Complex division, imag part

Percentage Accurate: 61.5% → 85.6%

Time: 12.0s

Alternatives: 11

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return ((b * c) - (a * 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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(a, d, b \cdot \left(-c\right)\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= (/ (- (* b c) (* a d)) (+ (* c c) (* d d))) 2e+300)
   (/ (/ (- (fma a d (* b (- c)))) (hypot d c)) (hypot d c))
   (- (/ b c) (/ (* d (/ a c)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((((b * c) - (a * d)) / ((c * c) + (d * d))) <= 2e+300) {
		tmp = (-fma(a, d, (b * -c)) / hypot(d, c)) / hypot(d, c);
	} else {
		tmp = (b / c) - ((d * (a / c)) / c);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d))) <= 2e+300)
		tmp = Float64(Float64(Float64(-fma(a, d, Float64(b * Float64(-c)))) / hypot(d, c)) / hypot(d, c));
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+300], N[(N[((-N[(a * d + N[(b * (-c)), $MachinePrecision]), $MachinePrecision]) / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[d ^ 2 + c ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d))) < 2.0000000000000001e300

   1. Initial program 75.5%

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

      1. frac-2neg 75.5%

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

      2. div-inv 75.5%

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

   3. Applied egg-rr 75.5%

      \[\leadsto \color{blue}{\left(a \cdot d - b \cdot c\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]
   4. Step-by-step derivation

      1. *-commutative 75.5%

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

      2. neg-mul-1 75.5%

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

      3. associate-/r* 75.5%

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

      4. metadata-eval 75.5%

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

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\left(a \cdot d - c \cdot b\right) \cdot \frac{-1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 75.5%

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

      2. unpow2 75.5%

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

      3. associate-/r* 96.3%

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

      4. fma-neg 96.3%

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

      5. distribute-rgt-neg-in 96.3%

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

      6. hypot-udef 75.7%

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

      7. unpow2 75.7%

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

      8. unpow2 75.7%

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

      9. unpow2 75.7%

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

      10. unpow2 75.7%

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

      11. +-commutative 75.7%

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

      12. hypot-def 96.3%

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

      13. hypot-udef 75.7%

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

      14. unpow2 75.7%

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

      15. unpow2 75.7%

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

      16. unpow2 75.7%

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

      17. unpow2 75.7%

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

   7. Applied egg-rr 96.3%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (*.f64 b c) (*.f64 a d)) (+.f64 (*.f64 c c) (*.f64 d d)))

   1. Initial program 12.0%

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

   2. Taylor expanded in c around inf 49.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 49.9%

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

      2. metadata-eval 49.9%

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

      3. unpow2 49.9%

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

      4. times-frac 56.9%

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

      5. metadata-eval 56.9%

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

   4. Applied egg-rr 56.9%

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

      1. associate-*l/ 56.8%

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

      2. *-un-lft-identity 56.8%

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

      3. associate-/l* 66.5%

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

   6. Applied egg-rr 66.5%

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

      1. associate-/r/ 66.6%

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

   8. Applied egg-rr 66.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(a, d, b \cdot \left(-c\right)\right)}{\mathsf{hypot}\left(d, c\right)}}{\mathsf{hypot}\left(d, c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 2: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \mathbf{if}\;c \leq -7.4 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (+ (/ b c) (* (* d (/ a c)) (/ -1.0 c)))))
   (if (<= c -7.4e+102)
     t_0
     (if (<= c -1.8e-168)
       (/ (fma b c (* a (- d))) (fma c c (* d d)))
       (if (<= c 8.6e-188)
         (/ (- a) d)
         (if (<= c 2.4e+42)
           (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b / c) + ((d * (a / c)) * (-1.0 / c));
	double tmp;
	if (c <= -7.4e+102) {
		tmp = t_0;
	} else if (c <= -1.8e-168) {
		tmp = fma(b, c, (a * -d)) / fma(c, c, (d * d));
	} else if (c <= 8.6e-188) {
		tmp = -a / d;
	} else if (c <= 2.4e+42) {
		tmp = ((b * c) - (a * d)) / ((c * c) + (d * d));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b / c) + Float64(Float64(d * Float64(a / c)) * Float64(-1.0 / c)))
	tmp = 0.0
	if (c <= -7.4e+102)
		tmp = t_0;
	elseif (c <= -1.8e-168)
		tmp = Float64(fma(b, c, Float64(a * Float64(-d))) / fma(c, c, Float64(d * d)));
	elseif (c <= 8.6e-188)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.4e+42)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b / c), $MachinePrecision] + N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.4e+102], t$95$0, If[LessEqual[c, -1.8e-168], N[(N[(b * c + N[(a * (-d)), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 8.6e-188], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.4e+42], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.40000000000000045e102 or 2.3999999999999999e42 < c

   1. Initial program 34.3%

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

   2. Taylor expanded in c around inf 73.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 73.7%

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

      2. metadata-eval 73.7%

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

      3. unpow2 73.7%

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

      4. times-frac 81.5%

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

      5. metadata-eval 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. associate-/l* 86.7%

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

      2. associate-/r/ 87.7%

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

   6. Applied egg-rr 87.7%

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

   if -7.40000000000000045e102 < c < -1.7999999999999999e-168

   1. Initial program 88.1%

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

      1. fma-neg 88.1%

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

      2. distribute-rgt-neg-out 88.1%

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

      3. fma-def 88.1%

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

   3. Simplified 88.1%

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

   if -1.7999999999999999e-168 < c < 8.59999999999999975e-188

   1. Initial program 56.5%

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

   2. Taylor expanded in c around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 83.8%

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

   4. Simplified 83.8%

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

   if 8.59999999999999975e-188 < c < 2.3999999999999999e42

   1. Initial program 81.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \mathbf{elif}\;c \leq -1.8 \cdot 10^{-168}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, c, a \cdot \left(-d\right)\right)}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 8.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \end{array} \]

Alternative 3: 79.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \mathbf{if}\;c \leq -8.2 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))
        (t_1 (+ (/ b c) (* (* d (/ a c)) (/ -1.0 c)))))
   (if (<= c -8.2e+102)
     t_1
     (if (<= c -9.5e-170)
       t_0
       (if (<= c 4.5e-184) (/ (- a) d) (if (<= c 2.4e+42) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) + ((d * (a / c)) * (-1.0 / c));
	double tmp;
	if (c <= -8.2e+102) {
		tmp = t_1;
	} else if (c <= -9.5e-170) {
		tmp = t_0;
	} else if (c <= 4.5e-184) {
		tmp = -a / d;
	} else if (c <= 2.4e+42) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
    t_1 = (b / c) + ((d * (a / c)) * ((-1.0d0) / c))
    if (c <= (-8.2d+102)) then
        tmp = t_1
    else if (c <= (-9.5d-170)) then
        tmp = t_0
    else if (c <= 4.5d-184) then
        tmp = -a / d
    else if (c <= 2.4d+42) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double t_1 = (b / c) + ((d * (a / c)) * (-1.0 / c));
	double tmp;
	if (c <= -8.2e+102) {
		tmp = t_1;
	} else if (c <= -9.5e-170) {
		tmp = t_0;
	} else if (c <= 4.5e-184) {
		tmp = -a / d;
	} else if (c <= 2.4e+42) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d))
	t_1 = (b / c) + ((d * (a / c)) * (-1.0 / c))
	tmp = 0
	if c <= -8.2e+102:
		tmp = t_1
	elif c <= -9.5e-170:
		tmp = t_0
	elif c <= 4.5e-184:
		tmp = -a / d
	elif c <= 2.4e+42:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) + Float64(Float64(d * Float64(a / c)) * Float64(-1.0 / c)))
	tmp = 0.0
	if (c <= -8.2e+102)
		tmp = t_1;
	elseif (c <= -9.5e-170)
		tmp = t_0;
	elseif (c <= 4.5e-184)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.4e+42)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	t_1 = (b / c) + ((d * (a / c)) * (-1.0 / c));
	tmp = 0.0;
	if (c <= -8.2e+102)
		tmp = t_1;
	elseif (c <= -9.5e-170)
		tmp = t_0;
	elseif (c <= 4.5e-184)
		tmp = -a / d;
	elseif (c <= 2.4e+42)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] + N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -8.2e+102], t$95$1, If[LessEqual[c, -9.5e-170], t$95$0, If[LessEqual[c, 4.5e-184], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.4e+42], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -8.1999999999999999e102 or 2.3999999999999999e42 < c

   1. Initial program 34.3%

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

   2. Taylor expanded in c around inf 73.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 73.7%

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

      2. metadata-eval 73.7%

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

      3. unpow2 73.7%

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

      4. times-frac 81.5%

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

      5. metadata-eval 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. associate-/l* 86.7%

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

      2. associate-/r/ 87.7%

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

   6. Applied egg-rr 87.7%

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

   if -8.1999999999999999e102 < c < -9.5000000000000001e-170 or 4.5000000000000001e-184 < c < 2.3999999999999999e42

   1. Initial program 85.2%

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

   if -9.5000000000000001e-170 < c < 4.5000000000000001e-184

   1. Initial program 56.5%

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

   2. Taylor expanded in c around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 83.8%

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

   4. Simplified 83.8%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \mathbf{elif}\;c \leq -9.5 \cdot 10^{-170}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{-184}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \left(d \cdot \frac{a}{c}\right) \cdot \frac{-1}{c}\\ \end{array} \]

Alternative 4: 66.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -9.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-84}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (+ (* c c) (* d d)))))
   (if (<= c -9.6e+94)
     (/ b c)
     (if (<= c -4.1e-84)
       t_0
       (if (<= c 2.5e-66) (/ (- a) d) (if (<= c 3.8e+40) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double tmp;
	if (c <= -9.6e+94) {
		tmp = b / c;
	} else if (c <= -4.1e-84) {
		tmp = t_0;
	} else if (c <= 2.5e-66) {
		tmp = -a / d;
	} else if (c <= 3.8e+40) {
		tmp = t_0;
	} else {
		tmp = 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) :: t_0
    real(8) :: tmp
    t_0 = (b * c) / ((c * c) + (d * d))
    if (c <= (-9.6d+94)) then
        tmp = b / c
    else if (c <= (-4.1d-84)) then
        tmp = t_0
    else if (c <= 2.5d-66) then
        tmp = -a / d
    else if (c <= 3.8d+40) then
        tmp = t_0
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double tmp;
	if (c <= -9.6e+94) {
		tmp = b / c;
	} else if (c <= -4.1e-84) {
		tmp = t_0;
	} else if (c <= 2.5e-66) {
		tmp = -a / d;
	} else if (c <= 3.8e+40) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) / ((c * c) + (d * d))
	tmp = 0
	if c <= -9.6e+94:
		tmp = b / c
	elif c <= -4.1e-84:
		tmp = t_0
	elif c <= 2.5e-66:
		tmp = -a / d
	elif c <= 3.8e+40:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -9.6e+94)
		tmp = Float64(b / c);
	elseif (c <= -4.1e-84)
		tmp = t_0;
	elseif (c <= 2.5e-66)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 3.8e+40)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) / ((c * c) + (d * d));
	tmp = 0.0;
	if (c <= -9.6e+94)
		tmp = b / c;
	elseif (c <= -4.1e-84)
		tmp = t_0;
	elseif (c <= 2.5e-66)
		tmp = -a / d;
	elseif (c <= 3.8e+40)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -9.6e+94], N[(b / c), $MachinePrecision], If[LessEqual[c, -4.1e-84], t$95$0, If[LessEqual[c, 2.5e-66], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 3.8e+40], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -9.5999999999999993e94 or 3.80000000000000004e40 < c

   1. Initial program 35.0%

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

   2. Taylor expanded in c around inf 73.5%

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

   if -9.5999999999999993e94 < c < -4.10000000000000005e-84 or 2.49999999999999981e-66 < c < 3.80000000000000004e40

   1. Initial program 90.3%

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

   2. Taylor expanded in b around inf 66.7%

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

      1. *-commutative 66.7%

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

   4. Simplified 66.7%

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

   if -4.10000000000000005e-84 < c < 2.49999999999999981e-66

   1. Initial program 66.4%

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

   2. Taylor expanded in c around 0 70.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.6 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -4.1 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]

Alternative 5: 70.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{if}\;c \leq -24000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (* (/ a c) (/ d c)))))
   (if (<= c -24000.0)
     t_1
     (if (<= c -2.8e-87)
       t_0
       (if (<= c 3.2e-66) (/ (- a) d) (if (<= c 3.9e+36) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -24000.0) {
		tmp = t_1;
	} else if (c <= -2.8e-87) {
		tmp = t_0;
	} else if (c <= 3.2e-66) {
		tmp = -a / d;
	} else if (c <= 3.9e+36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * c) / ((c * c) + (d * d))
    t_1 = (b / c) - ((a / c) * (d / c))
    if (c <= (-24000.0d0)) then
        tmp = t_1
    else if (c <= (-2.8d-87)) then
        tmp = t_0
    else if (c <= 3.2d-66) then
        tmp = -a / d
    else if (c <= 3.9d+36) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((a / c) * (d / c));
	double tmp;
	if (c <= -24000.0) {
		tmp = t_1;
	} else if (c <= -2.8e-87) {
		tmp = t_0;
	} else if (c <= 3.2e-66) {
		tmp = -a / d;
	} else if (c <= 3.9e+36) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) / ((c * c) + (d * d))
	t_1 = (b / c) - ((a / c) * (d / c))
	tmp = 0
	if c <= -24000.0:
		tmp = t_1
	elif c <= -2.8e-87:
		tmp = t_0
	elif c <= 3.2e-66:
		tmp = -a / d
	elif c <= 3.9e+36:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(a / c) * Float64(d / c)))
	tmp = 0.0
	if (c <= -24000.0)
		tmp = t_1;
	elseif (c <= -2.8e-87)
		tmp = t_0;
	elseif (c <= 3.2e-66)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 3.9e+36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) / ((c * c) + (d * d));
	t_1 = (b / c) - ((a / c) * (d / c));
	tmp = 0.0;
	if (c <= -24000.0)
		tmp = t_1;
	elseif (c <= -2.8e-87)
		tmp = t_0;
	elseif (c <= 3.2e-66)
		tmp = -a / d;
	elseif (c <= 3.9e+36)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(N[(a / c), $MachinePrecision] * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -24000.0], t$95$1, If[LessEqual[c, -2.8e-87], t$95$0, If[LessEqual[c, 3.2e-66], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 3.9e+36], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -24000 or 3.90000000000000021e36 < c

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 75.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. times-frac 86.4%

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

   4. Applied egg-rr 86.4%

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

   if -24000 < c < -2.8000000000000001e-87 or 3.19999999999999982e-66 < c < 3.90000000000000021e36

   1. Initial program 88.3%

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

   2. Taylor expanded in b around inf 68.5%

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

      1. *-commutative 68.5%

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

   4. Simplified 68.5%

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

   if -2.8000000000000001e-87 < c < 3.19999999999999982e-66

   1. Initial program 66.4%

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

   2. Taylor expanded in c around 0 70.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -24000:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \mathbf{elif}\;c \leq -2.8 \cdot 10^{-87}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{-66}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 3.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{a}{c} \cdot \frac{d}{c}\\ \end{array} \]

Alternative 6: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c}{c \cdot c + d \cdot d}\\ t_1 := \frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{if}\;c \leq -235:\\ \;\;\;\;t_1\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.02 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (* b c) (+ (* c c) (* d d))))
        (t_1 (- (/ b c) (/ (* d (/ a c)) c))))
   (if (<= c -235.0)
     t_1
     (if (<= c -2.2e-85)
       t_0
       (if (<= c 1.7e-67) (/ (- a) d) (if (<= c 2.02e+37) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((d * (a / c)) / c);
	double tmp;
	if (c <= -235.0) {
		tmp = t_1;
	} else if (c <= -2.2e-85) {
		tmp = t_0;
	} else if (c <= 1.7e-67) {
		tmp = -a / d;
	} else if (c <= 2.02e+37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (b * c) / ((c * c) + (d * d))
    t_1 = (b / c) - ((d * (a / c)) / c)
    if (c <= (-235.0d0)) then
        tmp = t_1
    else if (c <= (-2.2d-85)) then
        tmp = t_0
    else if (c <= 1.7d-67) then
        tmp = -a / d
    else if (c <= 2.02d+37) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b * c) / ((c * c) + (d * d));
	double t_1 = (b / c) - ((d * (a / c)) / c);
	double tmp;
	if (c <= -235.0) {
		tmp = t_1;
	} else if (c <= -2.2e-85) {
		tmp = t_0;
	} else if (c <= 1.7e-67) {
		tmp = -a / d;
	} else if (c <= 2.02e+37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b * c) / ((c * c) + (d * d))
	t_1 = (b / c) - ((d * (a / c)) / c)
	tmp = 0
	if c <= -235.0:
		tmp = t_1
	elif c <= -2.2e-85:
		tmp = t_0
	elif c <= 1.7e-67:
		tmp = -a / d
	elif c <= 2.02e+37:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b * c) / Float64(Float64(c * c) + Float64(d * d)))
	t_1 = Float64(Float64(b / c) - Float64(Float64(d * Float64(a / c)) / c))
	tmp = 0.0
	if (c <= -235.0)
		tmp = t_1;
	elseif (c <= -2.2e-85)
		tmp = t_0;
	elseif (c <= 1.7e-67)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.02e+37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b * c) / ((c * c) + (d * d));
	t_1 = (b / c) - ((d * (a / c)) / c);
	tmp = 0.0;
	if (c <= -235.0)
		tmp = t_1;
	elseif (c <= -2.2e-85)
		tmp = t_0;
	elseif (c <= 1.7e-67)
		tmp = -a / d;
	elseif (c <= 2.02e+37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b * c), $MachinePrecision] / N[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b / c), $MachinePrecision] - N[(N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -235.0], t$95$1, If[LessEqual[c, -2.2e-85], t$95$0, If[LessEqual[c, 1.7e-67], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.02e+37], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -235 or 2.0199999999999999e37 < c

   1. Initial program 45.2%

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

   2. Taylor expanded in c around inf 75.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 75.1%

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

      2. metadata-eval 75.1%

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

      3. unpow2 75.1%

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

      4. times-frac 81.5%

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

      5. metadata-eval 81.5%

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

   4. Applied egg-rr 81.5%

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

      1. associate-*l/ 81.5%

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

      2. *-un-lft-identity 81.5%

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

      3. associate-/l* 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. associate-/r/ 86.6%

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

   8. Applied egg-rr 86.6%

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

   if -235 < c < -2.2e-85 or 1.70000000000000005e-67 < c < 2.0199999999999999e37

   1. Initial program 88.3%

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

   2. Taylor expanded in b around inf 68.5%

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

      1. *-commutative 68.5%

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

   4. Simplified 68.5%

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

   if -2.2e-85 < c < 1.70000000000000005e-67

   1. Initial program 66.4%

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

   2. Taylor expanded in c around 0 70.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 70.7%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 70.7%

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

   4. Simplified 70.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -235:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \mathbf{elif}\;c \leq -2.2 \cdot 10^{-85}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.02 \cdot 10^{+37}:\\ \;\;\;\;\frac{b \cdot c}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{d \cdot \frac{a}{c}}{c}\\ \end{array} \]

Alternative 7: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+85} \lor \neg \left(d \leq 1.4 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -7.2e+85) (not (<= d 1.4e+23)))
   (/ (- a) d)
   (+ (/ b c) (* (/ (* a d) c) (/ -1.0 c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+85) || !(d <= 1.4e+23)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) + (((a * d) / c) * (-1.0 / 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 <= (-7.2d+85)) .or. (.not. (d <= 1.4d+23))) then
        tmp = -a / d
    else
        tmp = (b / c) + (((a * d) / c) * ((-1.0d0) / c))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -7.2e+85) || !(d <= 1.4e+23)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -7.2e+85) or not (d <= 1.4e+23):
		tmp = -a / d
	else:
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c))
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -7.2e+85) || !(d <= 1.4e+23))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b / c) + Float64(Float64(Float64(a * d) / c) * Float64(-1.0 / c)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -7.2e+85) || ~((d <= 1.4e+23)))
		tmp = -a / d;
	else
		tmp = (b / c) + (((a * d) / c) * (-1.0 / c));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -7.2e+85], N[Not[LessEqual[d, 1.4e+23]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] + N[(N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -7.1999999999999996e85 or 1.4e23 < d

   1. Initial program 40.4%

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

   2. Taylor expanded in c around 0 68.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 68.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 68.0%

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

   4. Simplified 68.0%

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

   if -7.1999999999999996e85 < d < 1.4e23

   1. Initial program 72.0%

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

   2. Taylor expanded in c around inf 72.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 72.3%

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

      2. metadata-eval 72.3%

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

      3. unpow2 72.3%

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

      4. times-frac 78.9%

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

      5. metadata-eval 78.9%

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

   4. Applied egg-rr 78.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -7.2 \cdot 10^{+85} \lor \neg \left(d \leq 1.4 \cdot 10^{+23}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} + \frac{a \cdot d}{c} \cdot \frac{-1}{c}\\ \end{array} \]

Alternative 8: 71.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+86} \lor \neg \left(d \leq 2.35 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.4e+86) (not (<= d 2.35e+20)))
   (/ (- a) d)
   (- (/ b c) (/ (/ a (/ c d)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e+86) || !(d <= 2.35e+20)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((a / (c / d)) / 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 <= (-1.4d+86)) .or. (.not. (d <= 2.35d+20))) then
        tmp = -a / d
    else
        tmp = (b / c) - ((a / (c / d)) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.4e+86) || !(d <= 2.35e+20)) {
		tmp = -a / d;
	} else {
		tmp = (b / c) - ((a / (c / d)) / c);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.4e+86) or not (d <= 2.35e+20):
		tmp = -a / d
	else:
		tmp = (b / c) - ((a / (c / d)) / c)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.4e+86) || !(d <= 2.35e+20))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(Float64(b / c) - Float64(Float64(a / Float64(c / d)) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.4e+86) || ~((d <= 2.35e+20)))
		tmp = -a / d;
	else
		tmp = (b / c) - ((a / (c / d)) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.4e+86], N[Not[LessEqual[d, 2.35e+20]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(N[(b / c), $MachinePrecision] - N[(N[(a / N[(c / d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.40000000000000002e86 or 2.35e20 < d

   1. Initial program 40.4%

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

   2. Taylor expanded in c around 0 68.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 68.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 68.0%

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

   4. Simplified 68.0%

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

   if -1.40000000000000002e86 < d < 2.35e20

   1. Initial program 72.0%

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

   2. Taylor expanded in c around inf 72.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot d}{{c}^{2}} + \frac{b}{c}} \]
   3. Step-by-step derivation

      1. *-un-lft-identity 72.3%

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

      2. metadata-eval 72.3%

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

      3. unpow2 72.3%

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

      4. times-frac 78.9%

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

      5. metadata-eval 78.9%

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

   4. Applied egg-rr 78.9%

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

      1. associate-*l/ 78.8%

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

      2. *-un-lft-identity 78.8%

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

      3. associate-/l* 78.3%

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

   6. Applied egg-rr 78.3%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.4 \cdot 10^{+86} \lor \neg \left(d \leq 2.35 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c} - \frac{\frac{a}{\frac{c}{d}}}{c}\\ \end{array} \]

Alternative 9: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-83} \lor \neg \left(c \leq 6.4 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.15e-83) (not (<= c 6.4e-41))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.15e-83) || !(c <= 6.4e-41)) {
		tmp = b / 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 ((c <= (-1.15d-83)) .or. (.not. (c <= 6.4d-41))) then
        tmp = b / 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 ((c <= -1.15e-83) || !(c <= 6.4e-41)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.15e-83) or not (c <= 6.4e-41):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.15e-83) || !(c <= 6.4e-41))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.15e-83) || ~((c <= 6.4e-41)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.15e-83], N[Not[LessEqual[c, 6.4e-41]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.14999999999999995e-83 or 6.40000000000000024e-41 < c

   1. Initial program 55.3%

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

   2. Taylor expanded in c around inf 64.1%

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

   if -1.14999999999999995e-83 < c < 6.40000000000000024e-41

   1. Initial program 67.7%

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

   2. Taylor expanded in c around 0 69.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   3. Step-by-step derivation

      1. associate-*r/ 69.0%

         \[\leadsto \color{blue}{\frac{-1 \cdot a}{d}} \]

      2. neg-mul-1 69.0%

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

   4. Simplified 69.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.15 \cdot 10^{-83} \lor \neg \left(c \leq 6.4 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]

Alternative 10: 9.8% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

   1. frac-2neg 60.4%

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

   2. div-inv 60.3%

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

3. Applied egg-rr 60.3%

   \[\leadsto \color{blue}{\left(a \cdot d - b \cdot c\right) \cdot \frac{1}{-{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]
4. Step-by-step derivation

   1. *-commutative 60.3%

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

   2. neg-mul-1 60.3%

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

   3. associate-/r* 60.3%

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

   4. metadata-eval 60.3%

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

5. Simplified 60.3%

   \[\leadsto \color{blue}{\left(a \cdot d - c \cdot b\right) \cdot \frac{-1}{{\left(\mathsf{hypot}\left(c, d\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 60.4%

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

   2. unpow2 60.4%

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

   3. associate-/r* 77.5%

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

   4. fma-neg 77.5%

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

   5. distribute-rgt-neg-in 77.5%

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

   6. hypot-udef 60.5%

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

   7. unpow2 60.5%

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

   8. unpow2 60.5%

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

   9. unpow2 60.5%

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

   10. unpow2 60.5%

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

   11. +-commutative 60.5%

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

   12. hypot-def 77.5%

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

   13. hypot-udef 60.5%

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

   14. unpow2 60.5%

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

   15. unpow2 60.5%

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

   16. unpow2 60.5%

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

   17. unpow2 60.5%

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

7. Applied egg-rr 77.5%

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

8. Taylor expanded in c around -inf 31.2%

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

9. Taylor expanded in d around inf 11.6%

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

10. Final simplification 11.6%

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

Alternative 11: 42.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c, double d) {
	return b / 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 = b / c
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.4%

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

2. Taylor expanded in c around inf 43.6%

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

3. Final simplification 43.6%

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

Developer target: 99.2% accurate, 0.1× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (- b (* a (/ d c))) (+ c (* d (/ d c))))
   (/ (+ (- a) (* b (/ 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (-a + (b * (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 = (b - (a * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (-a + (b * (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(b - Float64(a * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(Float64(-a) + Float64(b * 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 = (b - (a * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (-a + (b * (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[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) + N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023305 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :herbie-target
  (if (< (fabs d) (fabs c)) (/ (- b (* a (/ d c))) (+ c (* d (/ d c)))) (/ (+ (- a) (* b (/ c d))) (+ d (* c (/ c d)))))

  (/ (- (* b c) (* a d)) (+ (* c c) (* d d))))