Complex division, imag part

Percentage Accurate: 61.4% → 79.4%

Time: 6.5s

Alternatives: 7

Speedup: 1.5×

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: 61.4% 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: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, \frac{a}{t\_0} \cdot \left(-d\right)\right)\\ \mathbf{if}\;c \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (/ c t_0) b (* (/ a t_0) (- d)))))
   (if (<= c -4e+150)
     (/ (- b (/ (* d a) c)) c)
     (if (<= c -9e+19)
       t_1
       (if (<= c 6.8e-10)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 2.5e+132) t_1 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((c / t_0), b, ((a / t_0) * -d));
	double tmp;
	if (c <= -4e+150) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= -9e+19) {
		tmp = t_1;
	} else if (c <= 6.8e-10) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 2.5e+132) {
		tmp = t_1;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), b, Float64(Float64(a / t_0) * Float64(-d)))
	tmp = 0.0
	if (c <= -4e+150)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= -9e+19)
		tmp = t_1;
	elseif (c <= 6.8e-10)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 2.5e+132)
		tmp = t_1;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c / t$95$0), $MachinePrecision] * b + N[(N[(a / t$95$0), $MachinePrecision] * (-d)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -4e+150], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -9e+19], t$95$1, If[LessEqual[c, 6.8e-10], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.5e+132], t$95$1, N[(b / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.99999999999999992e150

   1. Initial program 35.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -3.99999999999999992e150 < c < -9e19 or 6.8000000000000003e-10 < c < 2.5000000000000001e132

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

   4. Applied rewrites 84.9%

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

   if -9e19 < c < 6.8000000000000003e-10

   1. Initial program 70.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if 2.5000000000000001e132 < c

   1. Initial program 35.7%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq -9 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 2.5 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-d\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.38e+20)
   (/ (- b (/ (* d a) c)) c)
   (if (<= c 7.2e-59)
     (/ (- (/ (* b c) d) a) d)
     (if (<= c 1.9e+132)
       (/ (- (* b c) (* d a)) (+ (* d d) (* c c)))
       (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.38e+20) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 7.2e-59) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.9e+132) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} 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) :: tmp
    if (c <= (-1.38d+20)) then
        tmp = (b - ((d * a) / c)) / c
    else if (c <= 7.2d-59) then
        tmp = (((b * c) / d) - a) / d
    else if (c <= 1.9d+132) then
        tmp = ((b * c) - (d * a)) / ((d * d) + (c * c))
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.38e+20) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (c <= 7.2e-59) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 1.9e+132) {
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.38e+20:
		tmp = (b - ((d * a) / c)) / c
	elif c <= 7.2e-59:
		tmp = (((b * c) / d) - a) / d
	elif c <= 1.9e+132:
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c))
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.38e+20)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (c <= 7.2e-59)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 1.9e+132)
		tmp = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(Float64(d * d) + Float64(c * c)));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.38e+20)
		tmp = (b - ((d * a) / c)) / c;
	elseif (c <= 7.2e-59)
		tmp = (((b * c) / d) - a) / d;
	elseif (c <= 1.9e+132)
		tmp = ((b * c) - (d * a)) / ((d * d) + (c * c));
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.38e+20], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.2e-59], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.9e+132], N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.38e20

   1. Initial program 64.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -1.38e20 < c < 7.20000000000000001e-59

   1. Initial program 69.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 91.8

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

   5. Applied rewrites 91.8%

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

   if 7.20000000000000001e-59 < c < 1.90000000000000003e132

   1. Initial program 77.1%

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

   if 1.90000000000000003e132 < c

   1. Initial program 35.7%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 93.8

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

   5. Applied rewrites 93.8%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 7.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.9 \cdot 10^{+132}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{if}\;c \leq -1.22 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq 10^{+130}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* d a)) (* c c))))
   (if (<= c -1.22e+178)
     (/ b c)
     (if (<= c -9.8e+19)
       t_0
       (if (<= c 1.05e-8) (/ a (- d)) (if (<= c 1e+130) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (d * a)) / (c * c);
	double tmp;
	if (c <= -1.22e+178) {
		tmp = b / c;
	} else if (c <= -9.8e+19) {
		tmp = t_0;
	} else if (c <= 1.05e-8) {
		tmp = a / -d;
	} else if (c <= 1e+130) {
		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) - (d * a)) / (c * c)
    if (c <= (-1.22d+178)) then
        tmp = b / c
    else if (c <= (-9.8d+19)) then
        tmp = t_0
    else if (c <= 1.05d-8) then
        tmp = a / -d
    else if (c <= 1d+130) 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) - (d * a)) / (c * c);
	double tmp;
	if (c <= -1.22e+178) {
		tmp = b / c;
	} else if (c <= -9.8e+19) {
		tmp = t_0;
	} else if (c <= 1.05e-8) {
		tmp = a / -d;
	} else if (c <= 1e+130) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = ((b * c) - (d * a)) / (c * c)
	tmp = 0
	if c <= -1.22e+178:
		tmp = b / c
	elif c <= -9.8e+19:
		tmp = t_0
	elif c <= 1.05e-8:
		tmp = a / -d
	elif c <= 1e+130:
		tmp = t_0
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(d * a)) / Float64(c * c))
	tmp = 0.0
	if (c <= -1.22e+178)
		tmp = Float64(b / c);
	elseif (c <= -9.8e+19)
		tmp = t_0;
	elseif (c <= 1.05e-8)
		tmp = Float64(a / Float64(-d));
	elseif (c <= 1e+130)
		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) - (d * a)) / (c * c);
	tmp = 0.0;
	if (c <= -1.22e+178)
		tmp = b / c;
	elseif (c <= -9.8e+19)
		tmp = t_0;
	elseif (c <= 1.05e-8)
		tmp = a / -d;
	elseif (c <= 1e+130)
		tmp = t_0;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.22e+178], N[(b / c), $MachinePrecision], If[LessEqual[c, -9.8e+19], t$95$0, If[LessEqual[c, 1.05e-8], N[(a / (-d)), $MachinePrecision], If[LessEqual[c, 1e+130], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.2199999999999999e178 or 1.0000000000000001e130 < c

   1. Initial program 31.7%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if -1.2199999999999999e178 < c < -9.8e19 or 1.04999999999999997e-8 < c < 1.0000000000000001e130

   1. Initial program 80.8%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.8

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

   5. Applied rewrites 70.8%

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

   if -9.8e19 < c < 1.04999999999999997e-8

   1. Initial program 70.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.22 \cdot 10^{+178}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -9.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;c \leq 10^{+130}:\\ \;\;\;\;\frac{b \cdot c - d \cdot a}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -1.38e+20) t_0 (if (<= c 4e+26) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -1.38e+20) {
		tmp = t_0;
	} else if (c <= 4e+26) {
		tmp = (((b * c) / d) - a) / 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 * a) / c)) / c
    if (c <= (-1.38d+20)) then
        tmp = t_0
    else if (c <= 4d+26) then
        tmp = (((b * c) / d) - a) / 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 * a) / c)) / c;
	double tmp;
	if (c <= -1.38e+20) {
		tmp = t_0;
	} else if (c <= 4e+26) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((d * a) / c)) / c
	tmp = 0
	if c <= -1.38e+20:
		tmp = t_0
	elif c <= 4e+26:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -1.38e+20)
		tmp = t_0;
	elseif (c <= 4e+26)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((d * a) / c)) / c;
	tmp = 0.0;
	if (c <= -1.38e+20)
		tmp = t_0;
	elseif (c <= 4e+26)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.38e+20], t$95$0, If[LessEqual[c, 4e+26], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.38e20 or 4.00000000000000019e26 < c

   1. Initial program 60.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -1.38e20 < c < 4.00000000000000019e26

   1. Initial program 69.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 87.9

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

   5. Applied rewrites 87.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{if}\;c \leq -9.8 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* d a) c)) c)))
   (if (<= c -9.8e+19) t_0 (if (<= c 1.05e-8) (/ a (- d)) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((d * a) / c)) / c;
	double tmp;
	if (c <= -9.8e+19) {
		tmp = t_0;
	} else if (c <= 1.05e-8) {
		tmp = a / -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 * a) / c)) / c
    if (c <= (-9.8d+19)) then
        tmp = t_0
    else if (c <= 1.05d-8) then
        tmp = a / -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 * a) / c)) / c;
	double tmp;
	if (c <= -9.8e+19) {
		tmp = t_0;
	} else if (c <= 1.05e-8) {
		tmp = a / -d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((d * a) / c)) / c
	tmp = 0
	if c <= -9.8e+19:
		tmp = t_0
	elif c <= 1.05e-8:
		tmp = a / -d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(d * a) / c)) / c)
	tmp = 0.0
	if (c <= -9.8e+19)
		tmp = t_0;
	elseif (c <= 1.05e-8)
		tmp = Float64(a / Float64(-d));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((d * a) / c)) / c;
	tmp = 0.0;
	if (c <= -9.8e+19)
		tmp = t_0;
	elseif (c <= 1.05e-8)
		tmp = a / -d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -9.8e+19], t$95$0, If[LessEqual[c, 1.05e-8], N[(a / (-d)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -9.8e19 or 1.04999999999999997e-8 < c

   1. Initial program 60.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -9.8e19 < c < 1.04999999999999997e-8

   1. Initial program 70.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 74.1

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

   5. Applied rewrites 74.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -9.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;c \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.38e+20) (/ b c) (if (<= c 4e+26) (/ a (- d)) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.38e+20) {
		tmp = b / c;
	} else if (c <= 4e+26) {
		tmp = a / -d;
	} 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) :: tmp
    if (c <= (-1.38d+20)) then
        tmp = b / c
    else if (c <= 4d+26) then
        tmp = a / -d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.38e+20) {
		tmp = b / c;
	} else if (c <= 4e+26) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.38e+20:
		tmp = b / c
	elif c <= 4e+26:
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.38e+20)
		tmp = Float64(b / c);
	elseif (c <= 4e+26)
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.38e+20)
		tmp = b / c;
	elseif (c <= 4e+26)
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.38e+20], N[(b / c), $MachinePrecision], If[LessEqual[c, 4e+26], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.38e20 or 4.00000000000000019e26 < c

   1. Initial program 60.9%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   4. Step-by-step derivation

      1. lower-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -1.38e20 < c < 4.00000000000000019e26

   1. Initial program 69.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{a}{d}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{a}{\mathsf{neg}\left(d\right)}} \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

         \[\leadsto \frac{a}{\color{blue}{\mathsf{neg}\left(d\right)}} \]

      6. lower-neg.f64 72.3

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

   5. Applied rewrites 72.3%

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

Alternative 7: 42.3% accurate, 3.2× 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 66.0%

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

3. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
4. Step-by-step derivation

   1. lower-/.f64 37.6

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

5. Applied rewrites 37.6%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
6. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× 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 2024327 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform default (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))))