Complex division, real part

Percentage Accurate: 61.2% → 85.2%

Time: 9.5s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) INFINITY)
   (/ (/ (fma b d (* a c)) (hypot c 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))) <= ((double) INFINITY)) {
		tmp = (fma(b, d, (a * c)) / hypot(c, 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))) <= Inf)
		tmp = Float64(Float64(fma(b, d, Float64(a * c)) / hypot(c, 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], Infinity], N[(N[(N[(b * d + N[(a * c), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $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))) < +inf.0

   1. Initial program 73.7%

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

      1. *-un-lft-identity 73.7%

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

      2. add-sqr-sqrt 73.7%

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

      3. times-frac 73.6%

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

      4. hypot-define 73.6%

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

      5. fma-define 73.6%

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

      6. hypot-define 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. frac-times 73.7%

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

      2. *-un-lft-identity 73.7%

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

      3. fma-undefine 73.7%

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

      4. +-commutative 73.7%

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

      5. *-commutative 73.7%

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

      6. fma-undefine 73.7%

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

      7. associate-/r* 93.8%

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

      8. fma-undefine 93.8%

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

      9. *-commutative 93.8%

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

      10. fma-define 93.8%

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

   6. Applied egg-rr 93.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in c around inf 51.7%

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

      1. associate-/l* 56.7%

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

   5. Simplified 56.7%

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

      1. pow2 56.7%

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

      2. associate-*r/ 51.7%

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

      3. associate-/r* 52.2%

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

   7. Applied egg-rr 52.2%

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

      1. associate-/l/ 51.7%

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

      2. *-commutative 51.7%

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

      3. times-frac 66.4%

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

   9. Applied egg-rr 66.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq \infty:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(b, d, a \cdot c\right)}{\mathsf{hypot}\left(c, d\right)}}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -2.05 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-151}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{{d}^{2}}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2.05e+71)
     t_1
     (if (<= c -1.5e-107)
       t_0
       (if (<= c 1.75e-151)
         (+ (/ b d) (* a (/ c (pow d 2.0))))
         (if (<= c 1.8e+112) 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 <= -2.05e+71) {
		tmp = t_1;
	} else if (c <= -1.5e-107) {
		tmp = t_0;
	} else if (c <= 1.75e-151) {
		tmp = (b / d) + (a * (c / pow(d, 2.0)));
	} else if (c <= 1.8e+112) {
		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 <= (-2.05d+71)) then
        tmp = t_1
    else if (c <= (-1.5d-107)) then
        tmp = t_0
    else if (c <= 1.75d-151) then
        tmp = (b / d) + (a * (c / (d ** 2.0d0)))
    else if (c <= 1.8d+112) 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 <= -2.05e+71) {
		tmp = t_1;
	} else if (c <= -1.5e-107) {
		tmp = t_0;
	} else if (c <= 1.75e-151) {
		tmp = (b / d) + (a * (c / Math.pow(d, 2.0)));
	} else if (c <= 1.8e+112) {
		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 <= -2.05e+71:
		tmp = t_1
	elif c <= -1.5e-107:
		tmp = t_0
	elif c <= 1.75e-151:
		tmp = (b / d) + (a * (c / math.pow(d, 2.0)))
	elif c <= 1.8e+112:
		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 <= -2.05e+71)
		tmp = t_1;
	elseif (c <= -1.5e-107)
		tmp = t_0;
	elseif (c <= 1.75e-151)
		tmp = Float64(Float64(b / d) + Float64(a * Float64(c / (d ^ 2.0))));
	elseif (c <= 1.8e+112)
		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 <= -2.05e+71)
		tmp = t_1;
	elseif (c <= -1.5e-107)
		tmp = t_0;
	elseif (c <= 1.75e-151)
		tmp = (b / d) + (a * (c / (d ^ 2.0)));
	elseif (c <= 1.8e+112)
		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, -2.05e+71], t$95$1, If[LessEqual[c, -1.5e-107], t$95$0, If[LessEqual[c, 1.75e-151], N[(N[(b / d), $MachinePrecision] + N[(a * N[(c / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.8e+112], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.0500000000000001e71 or 1.8e112 < c

   1. Initial program 34.0%

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

   3. Taylor expanded in c around inf 75.7%

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

      1. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

      1. pow2 77.3%

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

      2. associate-*r/ 75.7%

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

      3. associate-/r* 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. associate-/l/ 75.7%

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

      2. *-commutative 75.7%

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

      3. times-frac 86.5%

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

   9. Applied egg-rr 86.5%

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

   if -2.0500000000000001e71 < c < -1.4999999999999999e-107 or 1.74999999999999998e-151 < c < 1.8e112

   1. Initial program 78.0%

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

   if -1.4999999999999999e-107 < c < 1.74999999999999998e-151

   1. Initial program 68.7%

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

   3. Taylor expanded in c around 0 89.4%

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

      1. associate-/l* 85.2%

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

   5. Simplified 85.2%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -1.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.75 \cdot 10^{-151}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{{d}^{2}}\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{{d}^{2}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.25e-21)
   (+ (/ b d) (* a (/ c (pow d 2.0))))
   (if (<= d 1.9e+67)
     (+ (/ a c) (/ (/ (* b d) c) c))
     (/ (+ b (* a (/ c d))) (hypot c d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e-21) {
		tmp = (b / d) + (a * (c / pow(d, 2.0)));
	} else if (d <= 1.9e+67) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (b + (a * (c / d))) / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.25e-21) {
		tmp = (b / d) + (a * (c / Math.pow(d, 2.0)));
	} else if (d <= 1.9e+67) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = (b + (a * (c / d))) / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.25e-21:
		tmp = (b / d) + (a * (c / math.pow(d, 2.0)))
	elif d <= 1.9e+67:
		tmp = (a / c) + (((b * d) / c) / c)
	else:
		tmp = (b + (a * (c / d))) / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.25e-21)
		tmp = Float64(Float64(b / d) + Float64(a * Float64(c / (d ^ 2.0))));
	elseif (d <= 1.9e+67)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	else
		tmp = Float64(Float64(b + Float64(a * Float64(c / d))) / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.25e-21)
		tmp = (b / d) + (a * (c / (d ^ 2.0)));
	elseif (d <= 1.9e+67)
		tmp = (a / c) + (((b * d) / c) / c);
	else
		tmp = (b + (a * (c / d))) / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.25e-21], N[(N[(b / d), $MachinePrecision] + N[(a * N[(c / N[Power[d, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.9e+67], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.24999999999999993e-21

   1. Initial program 50.3%

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

   3. Taylor expanded in c around 0 70.0%

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

      1. associate-/l* 71.4%

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

   5. Simplified 71.4%

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

   if -1.24999999999999993e-21 < d < 1.9000000000000001e67

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 78.2%

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

      3. associate-/r* 84.1%

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

   7. Applied egg-rr 84.1%

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

   if 1.9000000000000001e67 < d

   1. Initial program 39.1%

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

      1. *-un-lft-identity 39.1%

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

      2. add-sqr-sqrt 39.1%

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

      3. times-frac 39.1%

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

      4. hypot-define 39.1%

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

      5. fma-define 39.1%

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

      6. hypot-define 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. frac-times 39.1%

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

      2. *-un-lft-identity 39.1%

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

      3. fma-undefine 39.1%

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

      4. +-commutative 39.1%

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

      5. *-commutative 39.1%

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

      6. fma-undefine 39.1%

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

      7. associate-/r* 63.4%

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

      8. fma-undefine 63.3%

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

      9. *-commutative 63.3%

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

      10. fma-define 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in d around inf 82.0%

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

      1. associate-/l* 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.25 \cdot 10^{-21}:\\ \;\;\;\;\frac{b}{d} + a \cdot \frac{c}{{d}^{2}}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b + a \cdot \frac{c}{d}\\ \mathbf{if}\;d \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\frac{t\_0}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ b (* a (/ c d)))))
   (if (<= d -2e-21)
     (/ t_0 (- (hypot c d)))
     (if (<= d 1.9e+67) (+ (/ a c) (/ (/ (* b d) c) c)) (/ t_0 (hypot c d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b + (a * (c / d));
	double tmp;
	if (d <= -2e-21) {
		tmp = t_0 / -hypot(c, d);
	} else if (d <= 1.9e+67) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = t_0 / hypot(c, d);
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = b + (a * (c / d));
	double tmp;
	if (d <= -2e-21) {
		tmp = t_0 / -Math.hypot(c, d);
	} else if (d <= 1.9e+67) {
		tmp = (a / c) + (((b * d) / c) / c);
	} else {
		tmp = t_0 / Math.hypot(c, d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = b + (a * (c / d))
	tmp = 0
	if d <= -2e-21:
		tmp = t_0 / -math.hypot(c, d)
	elif d <= 1.9e+67:
		tmp = (a / c) + (((b * d) / c) / c)
	else:
		tmp = t_0 / math.hypot(c, d)
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(b + Float64(a * Float64(c / d)))
	tmp = 0.0
	if (d <= -2e-21)
		tmp = Float64(t_0 / Float64(-hypot(c, d)));
	elseif (d <= 1.9e+67)
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	else
		tmp = Float64(t_0 / hypot(c, d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = b + (a * (c / d));
	tmp = 0.0;
	if (d <= -2e-21)
		tmp = t_0 / -hypot(c, d);
	elseif (d <= 1.9e+67)
		tmp = (a / c) + (((b * d) / c) / c);
	else
		tmp = t_0 / hypot(c, d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2e-21], N[(t$95$0 / (-N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision])), $MachinePrecision], If[LessEqual[d, 1.9e+67], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.99999999999999982e-21

   1. Initial program 50.3%

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

      1. *-un-lft-identity 50.3%

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

      2. add-sqr-sqrt 50.3%

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

      3. times-frac 50.3%

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

      4. hypot-define 50.3%

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

      5. fma-define 50.3%

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

      6. hypot-define 70.7%

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

   4. Applied egg-rr 70.7%

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

      1. frac-times 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. fma-undefine 50.3%

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

      4. +-commutative 50.3%

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

      5. *-commutative 50.3%

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

      6. fma-undefine 50.3%

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

      7. associate-/r* 70.8%

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

      8. fma-undefine 70.8%

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

      9. *-commutative 70.8%

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

      10. fma-define 70.8%

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

   6. Applied egg-rr 70.8%

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

   7. Taylor expanded in d around -inf 75.5%

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

      1. distribute-lft-out 75.5%

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

      2. associate-/l* 78.4%

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

   9. Simplified 78.4%

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

   if -1.99999999999999982e-21 < d < 1.9000000000000001e67

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 78.2%

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

      3. associate-/r* 84.1%

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

   7. Applied egg-rr 84.1%

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

   if 1.9000000000000001e67 < d

   1. Initial program 39.1%

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

      1. *-un-lft-identity 39.1%

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

      2. add-sqr-sqrt 39.1%

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

      3. times-frac 39.1%

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

      4. hypot-define 39.1%

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

      5. fma-define 39.1%

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

      6. hypot-define 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. frac-times 39.1%

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

      2. *-un-lft-identity 39.1%

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

      3. fma-undefine 39.1%

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

      4. +-commutative 39.1%

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

      5. *-commutative 39.1%

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

      6. fma-undefine 39.1%

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

      7. associate-/r* 63.4%

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

      8. fma-undefine 63.3%

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

      9. *-commutative 63.3%

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

      10. fma-define 63.4%

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

   6. Applied egg-rr 63.4%

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

   7. Taylor expanded in d around inf 82.0%

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

      1. associate-/l* 86.3%

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

   9. Simplified 86.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2 \cdot 10^{-21}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{-\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+67}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a \cdot \frac{c}{d}}{\mathsf{hypot}\left(c, d\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{if}\;c \leq -8.8 \cdot 10^{+70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.76 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -8.8e+70)
     t_1
     (if (<= c -3.1e-107)
       t_0
       (if (<= c 4e-152) (/ b d) (if (<= c 1.76e+112) 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 <= -8.8e+70) {
		tmp = t_1;
	} else if (c <= -3.1e-107) {
		tmp = t_0;
	} else if (c <= 4e-152) {
		tmp = b / d;
	} else if (c <= 1.76e+112) {
		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 <= (-8.8d+70)) then
        tmp = t_1
    else if (c <= (-3.1d-107)) then
        tmp = t_0
    else if (c <= 4d-152) then
        tmp = b / d
    else if (c <= 1.76d+112) 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 <= -8.8e+70) {
		tmp = t_1;
	} else if (c <= -3.1e-107) {
		tmp = t_0;
	} else if (c <= 4e-152) {
		tmp = b / d;
	} else if (c <= 1.76e+112) {
		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 <= -8.8e+70:
		tmp = t_1
	elif c <= -3.1e-107:
		tmp = t_0
	elif c <= 4e-152:
		tmp = b / d
	elif c <= 1.76e+112:
		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 <= -8.8e+70)
		tmp = t_1;
	elseif (c <= -3.1e-107)
		tmp = t_0;
	elseif (c <= 4e-152)
		tmp = Float64(b / d);
	elseif (c <= 1.76e+112)
		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 <= -8.8e+70)
		tmp = t_1;
	elseif (c <= -3.1e-107)
		tmp = t_0;
	elseif (c <= 4e-152)
		tmp = b / d;
	elseif (c <= 1.76e+112)
		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, -8.8e+70], t$95$1, If[LessEqual[c, -3.1e-107], t$95$0, If[LessEqual[c, 4e-152], N[(b / d), $MachinePrecision], If[LessEqual[c, 1.76e+112], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.80000000000000003e70 or 1.75999999999999999e112 < c

   1. Initial program 34.0%

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

   3. Taylor expanded in c around inf 75.7%

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

      1. associate-/l* 77.3%

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

   5. Simplified 77.3%

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

      1. pow2 77.3%

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

      2. associate-*r/ 75.7%

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

      3. associate-/r* 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. associate-/l/ 75.7%

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

      2. *-commutative 75.7%

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

      3. times-frac 86.5%

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

   9. Applied egg-rr 86.5%

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

   if -8.80000000000000003e70 < c < -3.10000000000000022e-107 or 4.00000000000000026e-152 < c < 1.75999999999999999e112

   1. Initial program 78.0%

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

   if -3.10000000000000022e-107 < c < 4.00000000000000026e-152

   1. Initial program 68.7%

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

   3. Taylor expanded in c around 0 77.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{-152}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 1.76 \cdot 10^{+112}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-21} \lor \neg \left(d \leq 3.4 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.3e-21) (not (<= d 3.4e+67)))
   (/ b d)
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.3e-21) || !(d <= 3.4e+67)) {
		tmp = b / d;
	} 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) :: tmp
    if ((d <= (-2.3d-21)) .or. (.not. (d <= 3.4d+67))) then
        tmp = b / d
    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 tmp;
	if ((d <= -2.3e-21) || !(d <= 3.4e+67)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -2.3e-21) or not (d <= 3.4e+67):
		tmp = b / d
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.3e-21) || !(d <= 3.4e+67))
		tmp = Float64(b / d);
	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)
	tmp = 0.0;
	if ((d <= -2.3e-21) || ~((d <= 3.4e+67)))
		tmp = b / d;
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.3e-21], N[Not[LessEqual[d, 3.4e+67]], $MachinePrecision]], N[(b / d), $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 d < -2.29999999999999999e-21 or 3.4000000000000002e67 < d

   1. Initial program 45.8%

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

   3. Taylor expanded in c around 0 65.8%

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

   if -2.29999999999999999e-21 < d < 3.4000000000000002e67

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 78.2%

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

      3. associate-/r* 84.1%

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

   7. Applied egg-rr 84.1%

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

      1. associate-/l/ 78.2%

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

      2. *-commutative 78.2%

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

      3. times-frac 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.3 \cdot 10^{-21} \lor \neg \left(d \leq 3.4 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-21} \lor \neg \left(d \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.9e-21) (not (<= d 1.8e+67)))
   (/ b d)
   (+ (/ a c) (/ (/ (* b d) c) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e-21) || !(d <= 1.8e+67)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.9d-21)) .or. (.not. (d <= 1.8d+67))) then
        tmp = b / d
    else
        tmp = (a / c) + (((b * d) / c) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e-21) || !(d <= 1.8e+67)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + (((b * d) / c) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.9e-21) or not (d <= 1.8e+67):
		tmp = b / d
	else:
		tmp = (a / c) + (((b * d) / c) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.9e-21) || !(d <= 1.8e+67))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(Float64(b * d) / c) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.9e-21) || ~((d <= 1.8e+67)))
		tmp = b / d;
	else
		tmp = (a / c) + (((b * d) / c) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.9e-21], N[Not[LessEqual[d, 1.8e+67]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(N[(b * d), $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.8999999999999999e-21 or 1.7999999999999999e67 < d

   1. Initial program 45.8%

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

   3. Taylor expanded in c around 0 65.8%

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

   if -1.8999999999999999e-21 < d < 1.7999999999999999e67

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf 78.2%

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

      1. associate-/l* 77.1%

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

   5. Simplified 77.1%

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

      1. pow2 77.1%

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

      2. associate-*r/ 78.2%

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

      3. associate-/r* 84.1%

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

   7. Applied egg-rr 84.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-21} \lor \neg \left(d \leq 1.8 \cdot 10^{+67}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-22} \lor \neg \left(d \leq 3.4 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.2e-22) (not (<= d 3.4e+68))) (/ b d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.2e-22) || !(d <= 3.4e+68)) {
		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 ((d <= (-3.2d-22)) .or. (.not. (d <= 3.4d+68))) 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 ((d <= -3.2e-22) || !(d <= 3.4e+68)) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.2e-22) or not (d <= 3.4e+68):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.2e-22) || !(d <= 3.4e+68))
		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 ((d <= -3.2e-22) || ~((d <= 3.4e+68)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.2e-22], N[Not[LessEqual[d, 3.4e+68]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.19999999999999987e-22 or 3.40000000000000015e68 < d

   1. Initial program 45.8%

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

   3. Taylor expanded in c around 0 65.8%

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

   if -3.19999999999999987e-22 < d < 3.40000000000000015e68

   1. Initial program 71.4%

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

   3. Taylor expanded in c around inf 67.1%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.2 \cdot 10^{-22} \lor \neg \left(d \leq 3.4 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.8% 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 59.6%

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

3. Taylor expanded in c around inf 45.0%

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

4. Final simplification 45.0%

   \[\leadsto \frac{a}{c} \]
5. Add Preprocessing

Developer target: 99.3% accurate, 0.1× speedup

Math

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

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 2024039 
(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))))