Complex division, real part

Percentage Accurate: 61.2% → 85.1%

Time: 12.2s

Alternatives: 11

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: 61.2% 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: 85.1% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) 5e+305)
   (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
   (+ (/ b d) (* (/ c d) (/ a d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((a * c) + (b * d)) / ((c * c) + (d * d))) <= 5e+305) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 5e+305)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(a * c + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $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))) < 5.00000000000000009e305

   1. Initial program 82.9%

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

      1. *-un-lft-identity 82.9%

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

      2. add-sqr-sqrt 82.9%

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

      3. times-frac 82.8%

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

      4. hypot-def 82.8%

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

      5. fma-def 82.8%

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

      6. hypot-def 98.1%

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

   3. Applied egg-rr 98.1%

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

   if 5.00000000000000009e305 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.0%

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

   2. Taylor expanded in c around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 60.1%

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

   4. Simplified 60.1%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 2: 77.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 -5e-281)
     t_0
     (if (<= t_0 0.0)
       (* (/ c (hypot c d)) (/ a (hypot c d)))
       (if (<= t_0 5e+305) t_0 (+ (/ b d) (* (/ c d) (/ a d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (t_0 <= -5e-281) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	} else if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

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 tmp;
	if (t_0 <= -5e-281) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (c / Math.hypot(c, d)) * (a / Math.hypot(c, d));
	} else if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if t_0 <= -5e-281:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (c / math.hypot(c, d)) * (a / math.hypot(c, d))
	elif t_0 <= 5e+305:
		tmp = t_0
	else:
		tmp = (b / d) + ((c / d) * (a / d))
	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)))
	tmp = 0.0
	if (t_0 <= -5e-281)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(c / hypot(c, d)) * Float64(a / hypot(c, d)));
	elseif (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (t_0 <= -5e-281)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (c / hypot(c, d)) * (a / hypot(c, d));
	elseif (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = (b / d) + ((c / d) * (a / d));
	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]}, If[LessEqual[t$95$0, -5e-281], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(c / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(a / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+305], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -4.9999999999999998e-281 or -0.0 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 5.00000000000000009e305

   1. Initial program 95.2%

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

   if -4.9999999999999998e-281 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d))) < -0.0

   1. Initial program 64.1%

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

   2. Taylor expanded in a around inf 64.1%

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

      1. *-commutative 64.1%

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

   4. Simplified 64.1%

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

      1. add-sqr-sqrt 64.1%

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

      2. hypot-udef 64.1%

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

      3. hypot-udef 64.1%

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

      4. times-frac 80.6%

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

   6. Applied egg-rr 80.6%

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

   if 5.00000000000000009e305 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.0%

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

   2. Taylor expanded in c around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 60.1%

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

   4. Simplified 60.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq -5 \cdot 10^{-281}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 0:\\ \;\;\;\;\frac{c}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{a}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{elif}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 3: 72.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq 5 \cdot 10^{+305}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 5e+305) t_0 (+ (/ b d) (* (/ c d) (/ a d))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	double tmp;
	if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (t_0 <= 5d+305) then
        tmp = t_0
    else
        tmp = (b / d) + ((c / d) * (a / d))
    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 tmp;
	if (t_0 <= 5e+305) {
		tmp = t_0;
	} else {
		tmp = (b / d) + ((c / d) * (a / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if t_0 <= 5e+305:
		tmp = t_0
	else:
		tmp = (b / d) + ((c / d) * (a / d))
	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)))
	tmp = 0.0
	if (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d));
	tmp = 0.0;
	if (t_0 <= 5e+305)
		tmp = t_0;
	else
		tmp = (b / d) + ((c / d) * (a / d));
	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]}, If[LessEqual[t$95$0, 5e+305], t$95$0, N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $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))) < 5.00000000000000009e305

   1. Initial program 82.9%

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

   if 5.00000000000000009e305 < (/.f64 (+.f64 (*.f64 a c) (*.f64 b d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 15.0%

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

   2. Taylor expanded in c around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 60.1%

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

   4. Simplified 60.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \end{array} \]

Alternative 4: 68.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+39} \lor \neg \left(c \leq 2.15 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e+39) (not (<= c 2.15e+46)))
   (+ (/ a c) (* (/ d c) (/ b c)))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+39) || !(c <= 2.15e+46)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.4d+39)) .or. (.not. (c <= 2.15d+46))) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+39) || !(c <= 2.15e+46)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e+39) or not (c <= 2.15e+46):
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e+39) || !(c <= 2.15e+46))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.4e+39) || ~((c <= 2.15e+46)))
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e+39], N[Not[LessEqual[c, 2.15e+46]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.4000000000000001e39 or 2.15000000000000002e46 < c

   1. Initial program 54.0%

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

   2. Taylor expanded in c around inf 72.1%

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

      1. *-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. times-frac 80.9%

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

   4. Simplified 80.9%

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

   if -2.4000000000000001e39 < c < 2.15000000000000002e46

   1. Initial program 76.9%

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

   2. Taylor expanded in c around 0 64.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+39} \lor \neg \left(c \leq 2.15 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+39} \lor \neg \left(c \leq 1.3 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.4e+39) (not (<= c 1.3e+46)))
   (+ (/ a c) (/ (* d (/ b c)) c))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+39) || !(c <= 1.3e+46)) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.4d+39)) .or. (.not. (c <= 1.3d+46))) then
        tmp = (a / c) + ((d * (b / c)) / c)
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.4e+39) || !(c <= 1.3e+46)) {
		tmp = (a / c) + ((d * (b / c)) / c);
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.4e+39) or not (c <= 1.3e+46):
		tmp = (a / c) + ((d * (b / c)) / c)
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.4e+39) || !(c <= 1.3e+46))
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.4e+39) || ~((c <= 1.3e+46)))
		tmp = (a / c) + ((d * (b / c)) / c);
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.4e+39], N[Not[LessEqual[c, 1.3e+46]], $MachinePrecision]], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.4000000000000001e39 or 1.30000000000000007e46 < c

   1. Initial program 54.0%

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

   2. Taylor expanded in c around inf 72.1%

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

      1. *-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. times-frac 80.9%

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

   4. Simplified 80.9%

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

      1. associate-*l/ 81.8%

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

   6. Applied egg-rr 81.8%

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

   if -2.4000000000000001e39 < c < 1.30000000000000007e46

   1. Initial program 76.9%

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

   2. Taylor expanded in c around 0 64.6%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.4 \cdot 10^{+39} \lor \neg \left(c \leq 1.3 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 6: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.02e-10)
   (+ (/ a c) (/ (* b (/ d c)) c))
   (if (<= c 3.8e+46) (/ b d) (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.02e-10) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 3.8e+46) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.02d-10)) then
        tmp = (a / c) + ((b * (d / c)) / c)
    else if (c <= 3.8d+46) then
        tmp = b / d
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.02e-10) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 3.8e+46) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.02e-10:
		tmp = (a / c) + ((b * (d / c)) / c)
	elif c <= 3.8e+46:
		tmp = b / d
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.02e-10)
		tmp = Float64(Float64(a / c) + Float64(Float64(b * Float64(d / c)) / c));
	elseif (c <= 3.8e+46)
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.02e-10)
		tmp = (a / c) + ((b * (d / c)) / c);
	elseif (c <= 3.8e+46)
		tmp = b / d;
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.02e-10], N[(N[(a / c), $MachinePrecision] + N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.8e+46], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.01999999999999997e-10

   1. Initial program 59.9%

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

   2. Taylor expanded in c around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. times-frac 77.3%

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

   4. Simplified 77.3%

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

      1. associate-*r/ 78.7%

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

   6. Applied egg-rr 78.7%

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

   if -1.01999999999999997e-10 < c < 3.7999999999999999e46

   1. Initial program 77.3%

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

   2. Taylor expanded in c around 0 65.5%

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

   if 3.7999999999999999e46 < c

   1. Initial program 49.2%

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

   2. Taylor expanded in c around inf 74.7%

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

      1. *-commutative 74.7%

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

      2. unpow2 74.7%

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

      3. times-frac 79.4%

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

   4. Simplified 79.4%

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

      1. associate-*l/ 81.5%

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

   6. Applied egg-rr 81.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 7: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -8.6e-11)
   (+ (/ a c) (/ (* b (/ d c)) c))
   (if (<= c 2.2e+47)
     (+ (/ b d) (* c (/ (/ a d) d)))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.6e-11) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 2.2e+47) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-8.6d-11)) then
        tmp = (a / c) + ((b * (d / c)) / c)
    else if (c <= 2.2d+47) then
        tmp = (b / d) + (c * ((a / d) / d))
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -8.6e-11) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 2.2e+47) {
		tmp = (b / d) + (c * ((a / d) / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -8.6e-11:
		tmp = (a / c) + ((b * (d / c)) / c)
	elif c <= 2.2e+47:
		tmp = (b / d) + (c * ((a / d) / d))
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -8.6e-11)
		tmp = Float64(Float64(a / c) + Float64(Float64(b * Float64(d / c)) / c));
	elseif (c <= 2.2e+47)
		tmp = Float64(Float64(b / d) + Float64(c * Float64(Float64(a / d) / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -8.6e-11)
		tmp = (a / c) + ((b * (d / c)) / c);
	elseif (c <= 2.2e+47)
		tmp = (b / d) + (c * ((a / d) / d));
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -8.6e-11], N[(N[(a / c), $MachinePrecision] + N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.2e+47], N[(N[(b / d), $MachinePrecision] + N[(c * N[(N[(a / d), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.60000000000000003e-11

   1. Initial program 59.9%

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

   2. Taylor expanded in c around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. times-frac 77.3%

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

   4. Simplified 77.3%

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

      1. associate-*r/ 78.7%

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

   6. Applied egg-rr 78.7%

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

   if -8.60000000000000003e-11 < c < 2.1999999999999999e47

   1. Initial program 77.4%

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

   2. Taylor expanded in c around 0 81.1%

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

      1. unpow2 81.1%

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

      2. associate-/l* 78.2%

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

   4. Simplified 78.2%

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

   5. Taylor expanded in a around 0 81.1%

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

      1. unpow2 81.1%

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

      2. associate-*l/ 74.5%

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

      3. *-commutative 74.5%

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

      4. associate-/r* 76.4%

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

   7. Simplified 76.4%

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

   if 2.1999999999999999e47 < c

   1. Initial program 48.3%

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

   2. Taylor expanded in c around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 80.8%

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

   4. Simplified 80.8%

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

      1. associate-*l/ 82.9%

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

   6. Applied egg-rr 82.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+47}:\\ \;\;\;\;\frac{b}{d} + c \cdot \frac{\frac{a}{d}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 8: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3e-10)
   (+ (/ a c) (/ (* b (/ d c)) c))
   (if (<= c 3.5e+48)
     (+ (/ b d) (* (/ c d) (/ a d)))
     (+ (/ a c) (/ (* d (/ b c)) c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-10) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 3.5e+48) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-3d-10)) then
        tmp = (a / c) + ((b * (d / c)) / c)
    else if (c <= 3.5d+48) then
        tmp = (b / d) + ((c / d) * (a / d))
    else
        tmp = (a / c) + ((d * (b / c)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3e-10) {
		tmp = (a / c) + ((b * (d / c)) / c);
	} else if (c <= 3.5e+48) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else {
		tmp = (a / c) + ((d * (b / c)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3e-10:
		tmp = (a / c) + ((b * (d / c)) / c)
	elif c <= 3.5e+48:
		tmp = (b / d) + ((c / d) * (a / d))
	else:
		tmp = (a / c) + ((d * (b / c)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3e-10)
		tmp = Float64(Float64(a / c) + Float64(Float64(b * Float64(d / c)) / c));
	elseif (c <= 3.5e+48)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d * Float64(b / c)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -3e-10)
		tmp = (a / c) + ((b * (d / c)) / c);
	elseif (c <= 3.5e+48)
		tmp = (b / d) + ((c / d) * (a / d));
	else
		tmp = (a / c) + ((d * (b / c)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3e-10], N[(N[(a / c), $MachinePrecision] + N[(N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.5e+48], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d * N[(b / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3e-10

   1. Initial program 59.9%

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

   2. Taylor expanded in c around inf 67.7%

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

      1. *-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. times-frac 77.3%

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

   4. Simplified 77.3%

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

      1. associate-*r/ 78.7%

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

   6. Applied egg-rr 78.7%

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

   if -3e-10 < c < 3.4999999999999997e48

   1. Initial program 77.4%

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

   2. Taylor expanded in c around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. unpow2 81.1%

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

      3. times-frac 81.4%

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

   4. Simplified 81.4%

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

   if 3.4999999999999997e48 < c

   1. Initial program 48.3%

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

   2. Taylor expanded in c around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. times-frac 80.8%

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

   4. Simplified 80.8%

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

      1. associate-*l/ 82.9%

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

   6. Applied egg-rr 82.9%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{c} + \frac{b \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq 3.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d \cdot \frac{b}{c}}{c}\\ \end{array} \]

Alternative 9: 63.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.75e-30) (/ a c) (if (<= c 4.2e+91) (/ b d) (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.75e-30) {
		tmp = a / c;
	} else if (c <= 4.2e+91) {
		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 <= (-2.75d-30)) then
        tmp = a / c
    else if (c <= 4.2d+91) 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 <= -2.75e-30) {
		tmp = a / c;
	} else if (c <= 4.2e+91) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -2.75e-30:
		tmp = a / c
	elif c <= 4.2e+91:
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.75e-30)
		tmp = Float64(a / c);
	elseif (c <= 4.2e+91)
		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 <= -2.75e-30)
		tmp = a / c;
	elseif (c <= 4.2e+91)
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.75e-30], N[(a / c), $MachinePrecision], If[LessEqual[c, 4.2e+91], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -2.74999999999999988e-30 or 4.20000000000000015e91 < c

   1. Initial program 53.4%

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

   2. Taylor expanded in c around inf 67.9%

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

   if -2.74999999999999988e-30 < c < 4.20000000000000015e91

   1. Initial program 77.7%

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

   2. Taylor expanded in c around 0 63.8%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.75 \cdot 10^{-30}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 10: 43.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (if (<= d -9.2e+196) (/ a d) (/ a c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -9.2e+196) {
		tmp = a / 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 <= (-9.2d+196)) then
        tmp = a / 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 <= -9.2e+196) {
		tmp = a / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -9.2e+196:
		tmp = a / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -9.2e+196)
		tmp = Float64(a / d);
	else
		tmp = Float64(a / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -9.2e+196)
		tmp = a / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -9.2e+196], N[(a / d), $MachinePrecision], N[(a / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.19999999999999922e196

   1. Initial program 50.3%

      \[\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \]
   2. 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-def 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-def 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-def 71.0%

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

   3. Applied egg-rr 71.0%

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

   4. Taylor expanded in c around -inf 26.8%

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

      1. neg-mul-1 26.8%

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

   6. Simplified 26.8%

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

   7. Taylor expanded in d around -inf 26.7%

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

   if -9.19999999999999922e196 < d

   1. Initial program 68.1%

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

   2. Taylor expanded in c around inf 47.4%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.2 \cdot 10^{+196}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 11: 42.5% 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 66.2%

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

2. Taylor expanded in c around inf 42.8%

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

3. Final simplification 42.8%

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

Developer target: 99.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (Math.abs(d) < Math.abs(c)) {
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if math.fabs(d) < math.fabs(c):
		tmp = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)))
	return tmp

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023275 
(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))))