Initial program 75.9%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. frac-times 79.0%

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

      3. associate-/l/ 87.4%

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

      4. div-inv 87.4%

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

      5. div-inv 87.2%

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

      6. distribute-rgt-out 88.0%

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

      7. div-inv 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l* 88.0%

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

      10. div-inv 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 1.3e-11 < c

   1. Initial program 47.2%

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

   2. Taylor expanded in c around inf 79.7%

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

      1. unpow2 79.7%

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

      2. associate-/l* 80.0%

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

   4. Simplified 80.0%

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

      1. associate-/r/ 83.0%

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

   6. Applied egg-rr 83.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.45 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \end{array} \]

Alternative 6: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -5.3 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* a (/ c d)))))
        (t_1 (+ (/ a c) (* (/ d c) (/ b c)))))
   (if (<= c -5.3e-29)
     t_1
     (if (<= c 6.8e-105)
       t_0
       (if (<= c 6.2e-37)
         (+ (/ a c) (* b (/ d (* c c))))
         (if (<= c 1.3e-11) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = (a / c) + ((d / c) * (b / c));
	double tmp;
	if (c <= -5.3e-29) {
		tmp = t_1;
	} else if (c <= 6.8e-105) {
		tmp = t_0;
	} else if (c <= 6.2e-37) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else if (c <= 1.3e-11) {
		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 = (1.0d0 / d) * (b + (a * (c / d)))
    t_1 = (a / c) + ((d / c) * (b / c))
    if (c <= (-5.3d-29)) then
        tmp = t_1
    else if (c <= 6.8d-105) then
        tmp = t_0
    else if (c <= 6.2d-37) then
        tmp = (a / c) + (b * (d / (c * c)))
    else if (c <= 1.3d-11) 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 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = (a / c) + ((d / c) * (b / c));
	double tmp;
	if (c <= -5.3e-29) {
		tmp = t_1;
	} else if (c <= 6.8e-105) {
		tmp = t_0;
	} else if (c <= 6.2e-37) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else if (c <= 1.3e-11) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a * (c / d)))
	t_1 = (a / c) + ((d / c) * (b / c))
	tmp = 0
	if c <= -5.3e-29:
		tmp = t_1
	elif c <= 6.8e-105:
		tmp = t_0
	elif c <= 6.2e-37:
		tmp = (a / c) + (b * (d / (c * c)))
	elif c <= 1.3e-11:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))))
	t_1 = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)))
	tmp = 0.0
	if (c <= -5.3e-29)
		tmp = t_1;
	elseif (c <= 6.8e-105)
		tmp = t_0;
	elseif (c <= 6.2e-37)
		tmp = Float64(Float64(a / c) + Float64(b * Float64(d / Float64(c * c))));
	elseif (c <= 1.3e-11)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a * (c / d)));
	t_1 = (a / c) + ((d / c) * (b / c));
	tmp = 0.0;
	if (c <= -5.3e-29)
		tmp = t_1;
	elseif (c <= 6.8e-105)
		tmp = t_0;
	elseif (c <= 6.2e-37)
		tmp = (a / c) + (b * (d / (c * c)));
	elseif (c <= 1.3e-11)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.3e-29], t$95$1, If[LessEqual[c, 6.8e-105], t$95$0, If[LessEqual[c, 6.2e-37], N[(N[(a / c), $MachinePrecision] + N[(b * N[(d / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.3e-11], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.2999999999999999e-29 or 1.3e-11 < c

   1. Initial program 48.8%

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

   2. Taylor expanded in c around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. unpow2 79.3%

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

      3. times-frac 84.2%

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

   4. Simplified 84.2%

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

   if -5.2999999999999999e-29 < c < 6.79999999999999984e-105 or 6.19999999999999987e-37 < c < 1.3e-11

   1. Initial program 75.9%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. frac-times 79.0%

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

      3. associate-/l/ 87.4%

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

      4. div-inv 87.4%

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

      5. div-inv 87.2%

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

      6. distribute-rgt-out 88.0%

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

      7. div-inv 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l* 88.0%

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

      10. div-inv 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 6.79999999999999984e-105 < c < 6.19999999999999987e-37

   1. Initial program 94.5%

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

   2. Taylor expanded in c around inf 64.7%

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

      1. unpow2 64.7%

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

      2. associate-/l* 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in b around 0 64.7%

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

      1. unpow2 64.7%

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

      2. *-lft-identity 64.7%

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

      3. times-frac 64.7%

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

      4. /-rgt-identity 64.7%

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

   7. Simplified 64.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 6.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 7: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{if}\;c \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* a (/ c d))))))
   (if (<= c -3.8e-28)
     (+ (/ a c) (/ (* b (/ d c)) c))
     (if (<= c 6.8e-105)
       t_0
       (if (<= c 5.6e-30)
         (+ (/ a c) (* b (/ d (* c c))))
         (if (<= c 1.2e-11) t_0 (+ (/ a c) (* (/ d c) (/ b c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double tmp;
	if (c <= -3.8e-28) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 6.8e-105) {
		tmp = t_0;
	} else if (c <= 5.6e-30) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else if (c <= 1.2e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / d) * (b + (a * (c / d)))
    if (c <= (-3.8d-28)) then
        tmp = (a / c) + ((b * (d / c)) / c)
    else if (c <= 6.8d-105) then
        tmp = t_0
    else if (c <= 5.6d-30) then
        tmp = (a / c) + (b * (d / (c * c)))
    else if (c <= 1.2d-11) then
        tmp = t_0
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double tmp;
	if (c <= -3.8e-28) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 6.8e-105) {
		tmp = t_0;
	} else if (c <= 5.6e-30) {
		tmp = (a / c) + (b * (d / (c * c)));
	} else if (c <= 1.2e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a * (c / d)))
	tmp = 0
	if c <= -3.8e-28:
		tmp = (a / c) + ((b * (d / c)) / c)
	elif c <= 6.8e-105:
		tmp = t_0
	elif c <= 5.6e-30:
		tmp = (a / c) + (b * (d / (c * c)))
	elif c <= 1.2e-11:
		tmp = t_0
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))))
	tmp = 0.0
	if (c <= -3.8e-28)
		tmp = Float64(Float64(a / c) + Float64(Float64(b * Float64(d / c)) / c));
	elseif (c <= 6.8e-105)
		tmp = t_0;
	elseif (c <= 5.6e-30)
		tmp = Float64(Float64(a / c) + Float64(b * Float64(d / Float64(c * c))));
	elseif (c <= 1.2e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a * (c / d)));
	tmp = 0.0;
	if (c <= -3.8e-28)
		tmp = (a / c) + ((b * (d / c)) / c);
	elseif (c <= 6.8e-105)
		tmp = t_0;
	elseif (c <= 5.6e-30)
		tmp = (a / c) + (b * (d / (c * c)));
	elseif (c <= 1.2e-11)
		tmp = t_0;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.8e-28], N[(N[(a / c), $MachinePrecision] + N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.8e-105], t$95$0, If[LessEqual[c, 5.6e-30], N[(N[(a / c), $MachinePrecision] + N[(b * N[(d / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.2e-11], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.80000000000000009e-28

   1. Initial program 50.3%

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

   2. Taylor expanded in c around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-/l* 79.2%

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

   4. Simplified 79.2%

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

      1. associate-/r/ 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. associate-*l/ 78.9%

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

      2. associate-/r* 80.7%

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

      3. associate-*r/ 82.4%

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

   8. Applied egg-rr 82.4%

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

   if -3.80000000000000009e-28 < c < 6.79999999999999984e-105 or 5.59999999999999977e-30 < c < 1.2000000000000001e-11

   1. Initial program 75.9%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. frac-times 79.0%

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

      3. associate-/l/ 87.4%

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

      4. div-inv 87.4%

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

      5. div-inv 87.2%

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

      6. distribute-rgt-out 88.0%

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

      7. div-inv 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l* 88.0%

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

      10. div-inv 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 6.79999999999999984e-105 < c < 5.59999999999999977e-30

   1. Initial program 94.5%

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

   2. Taylor expanded in c around inf 64.7%

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

      1. unpow2 64.7%

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

      2. associate-/l* 64.7%

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

   4. Simplified 64.7%

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

   5. Taylor expanded in b around 0 64.7%

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

      1. unpow2 64.7%

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

      2. *-lft-identity 64.7%

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

      3. times-frac 64.7%

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

      4. /-rgt-identity 64.7%

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

   7. Simplified 64.7%

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

   if 1.2000000000000001e-11 < c

   1. Initial program 47.2%

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

   2. Taylor expanded in c around inf 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. times-frac 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 5.6 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c} + b \cdot \frac{d}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 8: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ t_1 := b \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{a}{c} + \frac{t_1}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{t_1}}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 d) (+ b (* a (/ c d))))) (t_1 (* b (/ d c))))
   (if (<= c -2.8e-32)
     (+ (/ a c) (/ t_1 c))
     (if (<= c 4e-106)
       t_0
       (if (<= c 8.6e-31)
         (+ (/ a c) (/ 1.0 (/ c t_1)))
         (if (<= c 1.15e-11) t_0 (+ (/ a c) (* (/ d c) (/ b c)))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = b * (d / c);
	double tmp;
	if (c <= -2.8e-32) {
		tmp = (a / c) + (t_1 / c);
	} else if (c <= 4e-106) {
		tmp = t_0;
	} else if (c <= 8.6e-31) {
		tmp = (a / c) + (1.0 / (c / t_1));
	} else if (c <= 1.15e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (1.0d0 / d) * (b + (a * (c / d)))
    t_1 = b * (d / c)
    if (c <= (-2.8d-32)) then
        tmp = (a / c) + (t_1 / c)
    else if (c <= 4d-106) then
        tmp = t_0
    else if (c <= 8.6d-31) then
        tmp = (a / c) + (1.0d0 / (c / t_1))
    else if (c <= 1.15d-11) then
        tmp = t_0
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (1.0 / d) * (b + (a * (c / d)));
	double t_1 = b * (d / c);
	double tmp;
	if (c <= -2.8e-32) {
		tmp = (a / c) + (t_1 / c);
	} else if (c <= 4e-106) {
		tmp = t_0;
	} else if (c <= 8.6e-31) {
		tmp = (a / c) + (1.0 / (c / t_1));
	} else if (c <= 1.15e-11) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (1.0 / d) * (b + (a * (c / d)))
	t_1 = b * (d / c)
	tmp = 0
	if c <= -2.8e-32:
		tmp = (a / c) + (t_1 / c)
	elif c <= 4e-106:
		tmp = t_0
	elif c <= 8.6e-31:
		tmp = (a / c) + (1.0 / (c / t_1))
	elif c <= 1.15e-11:
		tmp = t_0
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))))
	t_1 = Float64(b * Float64(d / c))
	tmp = 0.0
	if (c <= -2.8e-32)
		tmp = Float64(Float64(a / c) + Float64(t_1 / c));
	elseif (c <= 4e-106)
		tmp = t_0;
	elseif (c <= 8.6e-31)
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(c / t_1)));
	elseif (c <= 1.15e-11)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (1.0 / d) * (b + (a * (c / d)));
	t_1 = b * (d / c);
	tmp = 0.0;
	if (c <= -2.8e-32)
		tmp = (a / c) + (t_1 / c);
	elseif (c <= 4e-106)
		tmp = t_0;
	elseif (c <= 8.6e-31)
		tmp = (a / c) + (1.0 / (c / t_1));
	elseif (c <= 1.15e-11)
		tmp = t_0;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.8e-32], N[(N[(a / c), $MachinePrecision] + N[(t$95$1 / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4e-106], t$95$0, If[LessEqual[c, 8.6e-31], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.15e-11], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.7999999999999999e-32

   1. Initial program 50.3%

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

   2. Taylor expanded in c around inf 78.9%

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

      1. unpow2 78.9%

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

      2. associate-/l* 79.2%

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

   4. Simplified 79.2%

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

      1. associate-/r/ 79.1%

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

   6. Applied egg-rr 79.1%

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

      1. associate-*l/ 78.9%

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

      2. associate-/r* 80.7%

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

      3. associate-*r/ 82.4%

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

   8. Applied egg-rr 82.4%

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

   if -2.7999999999999999e-32 < c < 3.99999999999999976e-106 or 8.6e-31 < c < 1.15000000000000007e-11

   1. Initial program 75.9%

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

   2. Taylor expanded in c around 0 79.0%

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

      1. *-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 86.6%

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

   4. Simplified 86.6%

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

      1. +-commutative 86.6%

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

      2. frac-times 79.0%

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

      3. associate-/l/ 87.4%

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

      4. div-inv 87.4%

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

      5. div-inv 87.2%

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

      6. distribute-rgt-out 88.0%

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

      7. div-inv 88.0%

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

      8. *-commutative 88.0%

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

      9. associate-*l* 88.0%

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

      10. div-inv 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 3.99999999999999976e-106 < c < 8.6e-31

   1. Initial program 94.5%

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

   2. Taylor expanded in c around inf 64.7%

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

      1. unpow2 64.7%

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

      2. associate-/l* 64.7%

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

   4. Simplified 64.7%

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

      1. clear-num 64.7%

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

      2. inv-pow 64.7%

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

      3. associate-/l* 64.6%

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

   6. Applied egg-rr 64.6%

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

      1. unpow-1 64.6%

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

      2. associate-/l/ 64.7%

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

   8. Simplified 64.7%

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

   if 1.15000000000000007e-11 < c

   1. Initial program 47.2%

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

   2. Taylor expanded in c around inf 79.7%

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

      1. *-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. times-frac 87.6%

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

   4. Simplified 87.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-31}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{\frac{c}{b \cdot \frac{d}{c}}}\\ \mathbf{elif}\;c \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 9: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-47} \lor \neg \left(c \leq 1.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.5e-47) (not (<= c 1.5e-11)))
   (/ a (+ c (/ (* d d) c)))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.5e-47) || !(c <= 1.5e-11)) {
		tmp = a / (c + ((d * d) / c));
	} else {
		tmp = (1.0 / d) * (b + (a * (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 ((c <= (-5.5d-47)) .or. (.not. (c <= 1.5d-11))) then
        tmp = a / (c + ((d * d) / c))
    else
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.5e-47) || !(c <= 1.5e-11)) {
		tmp = a / (c + ((d * d) / c));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.5e-47) or not (c <= 1.5e-11):
		tmp = a / (c + ((d * d) / c))
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.5e-47) || !(c <= 1.5e-11))
		tmp = Float64(a / Float64(c + Float64(Float64(d * d) / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.5e-47) || ~((c <= 1.5e-11)))
		tmp = a / (c + ((d * d) / c));
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.5e-47], N[Not[LessEqual[c, 1.5e-11]], $MachinePrecision]], N[(a / N[(c + N[(N[(d * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.5000000000000002e-47 or 1.5e-11 < c

   1. Initial program 49.6%

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

   2. Taylor expanded in a around inf 43.5%

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

      1. associate-/l* 50.4%

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

      2. unpow2 50.4%

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

      3. unpow2 50.4%

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

      4. +-commutative 50.4%

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

      5. fma-udef 50.4%

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

   4. Simplified 50.4%

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

   5. Taylor expanded in d around 0 76.1%

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

      1. unpow2 76.1%

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

   7. Simplified 76.1%

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

   if -5.5000000000000002e-47 < c < 1.5e-11

   1. Initial program 78.2%

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

   2. Taylor expanded in c around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. times-frac 80.3%

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

   4. Simplified 80.3%

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

      1. +-commutative 80.3%

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

      2. frac-times 73.6%

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

      3. associate-/l/ 81.0%

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

      4. div-inv 81.0%

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

      5. div-inv 80.8%

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

      6. distribute-rgt-out 81.5%

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

      7. div-inv 81.5%

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

      8. *-commutative 81.5%

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

      9. associate-*l* 81.5%

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

      10. div-inv 81.6%

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

   6. Applied egg-rr 81.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{-47} \lor \neg \left(c \leq 1.5 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 10: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{-94} \lor \neg \left(c \leq 8.5 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.1e-94) (not (<= c 8.5e-121)))
   (/ a (+ c (/ (* d d) c)))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.1e-94) || !(c <= 8.5e-121)) {
		tmp = a / (c + ((d * d) / 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 <= (-3.1d-94)) .or. (.not. (c <= 8.5d-121))) then
        tmp = a / (c + ((d * d) / 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 <= -3.1e-94) || !(c <= 8.5e-121)) {
		tmp = a / (c + ((d * d) / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -3.1e-94) or not (c <= 8.5e-121):
		tmp = a / (c + ((d * d) / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.1e-94) || !(c <= 8.5e-121))
		tmp = Float64(a / Float64(c + Float64(Float64(d * d) / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.1e-94) || ~((c <= 8.5e-121)))
		tmp = a / (c + ((d * d) / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.1e-94], N[Not[LessEqual[c, 8.5e-121]], $MachinePrecision]], N[(a / N[(c + N[(N[(d * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.0999999999999998e-94 or 8.50000000000000025e-121 < c

   1. Initial program 58.5%

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

   2. Taylor expanded in a around inf 45.8%

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

      1. associate-/l* 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

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

      4. +-commutative 51.1%

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

      5. fma-udef 51.1%

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

   4. Simplified 51.1%

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

   5. Taylor expanded in d around 0 70.7%

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

      1. unpow2 70.7%

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

   7. Simplified 70.7%

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

   if -3.0999999999999998e-94 < c < 8.50000000000000025e-121

   1. Initial program 74.5%

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

   2. Taylor expanded in c around 0 79.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.1 \cdot 10^{-94} \lor \neg \left(c \leq 8.5 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11: 61.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -6.8e-7) (/ a c) (if (<= c 9.5e-121) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.8e-7) {
		tmp = a / c;
	} else if (c <= 9.5e-121) {
		tmp = b / d;
	} else {
		tmp = a / 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 <= (-6.8d-7)) then
        tmp = a / c
    else if (c <= 9.5d-121) then
        tmp = b / d
    else
        tmp = a / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -6.8e-7) {
		tmp = a / c;
	} else if (c <= 9.5e-121) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -6.8e-7:
		tmp = a / c
	elif c <= 9.5e-121:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -6.8e-7)
		tmp = Float64(a / c);
	elseif (c <= 9.5e-121)
		tmp = Float64(b / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -6.8e-7)
		tmp = a / c;
	elseif (c <= 9.5e-121)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -6.8e-7], N[(a / c), $MachinePrecision], If[LessEqual[c, 9.5e-121], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -6.79999999999999948e-7 or 9.4999999999999994e-121 < c

   1. Initial program 55.2%

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

   2. Taylor expanded in c around inf 69.2%

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

   if -6.79999999999999948e-7 < c < 9.4999999999999994e-121

   1. Initial program 76.1%

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

   2. Taylor expanded in c around 0 72.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 12: 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 64.1%

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

2. Taylor expanded in c around inf 46.7%

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

3. Final simplification 46.7%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

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 2023279 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :herbie-target
  (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))))