Complex division, real part

Percentage Accurate: 62.0% → 85.9%

Time: 8.1s

Alternatives: 8

Speedup: 1.3×

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: 62.0% 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: 85.9% accurate, 0.0× speedup

Math

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

FPCore

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

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 = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (b / d) + ((c / d) / (d / a));
	}
	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(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(d / a)));
	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[(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[(d / a), $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 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. add-sqr-sqrt 78.7%

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

      3. times-frac 78.6%

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

      4. hypot-def 78.6%

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

      5. fma-def 78.6%

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

      6. hypot-def 95.8%

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

   3. Applied egg-rr 95.8%

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

   if +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. Taylor expanded in c around 0 49.7%

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

      1. unpow2 49.7%

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

      2. times-frac 57.8%

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

   4. Simplified 57.8%

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

      1. clear-num 57.9%

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

      2. un-div-inv 57.9%

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

   6. Applied egg-rr 57.9%

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

4. Final simplification 89.0%

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

Alternative 2: 80.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -3.25 \cdot 10^{-201}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= c -5.5e+84)
     (+ (/ a c) (* (/ d c) (/ b c)))
     (if (<= c -3.25e-201)
       t_0
       (if (<= c 2.3e-25)
         (+ (/ b d) (/ (* a (/ c d)) d))
         (if (<= c 1.02e+95) t_0 (+ (/ a c) (/ (/ b c) (/ c 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 (c <= -5.5e+84) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= -3.25e-201) {
		tmp = t_0;
	} else if (c <= 2.3e-25) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else if (c <= 1.02e+95) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((b / 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) :: t_0
    real(8) :: tmp
    t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
    if (c <= (-5.5d+84)) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (c <= (-3.25d-201)) then
        tmp = t_0
    else if (c <= 2.3d-25) then
        tmp = (b / d) + ((a * (c / d)) / d)
    else if (c <= 1.02d+95) then
        tmp = t_0
    else
        tmp = (a / c) + ((b / c) / (c / 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 (c <= -5.5e+84) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (c <= -3.25e-201) {
		tmp = t_0;
	} else if (c <= 2.3e-25) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else if (c <= 1.02e+95) {
		tmp = t_0;
	} else {
		tmp = (a / c) + ((b / c) / (c / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((a * c) + (b * d)) / ((c * c) + (d * d))
	tmp = 0
	if c <= -5.5e+84:
		tmp = (a / c) + ((d / c) * (b / c))
	elif c <= -3.25e-201:
		tmp = t_0
	elif c <= 2.3e-25:
		tmp = (b / d) + ((a * (c / d)) / d)
	elif c <= 1.02e+95:
		tmp = t_0
	else:
		tmp = (a / c) + ((b / c) / (c / 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 (c <= -5.5e+84)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (c <= -3.25e-201)
		tmp = t_0;
	elseif (c <= 2.3e-25)
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	elseif (c <= 1.02e+95)
		tmp = t_0;
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / 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 (c <= -5.5e+84)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (c <= -3.25e-201)
		tmp = t_0;
	elseif (c <= 2.3e-25)
		tmp = (b / d) + ((a * (c / d)) / d);
	elseif (c <= 1.02e+95)
		tmp = t_0;
	else
		tmp = (a / c) + ((b / c) / (c / 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[c, -5.5e+84], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, -3.25e-201], t$95$0, If[LessEqual[c, 2.3e-25], N[(N[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.02e+95], t$95$0, N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.5000000000000004e84

   1. Initial program 45.4%

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

   2. Taylor expanded in c around inf 90.0%

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

      1. unpow2 90.0%

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

      2. times-frac 97.6%

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

   4. Simplified 97.6%

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

   if -5.5000000000000004e84 < c < -3.24999999999999987e-201 or 2.2999999999999999e-25 < c < 1.0200000000000001e95

   1. Initial program 80.8%

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

   if -3.24999999999999987e-201 < c < 2.2999999999999999e-25

   1. Initial program 70.0%

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

   2. Taylor expanded in c around 0 85.5%

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

      1. unpow2 85.5%

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

      2. times-frac 92.0%

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

   4. Simplified 92.0%

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

      1. associate-*r/ 93.1%

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

   6. Applied egg-rr 93.1%

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

   if 1.0200000000000001e95 < c

   1. Initial program 36.3%

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

   2. Taylor expanded in c around inf 64.4%

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

      1. unpow2 64.4%

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

      2. times-frac 78.2%

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

   4. Simplified 78.2%

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

      1. *-commutative 78.2%

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

      2. clear-num 78.3%

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

      3. un-div-inv 78.4%

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

   6. Applied egg-rr 78.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;c \leq -3.25 \cdot 10^{-201}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 1.02 \cdot 10^{+95}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 3: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -0.35:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (/ b c) (/ c d)))))
   (if (<= c -0.35)
     t_0
     (if (<= c 1.2e-24)
       (+ (/ b d) (* (/ c d) (/ a d)))
       (if (<= c 3.4e+24)
         (+ (/ a c) (/ b (* c (/ c d))))
         (if (<= c 4e+53) (/ b d) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + ((b / c) / (c / d));
	double tmp;
	if (c <= -0.35) {
		tmp = t_0;
	} else if (c <= 1.2e-24) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= 3.4e+24) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= 4e+53) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	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 / c) / (c / d))
    if (c <= (-0.35d0)) then
        tmp = t_0
    else if (c <= 1.2d-24) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (c <= 3.4d+24) then
        tmp = (a / c) + (b / (c * (c / d)))
    else if (c <= 4d+53) then
        tmp = b / d
    else
        tmp = t_0
    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 / c) / (c / d));
	double tmp;
	if (c <= -0.35) {
		tmp = t_0;
	} else if (c <= 1.2e-24) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (c <= 3.4e+24) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= 4e+53) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + ((b / c) / (c / d))
	tmp = 0
	if c <= -0.35:
		tmp = t_0
	elif c <= 1.2e-24:
		tmp = (b / d) + ((c / d) * (a / d))
	elif c <= 3.4e+24:
		tmp = (a / c) + (b / (c * (c / d)))
	elif c <= 4e+53:
		tmp = b / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / d)))
	tmp = 0.0
	if (c <= -0.35)
		tmp = t_0;
	elseif (c <= 1.2e-24)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (c <= 3.4e+24)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (c <= 4e+53)
		tmp = Float64(b / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + ((b / c) / (c / d));
	tmp = 0.0;
	if (c <= -0.35)
		tmp = t_0;
	elseif (c <= 1.2e-24)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (c <= 3.4e+24)
		tmp = (a / c) + (b / (c * (c / d)));
	elseif (c <= 4e+53)
		tmp = b / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -0.35], t$95$0, If[LessEqual[c, 1.2e-24], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 3.4e+24], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 4e+53], N[(b / d), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -0.34999999999999998 or 4e53 < c

   1. Initial program 51.2%

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

   2. Taylor expanded in c around inf 74.4%

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

      1. unpow2 74.4%

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

      2. times-frac 81.9%

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

   4. Simplified 81.9%

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

      1. *-commutative 81.9%

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

      2. clear-num 81.9%

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

      3. un-div-inv 82.0%

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

   6. Applied egg-rr 82.0%

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

   if -0.34999999999999998 < c < 1.1999999999999999e-24

   1. Initial program 73.4%

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

   2. Taylor expanded in c around 0 81.9%

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

      1. unpow2 81.9%

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

      2. times-frac 85.9%

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

   4. Simplified 85.9%

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

   if 1.1999999999999999e-24 < c < 3.4000000000000001e24

   1. Initial program 99.8%

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

   2. Taylor expanded in c around inf 80.8%

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

      1. unpow2 80.8%

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

      2. times-frac 80.6%

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

   4. Simplified 80.6%

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

      1. clear-num 80.6%

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

      2. frac-times 80.8%

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

      3. *-un-lft-identity 80.8%

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

   6. Applied egg-rr 80.8%

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

   if 3.4000000000000001e24 < c < 4e53

   1. Initial program 77.6%

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

   2. Taylor expanded in c around 0 78.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -0.35:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq 1.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 4: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \mathbf{if}\;c \leq -6.6 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ a c) (/ (/ b c) (/ c d)))))
   (if (<= c -6.6e-8)
     t_0
     (if (<= c 1.9e-23)
       (+ (/ b d) (/ (* a (/ c d)) d))
       (if (<= c 2.2e+24)
         (+ (/ a c) (/ b (* c (/ c d))))
         (if (<= c 2.2e+52) (/ b d) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a / c) + ((b / c) / (c / d));
	double tmp;
	if (c <= -6.6e-8) {
		tmp = t_0;
	} else if (c <= 1.9e-23) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else if (c <= 2.2e+24) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= 2.2e+52) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	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 / c) / (c / d))
    if (c <= (-6.6d-8)) then
        tmp = t_0
    else if (c <= 1.9d-23) then
        tmp = (b / d) + ((a * (c / d)) / d)
    else if (c <= 2.2d+24) then
        tmp = (a / c) + (b / (c * (c / d)))
    else if (c <= 2.2d+52) then
        tmp = b / d
    else
        tmp = t_0
    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 / c) / (c / d));
	double tmp;
	if (c <= -6.6e-8) {
		tmp = t_0;
	} else if (c <= 1.9e-23) {
		tmp = (b / d) + ((a * (c / d)) / d);
	} else if (c <= 2.2e+24) {
		tmp = (a / c) + (b / (c * (c / d)));
	} else if (c <= 2.2e+52) {
		tmp = b / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a / c) + ((b / c) / (c / d))
	tmp = 0
	if c <= -6.6e-8:
		tmp = t_0
	elif c <= 1.9e-23:
		tmp = (b / d) + ((a * (c / d)) / d)
	elif c <= 2.2e+24:
		tmp = (a / c) + (b / (c * (c / d)))
	elif c <= 2.2e+52:
		tmp = b / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / d)))
	tmp = 0.0
	if (c <= -6.6e-8)
		tmp = t_0;
	elseif (c <= 1.9e-23)
		tmp = Float64(Float64(b / d) + Float64(Float64(a * Float64(c / d)) / d));
	elseif (c <= 2.2e+24)
		tmp = Float64(Float64(a / c) + Float64(b / Float64(c * Float64(c / d))));
	elseif (c <= 2.2e+52)
		tmp = Float64(b / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a / c) + ((b / c) / (c / d));
	tmp = 0.0;
	if (c <= -6.6e-8)
		tmp = t_0;
	elseif (c <= 1.9e-23)
		tmp = (b / d) + ((a * (c / d)) / d);
	elseif (c <= 2.2e+24)
		tmp = (a / c) + (b / (c * (c / d)));
	elseif (c <= 2.2e+52)
		tmp = b / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.6e-8], t$95$0, If[LessEqual[c, 1.9e-23], N[(N[(b / d), $MachinePrecision] + N[(N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.2e+24], N[(N[(a / c), $MachinePrecision] + N[(b / N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.2e+52], N[(b / d), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.59999999999999954e-8 or 2.2e52 < c

   1. Initial program 52.1%

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

   2. Taylor expanded in c around inf 74.0%

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

      1. unpow2 74.0%

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

      2. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.4%

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

      3. un-div-inv 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -6.59999999999999954e-8 < c < 1.90000000000000006e-23

   1. Initial program 73.4%

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

   2. Taylor expanded in c around 0 81.9%

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

      1. unpow2 81.9%

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

      2. times-frac 85.9%

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

   4. Simplified 85.9%

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

      1. associate-*r/ 86.6%

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

   6. Applied egg-rr 86.6%

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

   if 1.90000000000000006e-23 < c < 2.20000000000000002e24

   1. Initial program 99.8%

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

   2. Taylor expanded in c around inf 80.8%

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

      1. unpow2 80.8%

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

      2. times-frac 80.6%

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

   4. Simplified 80.6%

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

      1. clear-num 80.6%

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

      2. frac-times 80.8%

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

      3. *-un-lft-identity 80.8%

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

   6. Applied egg-rr 80.8%

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

   if 2.20000000000000002e24 < c < 2.2e52

   1. Initial program 71.2%

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

   2. Taylor expanded in c around 0 86.3%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{b}{d} + \frac{a \cdot \frac{c}{d}}{d}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\frac{a}{c} + \frac{b}{c \cdot \frac{c}{d}}\\ \mathbf{elif}\;c \leq 2.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 5: 70.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{-35} \lor \neg \left(d \leq 3.9 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.1e-35) (not (<= d 3.9e+118)))
   (/ b d)
   (+ (/ a c) (/ (/ b c) (/ c d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.1e-35) || !(d <= 3.9e+118)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((b / 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 ((d <= (-4.1d-35)) .or. (.not. (d <= 3.9d+118))) then
        tmp = b / d
    else
        tmp = (a / c) + ((b / 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 ((d <= -4.1e-35) || !(d <= 3.9e+118)) {
		tmp = b / d;
	} else {
		tmp = (a / c) + ((b / c) / (c / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.1e-35) or not (d <= 3.9e+118):
		tmp = b / d
	else:
		tmp = (a / c) + ((b / c) / (c / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.1e-35) || !(d <= 3.9e+118))
		tmp = Float64(b / d);
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(b / c) / Float64(c / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.1e-35) || ~((d <= 3.9e+118)))
		tmp = b / d;
	else
		tmp = (a / c) + ((b / c) / (c / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.1e-35], N[Not[LessEqual[d, 3.9e+118]], $MachinePrecision]], N[(b / d), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] / N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.10000000000000026e-35 or 3.9e118 < d

   1. Initial program 54.5%

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

   2. Taylor expanded in c around 0 74.9%

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

   if -4.10000000000000026e-35 < d < 3.9e118

   1. Initial program 73.8%

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

   2. Taylor expanded in c around inf 70.7%

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

      1. unpow2 70.7%

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

      2. times-frac 76.8%

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

   4. Simplified 76.8%

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

      1. *-commutative 76.8%

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

      2. clear-num 76.8%

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

      3. un-div-inv 76.8%

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

   6. Applied egg-rr 76.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.1 \cdot 10^{-35} \lor \neg \left(d \leq 3.9 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{\frac{b}{c}}{\frac{c}{d}}\\ \end{array} \]

Alternative 6: 70.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -5.2e-35)
   (/ b d)
   (if (<= d 6.6e+117) (+ (/ a c) (* (/ d c) (/ b c))) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -5.2e-35) {
		tmp = b / d;
	} else if (d <= 6.6e+117) {
		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 (d <= (-5.2d-35)) then
        tmp = b / d
    else if (d <= 6.6d+117) 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 (d <= -5.2e-35) {
		tmp = b / d;
	} else if (d <= 6.6e+117) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -5.2e-35:
		tmp = b / d
	elif d <= 6.6e+117:
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -5.2e-35)
		tmp = Float64(b / d);
	elseif (d <= 6.6e+117)
		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 (d <= -5.2e-35)
		tmp = b / d;
	elseif (d <= 6.6e+117)
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -5.2e-35], N[(b / d), $MachinePrecision], If[LessEqual[d, 6.6e+117], 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 d < -5.20000000000000009e-35 or 6.5999999999999996e117 < d

   1. Initial program 54.5%

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

   2. Taylor expanded in c around 0 74.9%

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

   if -5.20000000000000009e-35 < d < 6.5999999999999996e117

   1. Initial program 73.8%

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

   2. Taylor expanded in c around inf 70.7%

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

      1. unpow2 70.7%

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

      2. times-frac 76.8%

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

   4. Simplified 76.8%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 7: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-23} \lor \neg \left(c \leq 3.8 \cdot 10^{+24}\right) \land c \leq 2.65 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -3.7e+80)
   (/ a c)
   (if (or (<= c 1.7e-23) (and (not (<= c 3.8e+24)) (<= c 2.65e+53)))
     (/ b d)
     (/ a c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -3.7e+80) {
		tmp = a / c;
	} else if ((c <= 1.7e-23) || (!(c <= 3.8e+24) && (c <= 2.65e+53))) {
		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 <= (-3.7d+80)) then
        tmp = a / c
    else if ((c <= 1.7d-23) .or. (.not. (c <= 3.8d+24)) .and. (c <= 2.65d+53)) 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 <= -3.7e+80) {
		tmp = a / c;
	} else if ((c <= 1.7e-23) || (!(c <= 3.8e+24) && (c <= 2.65e+53))) {
		tmp = b / d;
	} else {
		tmp = a / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -3.7e+80:
		tmp = a / c
	elif (c <= 1.7e-23) or (not (c <= 3.8e+24) and (c <= 2.65e+53)):
		tmp = b / d
	else:
		tmp = a / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -3.7e+80)
		tmp = Float64(a / c);
	elseif ((c <= 1.7e-23) || (!(c <= 3.8e+24) && (c <= 2.65e+53)))
		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 <= -3.7e+80)
		tmp = a / c;
	elseif ((c <= 1.7e-23) || (~((c <= 3.8e+24)) && (c <= 2.65e+53)))
		tmp = b / d;
	else
		tmp = a / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -3.7e+80], N[(a / c), $MachinePrecision], If[Or[LessEqual[c, 1.7e-23], And[N[Not[LessEqual[c, 3.8e+24]], $MachinePrecision], LessEqual[c, 2.65e+53]]], N[(b / d), $MachinePrecision], N[(a / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -3.69999999999999996e80 or 1.7e-23 < c < 3.80000000000000015e24 or 2.6500000000000001e53 < c

   1. Initial program 51.8%

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

   2. Taylor expanded in c around inf 72.6%

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

   if -3.69999999999999996e80 < c < 1.7e-23 or 3.80000000000000015e24 < c < 2.6500000000000001e53

   1. Initial program 73.7%

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

   2. Taylor expanded in c around 0 69.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.7 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-23} \lor \neg \left(c \leq 3.8 \cdot 10^{+24}\right) \land c \leq 2.65 \cdot 10^{+53}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \]

Alternative 8: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

2. Taylor expanded in c around inf 41.6%

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

3. Final simplification 41.6%

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

Developer target: 99.5% 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 2023274 
(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))))