Complex division, real part

Percentage Accurate: 61.6% → 80.0%

Time: 9.2s

Alternatives: 8

Speedup: 2.1×

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: 61.6% 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: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{if}\;c \leq -2.35 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ 1.0 (* c (/ (/ c b) d))))))
   (if (<= c -2.35e+98)
     t_0
     (if (<= c -1.6e-9)
       (* (fma a c (* b d)) (/ 1.0 (fma c c (* d d))))
       (if (<= c 5.2e-123)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 9.4e-54)
           (/ (+ (* b d) (* c a)) (+ (* d d) (* c c)))
           (if (<= c 1.85e+56) (+ (/ b d) (* (/ c d) (/ a d))) t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + (1.0 / (c * ((c / b) / d)));
	double tmp;
	if (c <= -2.35e+98) {
		tmp = t_0;
	} else if (c <= -1.6e-9) {
		tmp = fma(a, c, (b * d)) * (1.0 / fma(c, c, (d * d)));
	} else if (c <= 5.2e-123) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 9.4e-54) {
		tmp = ((b * d) + (c * a)) / ((d * d) + (c * c));
	} else if (c <= 1.85e+56) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(1.0 / Float64(c * Float64(Float64(c / b) / d))))
	tmp = 0.0
	if (c <= -2.35e+98)
		tmp = t_0;
	elseif (c <= -1.6e-9)
		tmp = Float64(fma(a, c, Float64(b * d)) * Float64(1.0 / fma(c, c, Float64(d * d))));
	elseif (c <= 5.2e-123)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 9.4e-54)
		tmp = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)));
	elseif (c <= 1.85e+56)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c * N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.35e+98], t$95$0, If[LessEqual[c, -1.6e-9], N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.2e-123], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.4e-54], N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.85e+56], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -2.34999999999999985e98 or 1.84999999999999998e56 < c

   1. Initial program 35.1%

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

   2. Taylor expanded in c around inf 78.5%

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

      1. unpow2 78.5%

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

      2. associate-/l* 79.7%

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

   4. Simplified 79.7%

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

      1. clear-num 79.7%

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

      2. inv-pow 79.7%

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

      3. metadata-eval 79.7%

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

      4. sqr-pow 76.1%

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

      5. associate-/l/ 72.7%

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

      6. *-commutative 72.7%

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

      7. times-frac 73.7%

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

      8. metadata-eval 73.7%

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

      9. metadata-eval 73.7%

         \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{\frac{c \cdot c}{d}}{b}\right)}^{\left(\frac{-1}{2}\right)} \]

      10. associate-/l/ 73.6%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c \cdot c}{b \cdot d}\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. *-commutative 73.6%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c \cdot c}{\color{blue}{d \cdot b}}\right)}^{\left(\frac{-1}{2}\right)} \]

      12. times-frac 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}}^{\left(\frac{-1}{2}\right)} \]

      13. metadata-eval 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

      14. metadata-eval 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \]

   6. Applied egg-rr 78.9%

      \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. pow-sqr 87.7%

         \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 87.7%

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

      3. unpow-1 87.7%

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

      4. associate-*l/ 85.8%

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

      5. *-lft-identity 85.8%

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

      6. times-frac 88.7%

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

      7. /-rgt-identity 88.7%

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

   8. Simplified 88.7%

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

   if -2.34999999999999985e98 < c < -1.60000000000000006e-9

   1. Initial program 72.3%

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

      1. div-inv 72.4%

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

      2. fma-def 72.4%

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

      3. fma-def 72.4%

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

   3. Applied egg-rr 72.4%

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

   if -1.60000000000000006e-9 < c < 5.1999999999999999e-123

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 88.2%

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

      1. unpow2 88.2%

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

      2. associate-/l* 89.3%

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

   4. Simplified 89.3%

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

      1. associate-/l* 94.1%

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

      2. *-rgt-identity 94.1%

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

      3. associate-/r/ 94.0%

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

      4. times-frac 95.6%

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

   6. Applied egg-rr 95.6%

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

      1. +-commutative 95.6%

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

      2. div-inv 95.4%

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

      3. distribute-rgt-out 96.4%

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

      4. div-inv 96.3%

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

      5. clear-num 96.4%

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

   8. Applied egg-rr 96.4%

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

   if 5.1999999999999999e-123 < c < 9.4e-54

   1. Initial program 80.6%

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

   if 9.4e-54 < c < 1.84999999999999998e56

   1. Initial program 53.6%

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

   2. Taylor expanded in c around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-/l* 69.8%

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

   4. Simplified 69.8%

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

      1. associate-/r/ 65.2%

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

      2. associate-*l/ 69.9%

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

      3. *-commutative 69.9%

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

      4. times-frac 69.9%

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

   6. Applied egg-rr 69.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.35 \cdot 10^{+98}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(a, c, b \cdot d\right) \cdot \frac{1}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 9.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+56}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \end{array} \]

Alternative 2: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ t_1 := \frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{if}\;c \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* b d) (* c a)) (+ (* d d) (* c c))))
        (t_1 (+ (/ a c) (/ 1.0 (* c (/ (/ c b) d))))))
   (if (<= c -3.2e+97)
     t_1
     (if (<= c -1.6e-9)
       t_0
       (if (<= c 1.5e-122)
         (* (/ 1.0 d) (+ b (* a (/ c d))))
         (if (<= c 4.3e-44)
           t_0
           (if (<= c 1.5e+59) (+ (/ b d) (* (/ c d) (/ a d))) t_1)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c));
	double t_1 = (a / c) + (1.0 / (c * ((c / b) / d)));
	double tmp;
	if (c <= -3.2e+97) {
		tmp = t_1;
	} else if (c <= -1.6e-9) {
		tmp = t_0;
	} else if (c <= 1.5e-122) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 4.3e-44) {
		tmp = t_0;
	} else if (c <= 1.5e+59) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c))
    t_1 = (a / c) + (1.0d0 / (c * ((c / b) / d)))
    if (c <= (-3.2d+97)) then
        tmp = t_1
    else if (c <= (-1.6d-9)) then
        tmp = t_0
    else if (c <= 1.5d-122) then
        tmp = (1.0d0 / d) * (b + (a * (c / d)))
    else if (c <= 4.3d-44) then
        tmp = t_0
    else if (c <= 1.5d+59) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c));
	double t_1 = (a / c) + (1.0 / (c * ((c / b) / d)));
	double tmp;
	if (c <= -3.2e+97) {
		tmp = t_1;
	} else if (c <= -1.6e-9) {
		tmp = t_0;
	} else if (c <= 1.5e-122) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else if (c <= 4.3e-44) {
		tmp = t_0;
	} else if (c <= 1.5e+59) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c))
	t_1 = (a / c) + (1.0 / (c * ((c / b) / d)))
	tmp = 0
	if c <= -3.2e+97:
		tmp = t_1
	elif c <= -1.6e-9:
		tmp = t_0
	elif c <= 1.5e-122:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	elif c <= 4.3e-44:
		tmp = t_0
	elif c <= 1.5e+59:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * d) + Float64(c * a)) / Float64(Float64(d * d) + Float64(c * c)))
	t_1 = Float64(Float64(a / c) + Float64(1.0 / Float64(c * Float64(Float64(c / b) / d))))
	tmp = 0.0
	if (c <= -3.2e+97)
		tmp = t_1;
	elseif (c <= -1.6e-9)
		tmp = t_0;
	elseif (c <= 1.5e-122)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	elseif (c <= 4.3e-44)
		tmp = t_0;
	elseif (c <= 1.5e+59)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * d) + (c * a)) / ((d * d) + (c * c));
	t_1 = (a / c) + (1.0 / (c * ((c / b) / d)));
	tmp = 0.0;
	if (c <= -3.2e+97)
		tmp = t_1;
	elseif (c <= -1.6e-9)
		tmp = t_0;
	elseif (c <= 1.5e-122)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	elseif (c <= 4.3e-44)
		tmp = t_0;
	elseif (c <= 1.5e+59)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * d), $MachinePrecision] + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c * N[(N[(c / b), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -3.2e+97], t$95$1, If[LessEqual[c, -1.6e-9], t$95$0, If[LessEqual[c, 1.5e-122], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4.3e-44], t$95$0, If[LessEqual[c, 1.5e+59], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.20000000000000016e97 or 1.5e59 < c

   1. Initial program 35.1%

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

   2. Taylor expanded in c around inf 78.5%

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

      1. unpow2 78.5%

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

      2. associate-/l* 79.7%

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

   4. Simplified 79.7%

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

      1. clear-num 79.7%

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

      2. inv-pow 79.7%

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

      3. metadata-eval 79.7%

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

      4. sqr-pow 76.1%

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

      5. associate-/l/ 72.7%

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

      6. *-commutative 72.7%

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

      7. times-frac 73.7%

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

      8. metadata-eval 73.7%

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

      9. metadata-eval 73.7%

         \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{\frac{c \cdot c}{d}}{b}\right)}^{\left(\frac{-1}{2}\right)} \]

      10. associate-/l/ 73.6%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c \cdot c}{b \cdot d}\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. *-commutative 73.6%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c \cdot c}{\color{blue}{d \cdot b}}\right)}^{\left(\frac{-1}{2}\right)} \]

      12. times-frac 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}}^{\left(\frac{-1}{2}\right)} \]

      13. metadata-eval 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

      14. metadata-eval 78.9%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \]

   6. Applied egg-rr 78.9%

      \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. pow-sqr 87.7%

         \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 87.7%

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

      3. unpow-1 87.7%

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

      4. associate-*l/ 85.8%

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

      5. *-lft-identity 85.8%

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

      6. times-frac 88.7%

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

      7. /-rgt-identity 88.7%

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

   8. Simplified 88.7%

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

   if -3.20000000000000016e97 < c < -1.60000000000000006e-9 or 1.50000000000000002e-122 < c < 4.30000000000000013e-44

   1. Initial program 75.3%

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

   if -1.60000000000000006e-9 < c < 1.50000000000000002e-122

   1. Initial program 69.3%

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

   2. Taylor expanded in c around 0 88.2%

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

      1. unpow2 88.2%

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

      2. associate-/l* 89.3%

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

   4. Simplified 89.3%

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

      1. associate-/l* 94.1%

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

      2. *-rgt-identity 94.1%

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

      3. associate-/r/ 94.0%

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

      4. times-frac 95.6%

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

   6. Applied egg-rr 95.6%

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

      1. +-commutative 95.6%

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

      2. div-inv 95.4%

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

      3. distribute-rgt-out 96.4%

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

      4. div-inv 96.3%

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

      5. clear-num 96.4%

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

   8. Applied egg-rr 96.4%

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

   if 4.30000000000000013e-44 < c < 1.5e59

   1. Initial program 53.6%

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

   2. Taylor expanded in c around 0 69.9%

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

      1. unpow2 69.9%

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

      2. associate-/l* 69.8%

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

   4. Simplified 69.8%

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

      1. associate-/r/ 65.2%

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

      2. associate-*l/ 69.9%

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

      3. *-commutative 69.9%

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

      4. times-frac 69.9%

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

   6. Applied egg-rr 69.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{elif}\;c \leq -1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{b \cdot d + c \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;c \leq 1.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \end{array} \]

Alternative 3: 76.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+88} \lor \neg \left(c \leq 10^{+57}\right):\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.05e+88) (not (<= c 1e+57)))
   (+ (/ a c) (/ 1.0 (* c (/ (/ c b) d))))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.05e+88) || !(c <= 1e+57)) {
		tmp = (a / c) + (1.0 / (c * ((c / b) / d)));
	} 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 <= (-2.05d+88)) .or. (.not. (c <= 1d+57))) then
        tmp = (a / c) + (1.0d0 / (c * ((c / b) / d)))
    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 <= -2.05e+88) || !(c <= 1e+57)) {
		tmp = (a / c) + (1.0 / (c * ((c / b) / d)));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.05e+88) or not (c <= 1e+57):
		tmp = (a / c) + (1.0 / (c * ((c / b) / d)))
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.05e+88) || !(c <= 1e+57))
		tmp = Float64(Float64(a / c) + Float64(1.0 / Float64(c * Float64(Float64(c / b) / d))));
	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 <= -2.05e+88) || ~((c <= 1e+57)))
		tmp = (a / c) + (1.0 / (c * ((c / b) / d)));
	else
		tmp = (1.0 / d) * (b + (a * (c / d)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.05e+88], N[Not[LessEqual[c, 1e+57]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(1.0 / N[(c * N[(N[(c / b), $MachinePrecision] / d), $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 < -2.05000000000000014e88 or 1.00000000000000005e57 < c

   1. Initial program 38.1%

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

   2. Taylor expanded in c around inf 77.9%

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

      1. unpow2 77.9%

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

      2. associate-/l* 78.1%

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

   4. Simplified 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

      3. metadata-eval 78.1%

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

      4. sqr-pow 74.6%

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

      5. associate-/l/ 69.3%

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

      6. *-commutative 69.3%

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

      7. times-frac 70.3%

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

      8. metadata-eval 70.3%

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

      9. metadata-eval 70.3%

         \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \cdot {\left(\frac{\frac{c \cdot c}{d}}{b}\right)}^{\left(\frac{-1}{2}\right)} \]

      10. associate-/l/ 70.1%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c \cdot c}{b \cdot d}\right)}}^{\left(\frac{-1}{2}\right)} \]

      11. *-commutative 70.1%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c \cdot c}{\color{blue}{d \cdot b}}\right)}^{\left(\frac{-1}{2}\right)} \]

      12. times-frac 75.2%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\color{blue}{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}}^{\left(\frac{-1}{2}\right)} \]

      13. metadata-eval 75.2%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(\frac{\color{blue}{-1}}{2}\right)} \]

      14. metadata-eval 75.2%

          \[\leadsto \frac{a}{c} + {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\color{blue}{-0.5}} \]

   6. Applied egg-rr 75.2%

      \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5} \cdot {\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{-0.5}} \]
   7. Step-by-step derivation

      1. pow-sqr 85.6%

         \[\leadsto \frac{a}{c} + \color{blue}{{\left(\frac{c}{d} \cdot \frac{c}{b}\right)}^{\left(2 \cdot -0.5\right)}} \]

      2. metadata-eval 85.6%

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

      3. unpow-1 85.6%

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

      4. associate-*l/ 84.7%

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

      5. *-lft-identity 84.7%

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

      6. times-frac 87.4%

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

      7. /-rgt-identity 87.4%

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

   8. Simplified 87.4%

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

   if -2.05000000000000014e88 < c < 1.00000000000000005e57

   1. Initial program 68.6%

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

   2. Taylor expanded in c around 0 75.2%

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

      1. unpow2 75.2%

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

      2. associate-/l* 76.4%

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

   4. Simplified 76.4%

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

      1. associate-/l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/r/ 79.5%

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

      4. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. +-commutative 81.6%

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

      2. div-inv 81.5%

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

      3. distribute-rgt-out 82.1%

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

      4. div-inv 82.1%

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

      5. clear-num 82.1%

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

   8. Applied egg-rr 82.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+88} \lor \neg \left(c \leq 10^{+57}\right):\\ \;\;\;\;\frac{a}{c} + \frac{1}{c \cdot \frac{\frac{c}{b}}{d}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 4: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+90} \lor \neg \left(c \leq 3.3 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e+90) (not (<= c 3.3e+56)))
   (/ a c)
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+90) || !(c <= 3.3e+56)) {
		tmp = a / 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 <= (-2.6d+90)) .or. (.not. (c <= 3.3d+56))) then
        tmp = a / 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 <= -2.6e+90) || !(c <= 3.3e+56)) {
		tmp = a / c;
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.6e+90) or not (c <= 3.3e+56):
		tmp = a / c
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e+90) || !(c <= 3.3e+56))
		tmp = Float64(a / 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 <= -2.6e+90) || ~((c <= 3.3e+56)))
		tmp = a / 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, -2.6e+90], N[Not[LessEqual[c, 3.3e+56]], $MachinePrecision]], N[(a / c), $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 < -2.5999999999999998e90 or 3.30000000000000002e56 < c

   1. Initial program 38.1%

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

   2. Taylor expanded in c around inf 77.1%

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

   if -2.5999999999999998e90 < c < 3.30000000000000002e56

   1. Initial program 68.6%

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

   2. Taylor expanded in c around 0 75.2%

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

      1. unpow2 75.2%

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

      2. associate-/l* 76.4%

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

   4. Simplified 76.4%

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

      1. associate-/l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/r/ 79.5%

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

      4. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. +-commutative 81.6%

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

      2. div-inv 81.5%

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

      3. distribute-rgt-out 82.1%

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

      4. div-inv 82.1%

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

      5. clear-num 82.1%

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

   8. Applied egg-rr 82.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+90} \lor \neg \left(c \leq 3.3 \cdot 10^{+56}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 5: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+88} \lor \neg \left(c \leq 2.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -6.5e+88) (not (<= c 2.5e+57)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (* (/ 1.0 d) (+ b (* a (/ c d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -6.5e+88) || !(c <= 2.5e+57)) {
		tmp = (a / c) + ((d / c) * (b / 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 <= (-6.5d+88)) .or. (.not. (c <= 2.5d+57))) then
        tmp = (a / c) + ((d / c) * (b / 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 <= -6.5e+88) || !(c <= 2.5e+57)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -6.5e+88) or not (c <= 2.5e+57):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -6.5e+88) || !(c <= 2.5e+57))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / 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 <= -6.5e+88) || ~((c <= 2.5e+57)))
		tmp = (a / c) + ((d / c) * (b / 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, -6.5e+88], N[Not[LessEqual[c, 2.5e+57]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / 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 < -6.5000000000000002e88 or 2.49999999999999986e57 < c

   1. Initial program 38.1%

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

   2. Taylor expanded in c around inf 77.9%

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

      1. unpow2 77.9%

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

      2. associate-/l* 78.1%

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

   4. Simplified 78.1%

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

      1. associate-/r/ 80.9%

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

      2. associate-*l/ 77.9%

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

      3. *-commutative 77.9%

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

      4. times-frac 85.4%

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

   6. Applied egg-rr 85.4%

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

   if -6.5000000000000002e88 < c < 2.49999999999999986e57

   1. Initial program 68.6%

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

   2. Taylor expanded in c around 0 75.2%

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

      1. unpow2 75.2%

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

      2. associate-/l* 76.4%

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

   4. Simplified 76.4%

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

      1. associate-/l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/r/ 79.5%

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

      4. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. +-commutative 81.6%

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

      2. div-inv 81.5%

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

      3. distribute-rgt-out 82.1%

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

      4. div-inv 82.1%

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

      5. clear-num 82.1%

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

   8. Applied egg-rr 82.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.5 \cdot 10^{+88} \lor \neg \left(c \leq 2.5 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \end{array} \]

Alternative 6: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.8e+87)
   (+ (/ a c) (* d (/ b (* c c))))
   (if (<= c 3e+56) (* (/ 1.0 d) (+ b (* a (/ c d)))) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.8e+87) {
		tmp = (a / c) + (d * (b / (c * c)));
	} else if (c <= 3e+56) {
		tmp = (1.0 / d) * (b + (a * (c / 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 <= (-1.8d+87)) then
        tmp = (a / c) + (d * (b / (c * c)))
    else if (c <= 3d+56) then
        tmp = (1.0d0 / d) * (b + (a * (c / 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 <= -1.8e+87) {
		tmp = (a / c) + (d * (b / (c * c)));
	} else if (c <= 3e+56) {
		tmp = (1.0 / d) * (b + (a * (c / d)));
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.8e+87:
		tmp = (a / c) + (d * (b / (c * c)))
	elif c <= 3e+56:
		tmp = (1.0 / d) * (b + (a * (c / d)))
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.8e+87)
		tmp = Float64(Float64(a / c) + Float64(d * Float64(b / Float64(c * c))));
	elseif (c <= 3e+56)
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(a * Float64(c / d))));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.8e+87)
		tmp = (a / c) + (d * (b / (c * c)));
	elseif (c <= 3e+56)
		tmp = (1.0 / d) * (b + (a * (c / d)));
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.8e+87], N[(N[(a / c), $MachinePrecision] + N[(d * N[(b / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3e+56], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.79999999999999997e87

   1. Initial program 39.4%

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

   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* 79.5%

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

   4. Simplified 79.5%

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

      1. associate-/r/ 85.5%

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

   6. Applied egg-rr 85.5%

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

   if -1.79999999999999997e87 < c < 3.00000000000000006e56

   1. Initial program 68.6%

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

   2. Taylor expanded in c around 0 75.2%

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

      1. unpow2 75.2%

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

      2. associate-/l* 76.4%

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

   4. Simplified 76.4%

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

      1. associate-/l* 79.5%

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

      2. *-rgt-identity 79.5%

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

      3. associate-/r/ 79.5%

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

      4. times-frac 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. +-commutative 81.6%

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

      2. div-inv 81.5%

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

      3. distribute-rgt-out 82.1%

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

      4. div-inv 82.1%

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

      5. clear-num 82.1%

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

   8. Applied egg-rr 82.1%

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

   if 3.00000000000000006e56 < c

   1. Initial program 36.9%

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

   2. Taylor expanded in c around inf 77.1%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+87}:\\ \;\;\;\;\frac{a}{c} + d \cdot \frac{b}{c \cdot c}\\ \mathbf{elif}\;c \leq 3 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + a \cdot \frac{c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 7: 63.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.7e-9) (/ a c) (if (<= c 4.4e+19) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.7e-9) {
		tmp = a / c;
	} else if (c <= 4.4e+19) {
		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 <= (-1.7d-9)) then
        tmp = a / c
    else if (c <= 4.4d+19) 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 <= -1.7e-9) {
		tmp = a / c;
	} else if (c <= 4.4e+19) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.7e-9:
		tmp = a / c
	elif c <= 4.4e+19:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.7e-9)
		tmp = Float64(a / c);
	elseif (c <= 4.4e+19)
		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 <= -1.7e-9)
		tmp = a / c;
	elseif (c <= 4.4e+19)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.7e-9], N[(a / c), $MachinePrecision], If[LessEqual[c, 4.4e+19], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.6999999999999999e-9 or 4.4e19 < c

   1. Initial program 44.7%

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

   2. Taylor expanded in c around inf 70.0%

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

   if -1.6999999999999999e-9 < c < 4.4e19

   1. Initial program 69.2%

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

   2. Taylor expanded in c around 0 75.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

2. Taylor expanded in c around inf 43.4%

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

3. Final simplification 43.4%

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

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 2023297 
(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))))