Complex division, real part

Percentage Accurate: 61.8% → 85.4%

Time: 14.1s

Alternatives: 13

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: 61.8% 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: 85.4% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;t_0 \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{\mathsf{fma}\left(a, c, b \cdot d\right)}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))))
   (if (<= t_0 (- INFINITY))
     (+ (/ a c) (* (/ d c) (/ b c)))
     (if (<= t_0 INFINITY)
       (* (/ 1.0 (hypot c d)) (/ (fma a c (* b d)) (hypot c d)))
       (* (/ d (hypot d c)) (/ b (hypot d c)))))))

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 <= -((double) INFINITY)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (fma(a, c, (b * d)) / hypot(c, d));
	} else {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(fma(a, c, Float64(b * d)) / hypot(c, d)));
	else
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return 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, (-Infinity)], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 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[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $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))) < -inf.0

   1. Initial program 33.3%

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

   2. Taylor expanded in c around inf 48.7%

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

      1. unpow2 48.7%

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

      2. times-frac 67.4%

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

   4. Simplified 67.4%

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

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

   1. Initial program 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. add-sqr-sqrt 76.8%

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

      3. times-frac 76.8%

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

      4. hypot-def 76.8%

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

      5. fma-def 76.9%

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

      6. hypot-def 97.1%

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

   3. Applied egg-rr 97.1%

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

   if +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 a around 0 1.5%

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

      1. add-sqr-sqrt 1.5%

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

      2. hypot-udef 1.5%

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

      3. hypot-udef 1.5%

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

      4. times-frac 57.8%

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

      5. hypot-udef 4.1%

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

      6. +-commutative 4.1%

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

      7. hypot-def 57.8%

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

      8. hypot-udef 4.1%

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

      9. +-commutative 4.1%

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

      10. hypot-def 57.8%

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

   4. Applied egg-rr 57.8%

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

4. Final simplification 88.4%

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

Alternative 2: 85.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot c + b \cdot d\\ t_1 := \frac{t_0}{c \cdot c + d \cdot d}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(c, d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{d}{\mathsf{hypot}\left(d, c\right)} \cdot \frac{b}{\mathsf{hypot}\left(d, c\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (* a c) (* b d))) (t_1 (/ t_0 (+ (* c c) (* d d)))))
   (if (<= t_1 (- INFINITY))
     (+ (/ a c) (* (/ d c) (/ b c)))
     (if (<= t_1 INFINITY)
       (* (/ 1.0 (hypot c d)) (/ t_0 (hypot c d)))
       (* (/ d (hypot d c)) (/ b (hypot d c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	} else {
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (a * c) + (b * d);
	double t_1 = t_0 / ((c * c) + (d * d));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (1.0 / Math.hypot(c, d)) * (t_0 / Math.hypot(c, d));
	} else {
		tmp = (d / Math.hypot(d, c)) * (b / Math.hypot(d, c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (a * c) + (b * d)
	t_1 = t_0 / ((c * c) + (d * d))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (a / c) + ((d / c) * (b / c))
	elif t_1 <= math.inf:
		tmp = (1.0 / math.hypot(c, d)) * (t_0 / math.hypot(c, d))
	else:
		tmp = (d / math.hypot(d, c)) * (b / math.hypot(d, c))
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(a * c) + Float64(b * d))
	t_1 = Float64(t_0 / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(t_0 / hypot(c, d)));
	else
		tmp = Float64(Float64(d / hypot(d, c)) * Float64(b / hypot(d, c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (a * c) + (b * d);
	t_1 = t_0 / ((c * c) + (d * d));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (t_1 <= Inf)
		tmp = (1.0 / hypot(c, d)) * (t_0 / hypot(c, d));
	else
		tmp = (d / hypot(d, c)) * (b / hypot(d, c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(d / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b / N[Sqrt[d ^ 2 + c ^ 2], $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))) < -inf.0

   1. Initial program 33.3%

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

   2. Taylor expanded in c around inf 48.7%

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

      1. unpow2 48.7%

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

      2. times-frac 67.4%

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

   4. Simplified 67.4%

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

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

   1. Initial program 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. add-sqr-sqrt 76.8%

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

      3. times-frac 76.8%

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

      4. hypot-def 76.8%

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

      5. fma-def 76.9%

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

      6. hypot-def 97.1%

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

   3. Applied egg-rr 97.1%

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

      1. fma-def 97.1%

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

      2. +-commutative 97.1%

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

   5. Applied egg-rr 97.1%

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

   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 a around 0 1.5%

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

      1. add-sqr-sqrt 1.5%

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

      2. hypot-udef 1.5%

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

      3. hypot-udef 1.5%

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

      4. times-frac 57.8%

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

      5. hypot-udef 4.1%

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

      6. +-commutative 4.1%

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

      7. hypot-def 57.8%

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

      8. hypot-udef 4.1%

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

      9. +-commutative 4.1%

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

      10. hypot-def 57.8%

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

   4. Applied egg-rr 57.8%

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

4. Final simplification 88.4%

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

Alternative 3: 78.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -8.2e+69)
   (+ (/ b d) (/ c (* d (/ d a))))
   (if (<= d 1.85e-93)
     (+ (/ a c) (* (/ d c) (/ b c)))
     (if (<= d 2.25e+79)
       (/ (+ (* a c) (* b d)) (+ (* c c) (* d d)))
       (* (/ 1.0 (hypot c d)) (+ b (/ c (/ d a))))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8.2e+69) {
		tmp = (b / d) + (c / (d * (d / a)));
	} else if (d <= 1.85e-93) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 2.25e+79) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8.2e+69) {
		tmp = (b / d) + (c / (d * (d / a)));
	} else if (d <= 1.85e-93) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 2.25e+79) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = (1.0 / Math.hypot(c, d)) * (b + (c / (d / a)));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -8.2e+69:
		tmp = (b / d) + (c / (d * (d / a)))
	elif d <= 1.85e-93:
		tmp = (a / c) + ((d / c) * (b / c))
	elif d <= 2.25e+79:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = (1.0 / math.hypot(c, d)) * (b + (c / (d / a)))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -8.2e+69)
		tmp = Float64(Float64(b / d) + Float64(c / Float64(d * Float64(d / a))));
	elseif (d <= 1.85e-93)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (d <= 2.25e+79)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = Float64(Float64(1.0 / hypot(c, d)) * Float64(b + Float64(c / Float64(d / a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -8.2e+69)
		tmp = (b / d) + (c / (d * (d / a)));
	elseif (d <= 1.85e-93)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (d <= 2.25e+79)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = (1.0 / hypot(c, d)) * (b + (c / (d / a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -8.2e+69], N[(N[(b / d), $MachinePrecision] + N[(c / N[(d * N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.85e-93], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.25e+79], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[c ^ 2 + d ^ 2], $MachinePrecision]), $MachinePrecision] * N[(b + N[(c / N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -8.1999999999999998e69

   1. Initial program 51.0%

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

   2. Taylor expanded in c around 0 86.0%

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

      1. unpow2 86.0%

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

      2. times-frac 90.9%

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

   4. Simplified 90.9%

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

      1. *-commutative 90.9%

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

      2. clear-num 90.9%

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

      3. frac-times 91.2%

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

      4. *-un-lft-identity 91.2%

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

   6. Applied egg-rr 91.2%

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

   if -8.1999999999999998e69 < d < 1.85000000000000001e-93

   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 77.1%

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

      1. unpow2 77.1%

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

      2. times-frac 85.5%

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

   4. Simplified 85.5%

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

   if 1.85000000000000001e-93 < d < 2.24999999999999997e79

   1. Initial program 76.0%

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

   if 2.24999999999999997e79 < d

   1. Initial program 43.1%

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

      1. *-un-lft-identity 43.1%

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

      2. add-sqr-sqrt 43.1%

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

      3. times-frac 43.2%

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

      4. hypot-def 43.2%

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

      5. fma-def 43.2%

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

      6. hypot-def 64.1%

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

   3. Applied egg-rr 64.1%

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

   4. Taylor expanded in c around 0 80.5%

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

      1. associate-/l* 83.3%

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

   6. Simplified 83.3%

      \[\leadsto \frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \color{blue}{\left(b + \frac{c}{\frac{d}{a}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+69}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{-93}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 2.25 \cdot 10^{+79}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(c, d\right)} \cdot \left(b + \frac{c}{\frac{d}{a}}\right)\\ \end{array} \]

Alternative 4: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{if}\;d \leq -7 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ c (* d (/ d a))))))
   (if (<= d -7e+68)
     t_0
     (if (<= d 4.4e-95)
       (+ (/ a c) (* (/ d c) (/ b c)))
       (if (<= d 6.4e+62) (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (c / (d * (d / a)));
	double tmp;
	if (d <= -7e+68) {
		tmp = t_0;
	} else if (d <= 4.4e-95) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 6.4e+62) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * 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 = (b / d) + (c / (d * (d / a)))
    if (d <= (-7d+68)) then
        tmp = t_0
    else if (d <= 4.4d-95) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (d <= 6.4d+62) then
        tmp = ((a * c) + (b * d)) / ((c * c) + (d * 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 = (b / d) + (c / (d * (d / a)));
	double tmp;
	if (d <= -7e+68) {
		tmp = t_0;
	} else if (d <= 4.4e-95) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 6.4e+62) {
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (c / (d * (d / a)))
	tmp = 0
	if d <= -7e+68:
		tmp = t_0
	elif d <= 4.4e-95:
		tmp = (a / c) + ((d / c) * (b / c))
	elif d <= 6.4e+62:
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(c / Float64(d * Float64(d / a))))
	tmp = 0.0
	if (d <= -7e+68)
		tmp = t_0;
	elseif (d <= 4.4e-95)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (d <= 6.4e+62)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (c / (d * (d / a)));
	tmp = 0.0;
	if (d <= -7e+68)
		tmp = t_0;
	elseif (d <= 4.4e-95)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (d <= 6.4e+62)
		tmp = ((a * c) + (b * d)) / ((c * c) + (d * d));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(c / N[(d * N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -7e+68], t$95$0, If[LessEqual[d, 4.4e-95], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.4e+62], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -6.99999999999999955e68 or 6.39999999999999968e62 < d

   1. Initial program 48.0%

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

   2. Taylor expanded in c around 0 78.2%

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

      1. unpow2 78.2%

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

      2. times-frac 82.6%

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

   4. Simplified 82.6%

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

      1. *-commutative 82.6%

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

      2. clear-num 83.5%

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

      3. frac-times 86.2%

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

      4. *-un-lft-identity 86.2%

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

   6. Applied egg-rr 86.2%

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

   if -6.99999999999999955e68 < d < 4.3999999999999998e-95

   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 77.1%

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

      1. unpow2 77.1%

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

      2. times-frac 85.5%

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

   4. Simplified 85.5%

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

   if 4.3999999999999998e-95 < d < 6.39999999999999968e62

   1. Initial program 78.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7 \cdot 10^{+68}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;d \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 6.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \end{array} \]

Alternative 5: 74.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b d) (/ c (* d (/ d a))))))
   (if (<= d -1.15e+67)
     t_0
     (if (<= d 8.2e-74)
       (+ (/ a c) (* (/ d c) (/ b c)))
       (if (<= d 4.3e+33)
         (/ (* b d) (+ (* c c) (* d d)))
         (if (<= d 2.1e+80) (/ a (+ c (/ (* d d) c))) t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / d) + (c / (d * (d / a)));
	double tmp;
	if (d <= -1.15e+67) {
		tmp = t_0;
	} else if (d <= 8.2e-74) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 4.3e+33) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2.1e+80) {
		tmp = a / (c + ((d * d) / c));
	} 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 = (b / d) + (c / (d * (d / a)))
    if (d <= (-1.15d+67)) then
        tmp = t_0
    else if (d <= 8.2d-74) then
        tmp = (a / c) + ((d / c) * (b / c))
    else if (d <= 4.3d+33) then
        tmp = (b * d) / ((c * c) + (d * d))
    else if (d <= 2.1d+80) then
        tmp = a / (c + ((d * d) / c))
    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 = (b / d) + (c / (d * (d / a)));
	double tmp;
	if (d <= -1.15e+67) {
		tmp = t_0;
	} else if (d <= 8.2e-74) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else if (d <= 4.3e+33) {
		tmp = (b * d) / ((c * c) + (d * d));
	} else if (d <= 2.1e+80) {
		tmp = a / (c + ((d * d) / c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b / d) + (c / (d * (d / a)))
	tmp = 0
	if d <= -1.15e+67:
		tmp = t_0
	elif d <= 8.2e-74:
		tmp = (a / c) + ((d / c) * (b / c))
	elif d <= 4.3e+33:
		tmp = (b * d) / ((c * c) + (d * d))
	elif d <= 2.1e+80:
		tmp = a / (c + ((d * d) / c))
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / d) + Float64(c / Float64(d * Float64(d / a))))
	tmp = 0.0
	if (d <= -1.15e+67)
		tmp = t_0;
	elseif (d <= 8.2e-74)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	elseif (d <= 4.3e+33)
		tmp = Float64(Float64(b * d) / Float64(Float64(c * c) + Float64(d * d)));
	elseif (d <= 2.1e+80)
		tmp = Float64(a / Float64(c + Float64(Float64(d * d) / c)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b / d) + (c / (d * (d / a)));
	tmp = 0.0;
	if (d <= -1.15e+67)
		tmp = t_0;
	elseif (d <= 8.2e-74)
		tmp = (a / c) + ((d / c) * (b / c));
	elseif (d <= 4.3e+33)
		tmp = (b * d) / ((c * c) + (d * d));
	elseif (d <= 2.1e+80)
		tmp = a / (c + ((d * d) / c));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / d), $MachinePrecision] + N[(c / N[(d * N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.15e+67], t$95$0, If[LessEqual[d, 8.2e-74], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 4.3e+33], N[(N[(b * d), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.1e+80], N[(a / N[(c + N[(N[(d * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.1499999999999999e67 or 2.10000000000000001e80 < d

   1. Initial program 47.2%

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

   2. Taylor expanded in c around 0 79.5%

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

      1. unpow2 79.5%

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

      2. times-frac 84.2%

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

   4. Simplified 84.2%

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

      1. *-commutative 84.2%

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

      2. clear-num 85.1%

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

      3. frac-times 88.1%

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

      4. *-un-lft-identity 88.1%

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

   6. Applied egg-rr 88.1%

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

   if -1.1499999999999999e67 < d < 8.20000000000000063e-74

   1. Initial program 68.4%

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

   2. Taylor expanded in c around inf 77.3%

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

      1. unpow2 77.3%

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

      2. times-frac 85.6%

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

   4. Simplified 85.6%

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

   if 8.20000000000000063e-74 < d < 4.30000000000000028e33

   1. Initial program 82.2%

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

   2. Taylor expanded in a around 0 76.2%

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

   if 4.30000000000000028e33 < d < 2.10000000000000001e80

   1. Initial program 65.5%

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

   2. Taylor expanded in a around inf 46.5%

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

      1. *-commutative 46.5%

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

      2. associate-/l* 58.2%

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

      3. unpow2 58.2%

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

      4. unpow2 58.2%

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

      5. +-commutative 58.2%

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

      6. fma-udef 58.2%

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

   4. Simplified 58.2%

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

   5. Taylor expanded in c around 0 69.5%

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

      1. unpow2 69.5%

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

   7. Simplified 69.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{elif}\;d \leq 8.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{elif}\;d \leq 4.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{b \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{+80}:\\ \;\;\;\;\frac{a}{c + \frac{d \cdot d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \end{array} \]

Alternative 6: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-16} \lor \neg \left(c \leq 2.1 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.8e-16) (not (<= c 2.1e-7)))
   (/ a (+ c (* d (/ d c))))
   (* (/ 1.0 d) (+ b (/ (* a c) d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-16) || !(c <= 2.1e-7)) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-4.8d-16)) .or. (.not. (c <= 2.1d-7))) then
        tmp = a / (c + (d * (d / c)))
    else
        tmp = (1.0d0 / d) * (b + ((a * c) / d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.8e-16) || !(c <= 2.1e-7)) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -4.8e-16) or not (c <= 2.1e-7):
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = (1.0 / d) * (b + ((a * c) / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.8e-16) || !(c <= 2.1e-7))
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(Float64(a * c) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -4.8e-16) || ~((c <= 2.1e-7)))
		tmp = a / (c + (d * (d / c)));
	else
		tmp = (1.0 / d) * (b + ((a * c) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.8e-16], N[Not[LessEqual[c, 2.1e-7]], $MachinePrecision]], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.8000000000000001e-16 or 2.1e-7 < c

   1. Initial program 52.3%

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

   2. Taylor expanded in a around inf 45.1%

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

      1. *-commutative 45.1%

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

      2. associate-/l* 54.0%

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

      3. unpow2 54.0%

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

      4. unpow2 54.0%

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

      5. +-commutative 54.0%

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

      6. fma-udef 54.0%

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

   4. Simplified 54.0%

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

   5. Taylor expanded in c around 0 75.5%

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

      1. unpow2 75.5%

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

   7. Simplified 75.5%

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

   8. Taylor expanded in d around 0 75.5%

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

      1. unpow2 75.5%

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

      2. associate-*r/ 77.7%

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

   10. Simplified 77.7%

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

   if -4.8000000000000001e-16 < c < 2.1e-7

   1. Initial program 69.2%

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

      1. *-un-lft-identity 69.2%

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

      2. add-sqr-sqrt 69.2%

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

      3. times-frac 69.2%

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

      4. hypot-def 69.2%

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

      5. fma-def 69.2%

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

      6. hypot-def 84.4%

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

   3. Applied egg-rr 84.4%

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

   4. Taylor expanded in c around 0 46.8%

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

   5. Taylor expanded in c around 0 81.4%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{-16} \lor \neg \left(c \leq 2.1 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

Alternative 7: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+69} \lor \neg \left(d \leq 9.2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -8.2e+69) (not (<= d 9.2e-17)))
   (* (/ 1.0 d) (+ b (/ (* a c) d)))
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e+69) || !(d <= 9.2e-17)) {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-8.2d+69)) .or. (.not. (d <= 9.2d-17))) then
        tmp = (1.0d0 / d) * (b + ((a * c) / d))
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -8.2e+69) || !(d <= 9.2e-17)) {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -8.2e+69) or not (d <= 9.2e-17):
		tmp = (1.0 / d) * (b + ((a * c) / d))
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -8.2e+69) || !(d <= 9.2e-17))
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(Float64(a * c) / d)));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -8.2e+69) || ~((d <= 9.2e-17)))
		tmp = (1.0 / d) * (b + ((a * c) / d));
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -8.2e+69], N[Not[LessEqual[d, 9.2e-17]], $MachinePrecision]], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -8.1999999999999998e69 or 9.20000000000000035e-17 < d

   1. Initial program 52.2%

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

      1. *-un-lft-identity 52.2%

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

      2. add-sqr-sqrt 52.2%

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

      3. times-frac 52.2%

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

      4. hypot-def 52.2%

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

      5. fma-def 52.2%

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

      6. hypot-def 68.4%

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

   3. Applied egg-rr 68.4%

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

   4. Taylor expanded in c around 0 54.8%

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

   5. Taylor expanded in c around 0 79.1%

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

   if -8.1999999999999998e69 < d < 9.20000000000000035e-17

   1. Initial program 68.8%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. unpow2 75.5%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.2 \cdot 10^{+69} \lor \neg \left(d \leq 9.2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 8: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+67} \lor \neg \left(d \leq 3.7 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.15e+67) (not (<= d 3.7e-19)))
   (+ (/ b d) (/ c (* d (/ d a))))
   (+ (/ a c) (* (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e+67) || !(d <= 3.7e-19)) {
		tmp = (b / d) + (c / (d * (d / a)));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-1.15d+67)) .or. (.not. (d <= 3.7d-19))) then
        tmp = (b / d) + (c / (d * (d / a)))
    else
        tmp = (a / c) + ((d / c) * (b / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.15e+67) || !(d <= 3.7e-19)) {
		tmp = (b / d) + (c / (d * (d / a)));
	} else {
		tmp = (a / c) + ((d / c) * (b / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.15e+67) or not (d <= 3.7e-19):
		tmp = (b / d) + (c / (d * (d / a)))
	else:
		tmp = (a / c) + ((d / c) * (b / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.15e+67) || !(d <= 3.7e-19))
		tmp = Float64(Float64(b / d) + Float64(c / Float64(d * Float64(d / a))));
	else
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.15e+67) || ~((d <= 3.7e-19)))
		tmp = (b / d) + (c / (d * (d / a)));
	else
		tmp = (a / c) + ((d / c) * (b / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.15e+67], N[Not[LessEqual[d, 3.7e-19]], $MachinePrecision]], N[(N[(b / d), $MachinePrecision] + N[(c / N[(d * N[(d / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.1499999999999999e67 or 3.70000000000000005e-19 < d

   1. Initial program 52.2%

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

   2. Taylor expanded in c around 0 75.6%

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

      1. unpow2 75.6%

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

      2. times-frac 79.4%

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

   4. Simplified 79.4%

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

      1. *-commutative 79.4%

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

      2. clear-num 80.2%

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

      3. frac-times 82.6%

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

      4. *-un-lft-identity 82.6%

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

   6. Applied egg-rr 82.6%

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

   if -1.1499999999999999e67 < d < 3.70000000000000005e-19

   1. Initial program 68.8%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. unpow2 75.5%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+67} \lor \neg \left(d \leq 3.7 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{b}{d} + \frac{c}{d \cdot \frac{d}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \end{array} \]

Alternative 9: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.15e+67)
   (+ (/ b d) (* (/ c d) (/ a d)))
   (if (<= d 1.7e-17)
     (+ (/ a c) (* (/ d c) (/ b c)))
     (* (/ 1.0 d) (+ b (/ (* a c) d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e+67) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= 1.7e-17) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (d <= (-1.15d+67)) then
        tmp = (b / d) + ((c / d) * (a / d))
    else if (d <= 1.7d-17) then
        tmp = (a / c) + ((d / c) * (b / c))
    else
        tmp = (1.0d0 / d) * (b + ((a * c) / d))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.15e+67) {
		tmp = (b / d) + ((c / d) * (a / d));
	} else if (d <= 1.7e-17) {
		tmp = (a / c) + ((d / c) * (b / c));
	} else {
		tmp = (1.0 / d) * (b + ((a * c) / d));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.15e+67:
		tmp = (b / d) + ((c / d) * (a / d))
	elif d <= 1.7e-17:
		tmp = (a / c) + ((d / c) * (b / c))
	else:
		tmp = (1.0 / d) * (b + ((a * c) / d))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.15e+67)
		tmp = Float64(Float64(b / d) + Float64(Float64(c / d) * Float64(a / d)));
	elseif (d <= 1.7e-17)
		tmp = Float64(Float64(a / c) + Float64(Float64(d / c) * Float64(b / c)));
	else
		tmp = Float64(Float64(1.0 / d) * Float64(b + Float64(Float64(a * c) / d)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.15e+67)
		tmp = (b / d) + ((c / d) * (a / d));
	elseif (d <= 1.7e-17)
		tmp = (a / c) + ((d / c) * (b / c));
	else
		tmp = (1.0 / d) * (b + ((a * c) / d));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.15e+67], N[(N[(b / d), $MachinePrecision] + N[(N[(c / d), $MachinePrecision] * N[(a / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.7e-17], N[(N[(a / c), $MachinePrecision] + N[(N[(d / c), $MachinePrecision] * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / d), $MachinePrecision] * N[(b + N[(N[(a * c), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.1499999999999999e67

   1. Initial program 51.0%

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

   2. Taylor expanded in c around 0 86.0%

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

      1. unpow2 86.0%

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

      2. times-frac 90.9%

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

   4. Simplified 90.9%

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

   if -1.1499999999999999e67 < d < 1.6999999999999999e-17

   1. Initial program 68.8%

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

   2. Taylor expanded in c around inf 75.5%

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

      1. unpow2 75.5%

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

      2. times-frac 83.2%

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

   4. Simplified 83.2%

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

   if 1.6999999999999999e-17 < d

   1. Initial program 53.1%

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

      1. *-un-lft-identity 53.1%

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

      2. add-sqr-sqrt 53.1%

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

      3. times-frac 53.1%

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

      4. hypot-def 53.1%

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

      5. fma-def 53.1%

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

      6. hypot-def 67.9%

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

   3. Applied egg-rr 67.9%

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

   4. Taylor expanded in c around 0 74.1%

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

   5. Taylor expanded in c around 0 72.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;\frac{b}{d} + \frac{c}{d} \cdot \frac{a}{d}\\ \mathbf{elif}\;d \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{a}{c} + \frac{d}{c} \cdot \frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{d} \cdot \left(b + \frac{a \cdot c}{d}\right)\\ \end{array} \]

Alternative 10: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{-37} \lor \neg \left(c \leq 1.12 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -5.2e-37) (not (<= c 1.12e-25)))
   (/ a (+ c (* d (/ d c))))
   (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e-37) || !(c <= 1.12e-25)) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-5.2d-37)) .or. (.not. (c <= 1.12d-25))) then
        tmp = a / (c + (d * (d / c)))
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -5.2e-37) || !(c <= 1.12e-25)) {
		tmp = a / (c + (d * (d / c)));
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -5.2e-37) or not (c <= 1.12e-25):
		tmp = a / (c + (d * (d / c)))
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -5.2e-37) || !(c <= 1.12e-25))
		tmp = Float64(a / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -5.2e-37) || ~((c <= 1.12e-25)))
		tmp = a / (c + (d * (d / c)));
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -5.2e-37], N[Not[LessEqual[c, 1.12e-25]], $MachinePrecision]], N[(a / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -5.19999999999999959e-37 or 1.12e-25 < c

   1. Initial program 53.2%

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

   2. Taylor expanded in a around inf 44.6%

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

      1. *-commutative 44.6%

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

      2. associate-/l* 52.8%

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

      3. unpow2 52.8%

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

      4. unpow2 52.8%

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

      5. +-commutative 52.8%

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

      6. fma-udef 52.8%

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

   4. Simplified 52.8%

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

   5. Taylor expanded in c around 0 72.7%

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

      1. unpow2 72.7%

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

   7. Simplified 72.7%

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

   8. Taylor expanded in d around 0 72.7%

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

      1. unpow2 72.7%

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

      2. associate-*r/ 74.7%

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

   10. Simplified 74.7%

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

   if -5.19999999999999959e-37 < c < 1.12e-25

   1. Initial program 69.7%

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

   2. Taylor expanded in c around 0 66.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -5.2 \cdot 10^{-37} \lor \neg \left(c \leq 1.12 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{a}{c + d \cdot \frac{d}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 11: 45.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -8.5e+170) (/ a d) (if (<= d 2.7e+218) (/ a c) (/ a d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8.5e+170) {
		tmp = a / d;
	} else if (d <= 2.7e+218) {
		tmp = a / c;
	} else {
		tmp = 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) :: tmp
    if (d <= (-8.5d+170)) then
        tmp = a / d
    else if (d <= 2.7d+218) then
        tmp = a / c
    else
        tmp = a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -8.5e+170) {
		tmp = a / d;
	} else if (d <= 2.7e+218) {
		tmp = a / c;
	} else {
		tmp = a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -8.5e+170:
		tmp = a / d
	elif d <= 2.7e+218:
		tmp = a / c
	else:
		tmp = a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -8.5e+170)
		tmp = Float64(a / d);
	elseif (d <= 2.7e+218)
		tmp = Float64(a / c);
	else
		tmp = Float64(a / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -8.5e+170)
		tmp = a / d;
	elseif (d <= 2.7e+218)
		tmp = a / c;
	else
		tmp = a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -8.5e+170], N[(a / d), $MachinePrecision], If[LessEqual[d, 2.7e+218], N[(a / c), $MachinePrecision], N[(a / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -8.5000000000000004e170 or 2.70000000000000013e218 < d

   1. Initial program 41.4%

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

      1. *-un-lft-identity 41.4%

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

      2. add-sqr-sqrt 41.4%

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

      3. times-frac 41.4%

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

      4. hypot-def 41.4%

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

      5. fma-def 41.4%

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

      6. hypot-def 63.2%

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

   3. Applied egg-rr 63.2%

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

   4. Taylor expanded in d around -inf 80.5%

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

      1. neg-mul-1 80.5%

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

      2. +-commutative 80.5%

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

      3. unsub-neg 80.5%

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

      4. mul-1-neg 80.5%

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

      5. *-lft-identity 80.5%

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

      6. times-frac 80.5%

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

      7. /-rgt-identity 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in c around -inf 31.2%

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

   if -8.5000000000000004e170 < d < 2.70000000000000013e218

   1. Initial program 65.2%

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

   2. Taylor expanded in c around inf 50.7%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -8.5 \cdot 10^{+170}:\\ \;\;\;\;\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.7 \cdot 10^{+218}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{d}\\ \end{array} \]

Alternative 12: 62.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1e+57) (/ b d) (if (<= d 9e-79) (/ a c) (/ b d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1e+57) {
		tmp = b / d;
	} else if (d <= 9e-79) {
		tmp = a / 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 <= (-1d+57)) then
        tmp = b / d
    else if (d <= 9d-79) then
        tmp = a / 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 <= -1e+57) {
		tmp = b / d;
	} else if (d <= 9e-79) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1e+57:
		tmp = b / d
	elif d <= 9e-79:
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1e+57)
		tmp = Float64(b / d);
	elseif (d <= 9e-79)
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1e+57)
		tmp = b / d;
	elseif (d <= 9e-79)
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1e+57], N[(b / d), $MachinePrecision], If[LessEqual[d, 9e-79], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if d < -1.00000000000000005e57 or 9.0000000000000006e-79 < d

   1. Initial program 53.5%

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

   2. Taylor expanded in c around 0 67.4%

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

   if -1.00000000000000005e57 < d < 9.0000000000000006e-79

   1. Initial program 69.5%

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

   2. Taylor expanded in c around inf 68.3%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1 \cdot 10^{+57}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;d \leq 9 \cdot 10^{-79}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]

Alternative 13: 44.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.7%

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

2. Taylor expanded in c around inf 43.4%

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

3. Final simplification 43.4%

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

Developer target: 99.3% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023187 
(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))))