Complex division, imag part

Percentage Accurate: 61.1% → 78.9%

Time: 6.7s

Alternatives: 8

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: 61.1% 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: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{if}\;c \leq -1.36 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -8 \cdot 10^{-65}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;c \leq 2.6 \cdot 10^{+70}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- a) (/ d c) b) c)))
   (if (<= c -1.36e+124)
     t_0
     (if (<= c -8e-65)
       (/ (fma (- d) a (* b c)) (fma d d (* c c)))
       (if (<= c 2.6e+70) (/ (- (/ (* b c) d) a) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, (d / c), b) / c;
	double tmp;
	if (c <= -1.36e+124) {
		tmp = t_0;
	} else if (c <= -8e-65) {
		tmp = fma(-d, a, (b * c)) / fma(d, d, (c * c));
	} else if (c <= 2.6e+70) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-a), Float64(d / c), b) / c)
	tmp = 0.0
	if (c <= -1.36e+124)
		tmp = t_0;
	elseif (c <= -8e-65)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / fma(d, d, Float64(c * c)));
	elseif (c <= 2.6e+70)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) * N[(d / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.36e+124], t$95$0, If[LessEqual[c, -8e-65], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.6e+70], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.36e124 or 2.6e70 < c

   1. Initial program 34.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 88.5%

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

   if -1.36e124 < c < -7.99999999999999939e-65

   1. Initial program 92.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 92.2

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 92.2

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

   4. Applied rewrites 92.2%

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

   if -7.99999999999999939e-65 < c < 2.6e70

   1. Initial program 76.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 88.6

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

   5. Applied rewrites 88.6%

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

Alternative 2: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e+17) (not (<= c 2.6e+70)))
   (/ (fma (- a) (/ d c) b) c)
   (/ (- (/ (* b c) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+17) || !(c <= 2.6e+70)) {
		tmp = fma(-a, (d / c), b) / c;
	} else {
		tmp = (((b * c) / d) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e+17) || !(c <= 2.6e+70))
		tmp = Float64(fma(Float64(-a), Float64(d / c), b) / c);
	else
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.6e+17], N[Not[LessEqual[c, 2.6e+70]], $MachinePrecision]], N[(N[((-a) * N[(d / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6e17 or 2.6e70 < c

   1. Initial program 44.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   7. Applied rewrites 85.3%

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

   if -2.6e17 < c < 2.6e70

   1. Initial program 78.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, \frac{d}{c}, b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e+17) (not (<= c 2.6e+70)))
   (/ (- b (/ (* a d) c)) c)
   (/ (- (/ (* b c) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+17) || !(c <= 2.6e+70)) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = (((b * c) / d) - a) / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-2.6d+17)) .or. (.not. (c <= 2.6d+70))) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = (((b * c) / d) - 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 <= -2.6e+17) || !(c <= 2.6e+70)) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = (((b * c) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -2.6e+17) or not (c <= 2.6e+70):
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = (((b * c) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e+17) || !(c <= 2.6e+70))
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -2.6e+17) || ~((c <= 2.6e+70)))
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = (((b * c) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.6e+17], N[Not[LessEqual[c, 2.6e+70]], $MachinePrecision]], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6e17 or 2.6e70 < c

   1. Initial program 44.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -2.6e17 < c < 2.6e70

   1. Initial program 78.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.6e+17) (not (<= c 2.6e+70)))
   (/ (- b (/ (* a d) c)) c)
   (/ (fma b (/ c d) (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.6e+17) || !(c <= 2.6e+70)) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = fma(b, (c / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.6e+17) || !(c <= 2.6e+70))
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.6e+17], N[Not[LessEqual[c, 2.6e+70]], $MachinePrecision]], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.6e17 or 2.6e70 < c

   1. Initial program 44.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 77.9

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

   5. Applied rewrites 77.9%

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

   if -2.6e17 < c < 2.6e70

   1. Initial program 78.6%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 74.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 74.2

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

   6. Applied rewrites 74.2%

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

   7. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 84.7

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

   9. Applied rewrites 84.7%

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

   10. Taylor expanded in d around inf

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

   12. Applied rewrites 86.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.6 \cdot 10^{+17} \lor \neg \left(c \leq 2.6 \cdot 10^{+70}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+88} \lor \neg \left(c \leq 2.1 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -7.5e+88) (not (<= c 2.1e+53)))
   (/ b c)
   (/ (fma b (/ c d) (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -7.5e+88) || !(c <= 2.1e+53)) {
		tmp = b / c;
	} else {
		tmp = fma(b, (c / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -7.5e+88) || !(c <= 2.1e+53))
		tmp = Float64(b / c);
	else
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -7.5e+88], N[Not[LessEqual[c, 2.1e+53]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -7.50000000000000031e88 or 2.1000000000000002e53 < c

   1. Initial program 40.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -7.50000000000000031e88 < c < 2.1000000000000002e53

   1. Initial program 78.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. distribute-lft-neg-in N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lower-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

   4. Applied rewrites 75.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 75.5

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

   6. Applied rewrites 75.5%

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

   7. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 80.5

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

   9. Applied rewrites 80.5%

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

   10. Taylor expanded in d around inf

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

   12. Applied rewrites 82.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{+88} \lor \neg \left(c \leq 2.1 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.45 \cdot 10^{+131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.45e+131)
     t_0
     (if (<= d -2.6e-25)
       (/ (- (* b c) (* a d)) (* d d))
       (if (<= d 1.18e+94) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.45e+131) {
		tmp = t_0;
	} else if (d <= -2.6e-25) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 1.18e+94) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-1.45d+131)) then
        tmp = t_0
    else if (d <= (-2.6d-25)) then
        tmp = ((b * c) - (a * d)) / (d * d)
    else if (d <= 1.18d+94) then
        tmp = b / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.45e+131) {
		tmp = t_0;
	} else if (d <= -2.6e-25) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 1.18e+94) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -1.45e+131:
		tmp = t_0
	elif d <= -2.6e-25:
		tmp = ((b * c) - (a * d)) / (d * d)
	elif d <= 1.18e+94:
		tmp = b / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.45e+131)
		tmp = t_0;
	elseif (d <= -2.6e-25)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 1.18e+94)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -1.45e+131)
		tmp = t_0;
	elseif (d <= -2.6e-25)
		tmp = ((b * c) - (a * d)) / (d * d);
	elseif (d <= 1.18e+94)
		tmp = b / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.45e+131], t$95$0, If[LessEqual[d, -2.6e-25], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.18e+94], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.45000000000000005e131 or 1.18000000000000002e94 < d

   1. Initial program 51.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 73.7

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

   5. Applied rewrites 73.7%

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

   if -1.45000000000000005e131 < d < -2.6e-25

   1. Initial program 71.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if -2.6e-25 < d < 1.18000000000000002e94

   1. Initial program 66.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.4

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

   5. Applied rewrites 73.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.45 \cdot 10^{+131}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.18 \cdot 10^{+94}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+47} \lor \neg \left(d \leq 1.18 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -6.2e+47) (not (<= d 1.18e+94))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -6.2e+47) || !(d <= 1.18e+94)) {
		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 ((d <= (-6.2d+47)) .or. (.not. (d <= 1.18d+94))) 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 ((d <= -6.2e+47) || !(d <= 1.18e+94)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -6.2e+47) or not (d <= 1.18e+94):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -6.2e+47) || !(d <= 1.18e+94))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -6.2e+47) || ~((d <= 1.18e+94)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -6.2e+47], N[Not[LessEqual[d, 1.18e+94]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -6.2000000000000001e47 or 1.18000000000000002e94 < d

   1. Initial program 56.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 71.6

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

   5. Applied rewrites 71.6%

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

   if -6.2000000000000001e47 < d < 1.18000000000000002e94

   1. Initial program 67.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.2 \cdot 10^{+47} \lor \neg \left(d \leq 1.18 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.1% 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 62.4%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 45.5

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

5. Applied rewrites 45.5%

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

Developer Target 1: 99.4% 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 2024317 
(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))))