Complex division, imag part

Percentage Accurate: 62.0% → 83.9%

Time: 7.1s

Alternatives: 13

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: 62.0% 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: 83.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, \left(-d\right) \cdot \frac{a}{t\_0}\right)\\ \mathbf{if}\;c \leq -2.6 \cdot 10^{+118}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{elif}\;c \leq -4.6 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 8.2 \cdot 10^{-105}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (/ c t_0) b (* (- d) (/ a t_0)))))
   (if (<= c -2.6e+118)
     (/ (- b (* a (/ d c))) c)
     (if (<= c -4.6e-65)
       t_1
       (if (<= c 8.2e-105)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 8e+112)
           t_1
           (fma
            (fma
             (- (* (/ a (pow c 4.0)) d) (/ b (pow c 3.0)))
             d
             (/ (/ (- a) c) c))
            d
            (/ 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, (-d * (a / t_0)));
	double tmp;
	if (c <= -2.6e+118) {
		tmp = (b - (a * (d / c))) / c;
	} else if (c <= -4.6e-65) {
		tmp = t_1;
	} else if (c <= 8.2e-105) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 8e+112) {
		tmp = t_1;
	} else {
		tmp = fma(fma((((a / pow(c, 4.0)) * d) - (b / pow(c, 3.0))), d, ((-a / c) / c)), d, (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(-d) * Float64(a / t_0)))
	tmp = 0.0
	if (c <= -2.6e+118)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	elseif (c <= -4.6e-65)
		tmp = t_1;
	elseif (c <= 8.2e-105)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 8e+112)
		tmp = t_1;
	else
		tmp = fma(fma(Float64(Float64(Float64(a / (c ^ 4.0)) * d) - Float64(b / (c ^ 3.0))), d, Float64(Float64(Float64(-a) / c) / c)), d, 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[((-d) * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.6e+118], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, -4.6e-65], t$95$1, If[LessEqual[c, 8.2e-105], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 8e+112], t$95$1, N[(N[(N[(N[(N[(a / N[Power[c, 4.0], $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision] - N[(b / N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * d + N[(N[((-a) / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * d + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.60000000000000016e118

   1. Initial program 40.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 83.4%

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

   if -2.60000000000000016e118 < c < -4.5999999999999999e-65 or 8.20000000000000061e-105 < c < 7.9999999999999994e112

   1. Initial program 76.6%

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

   4. Applied rewrites 81.4%

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

   if -4.5999999999999999e-65 < c < 8.20000000000000061e-105

   1. Initial program 70.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 93.2

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

   5. Applied rewrites 93.2%

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

   if 7.9999999999999994e112 < c

   1. Initial program 31.2%

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

   3. Taylor expanded in d around 0

      \[\leadsto \color{blue}{d \cdot \left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) + \frac{b}{c}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right)\right) \cdot d} + \frac{b}{c} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \frac{a}{{c}^{2}} + d \cdot \left(\frac{a \cdot d}{{c}^{4}} - \frac{b}{{c}^{3}}\right), d, \frac{b}{c}\right)} \]

   5. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{a}{{c}^{4}} \cdot d - \frac{b}{{c}^{3}}, d, \frac{\frac{-a}{c}}{c}\right), d, \frac{b}{c}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+122}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \left(-d\right) \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{d}, \frac{c}{d}, -1\right), a, \left(-b\right) \cdot \left({\left(\frac{c}{d}\right)}^{3} - \frac{c}{d}\right)\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))))
   (if (<= d -5.8e+122)
     (/ (- (* b (/ c d)) a) d)
     (if (<= d -3e-83)
       (fma (/ c t_0) b (* (- d) (/ a t_0)))
       (if (<= d 6.6e-99)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 7.5e+42)
           (/ (- (* b c) (* a d)) (fma c c (* d d)))
           (/
            (fma
             (fma (/ c d) (/ c d) -1.0)
             a
             (* (- b) (- (pow (/ c d) 3.0) (/ c d))))
            d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double tmp;
	if (d <= -5.8e+122) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= -3e-83) {
		tmp = fma((c / t_0), b, (-d * (a / t_0)));
	} else if (d <= 6.6e-99) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 7.5e+42) {
		tmp = ((b * c) - (a * d)) / fma(c, c, (d * d));
	} else {
		tmp = fma(fma((c / d), (c / d), -1.0), a, (-b * (pow((c / d), 3.0) - (c / d)))) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	tmp = 0.0
	if (d <= -5.8e+122)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= -3e-83)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(-d) * Float64(a / t_0)));
	elseif (d <= 6.6e-99)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 7.5e+42)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = Float64(fma(fma(Float64(c / d), Float64(c / d), -1.0), a, Float64(Float64(-b) * Float64((Float64(c / d) ^ 3.0) - Float64(c / d)))) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5.8e+122], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -3e-83], N[(N[(c / t$95$0), $MachinePrecision] * b + N[((-d) * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e-99], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.5e+42], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(c / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + -1.0), $MachinePrecision] * a + N[((-b) * N[(N[Power[N[(c / d), $MachinePrecision], 3.0], $MachinePrecision] - N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -5.8000000000000002e122

   1. Initial program 29.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 81.1%

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

   if -5.8000000000000002e122 < d < -3.0000000000000001e-83

   1. Initial program 79.6%

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

   4. Applied rewrites 82.7%

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

   if -3.0000000000000001e-83 < d < 6.59999999999999973e-99

   1. Initial program 71.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 6.59999999999999973e-99 < d < 7.50000000000000041e42

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.1

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

   4. Applied rewrites 86.1%

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

   if 7.50000000000000041e42 < d

   1. Initial program 39.6%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{d}} \]

   5. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{c}{d}, \frac{c}{d}, -1\right), a, \left(-b\right) \cdot \left({\left(\frac{c}{d}\right)}^{3} - \frac{c}{d}\right)\right)}{d}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3 \cdot 10^{-83}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{t\_0}, b, \left(-d\right) \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (- (* b (/ c d)) a) d)))
   (if (<= d -5.8e+122)
     t_1
     (if (<= d -3e-83)
       (fma (/ c t_0) b (* (- d) (/ a t_0)))
       (if (<= d 6.6e-99)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 7.5e+42) (/ (- (* b c) (* a d)) (fma c c (* d d))) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -5.8e+122) {
		tmp = t_1;
	} else if (d <= -3e-83) {
		tmp = fma((c / t_0), b, (-d * (a / t_0)));
	} else if (d <= 6.6e-99) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 7.5e+42) {
		tmp = ((b * c) - (a * d)) / fma(c, c, (d * d));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(Float64(Float64(b * Float64(c / d)) - a) / d)
	tmp = 0.0
	if (d <= -5.8e+122)
		tmp = t_1;
	elseif (d <= -3e-83)
		tmp = fma(Float64(c / t_0), b, Float64(Float64(-d) * Float64(a / t_0)));
	elseif (d <= 6.6e-99)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 7.5e+42)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)));
	else
		tmp = t_1;
	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[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -5.8e+122], t$95$1, If[LessEqual[d, -3e-83], N[(N[(c / t$95$0), $MachinePrecision] * b + N[((-d) * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 6.6e-99], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.5e+42], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -5.8000000000000002e122 or 7.50000000000000041e42 < d

   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 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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 85.1%

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

   if -5.8000000000000002e122 < d < -3.0000000000000001e-83

   1. Initial program 79.6%

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

   4. Applied rewrites 82.7%

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

   if -3.0000000000000001e-83 < d < 6.59999999999999973e-99

   1. Initial program 71.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 6.59999999999999973e-99 < d < 7.50000000000000041e42

   1. Initial program 86.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 86.1

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

   4. Applied rewrites 86.1%

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

Alternative 4: 83.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{if}\;d \leq -2.5 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -3.3 \cdot 10^{-83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 6.6 \cdot 10^{-99}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{elif}\;d \leq 7.5 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (fma c c (* d d))))
        (t_1 (/ (- (* b (/ c d)) a) d)))
   (if (<= d -2.5e+116)
     t_1
     (if (<= d -3.3e-83)
       t_0
       (if (<= d 6.6e-99)
         (/ (- b (/ (* d a) c)) c)
         (if (<= d 7.5e+42) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / fma(c, c, (d * d));
	double t_1 = ((b * (c / d)) - a) / d;
	double tmp;
	if (d <= -2.5e+116) {
		tmp = t_1;
	} else if (d <= -3.3e-83) {
		tmp = t_0;
	} else if (d <= 6.6e-99) {
		tmp = (b - ((d * a) / c)) / c;
	} else if (d <= 7.5e+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)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(Float64(Float64(b * Float64(c / d)) - a) / d)
	tmp = 0.0
	if (d <= -2.5e+116)
		tmp = t_1;
	elseif (d <= -3.3e-83)
		tmp = t_0;
	elseif (d <= 6.6e-99)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	elseif (d <= 7.5e+42)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -2.5e+116], t$95$1, If[LessEqual[d, -3.3e-83], t$95$0, If[LessEqual[d, 6.6e-99], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 7.5e+42], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.50000000000000013e116 or 7.50000000000000041e42 < d

   1. Initial program 36.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 85.2%

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

   if -2.50000000000000013e116 < d < -3.2999999999999999e-83 or 6.59999999999999973e-99 < d < 7.50000000000000041e42

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 82.1

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

   4. Applied rewrites 82.1%

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

   if -3.2999999999999999e-83 < d < 6.59999999999999973e-99

   1. Initial program 71.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 91.2

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

   5. Applied rewrites 91.2%

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

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1720:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.75e+83)
     t_0
     (if (<= d -1e-58)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 1720.0) (/ (- b (/ (* d a) c)) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.75e+83) {
		tmp = t_0;
	} else if (d <= -1e-58) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 1720.0) {
		tmp = (b - ((d * a) / c)) / 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 <= -1.75e+83)
		tmp = t_0;
	elseif (d <= -1e-58)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 1720.0)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.75e+83], t$95$0, If[LessEqual[d, -1e-58], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1720.0], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.74999999999999989e83 or 1720 < d

   1. Initial program 43.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.74999999999999989e83 < d < -1e-58

   1. Initial program 82.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 61.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.6

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

   7. Applied rewrites 61.6%

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

   if -1e-58 < d < 1720

   1. Initial program 74.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1720:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1720:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.75e+83)
     t_0
     (if (<= d -1e-58)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 1720.0) (/ (- b (* a (/ d c))) c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.75e+83) {
		tmp = t_0;
	} else if (d <= -1e-58) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 1720.0) {
		tmp = (b - (a * (d / c))) / 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 <= -1.75e+83)
		tmp = t_0;
	elseif (d <= -1e-58)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 1720.0)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.75e+83], t$95$0, If[LessEqual[d, -1e-58], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1720.0], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.74999999999999989e83 or 1720 < d

   1. Initial program 43.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.74999999999999989e83 < d < -1e-58

   1. Initial program 82.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 61.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.6

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

   7. Applied rewrites 61.6%

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

   if -1e-58 < d < 1720

   1. Initial program 74.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.6

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

   5. Applied rewrites 85.6%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 1720:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+15} \lor \neg \left(c \leq 6.2 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.3e+15) (not (<= c 6.2e+16)))
   (/ (- b (* a (/ d c))) c)
   (/ (- (/ (* c b) d) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.3e+15) || !(c <= 6.2e+16)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / 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 <= (-1.3d+15)) .or. (.not. (c <= 6.2d+16))) then
        tmp = (b - (a * (d / c))) / c
    else
        tmp = (((c * b) / 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 <= -1.3e+15) || !(c <= 6.2e+16)) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = (((c * b) / d) - a) / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.3e+15) or not (c <= 6.2e+16):
		tmp = (b - (a * (d / c))) / c
	else:
		tmp = (((c * b) / d) - a) / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.3e+15) || !(c <= 6.2e+16))
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / c);
	else
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.3e+15) || ~((c <= 6.2e+16)))
		tmp = (b - (a * (d / c))) / c;
	else
		tmp = (((c * b) / d) - a) / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.3e+15], N[Not[LessEqual[c, 6.2e+16]], $MachinePrecision]], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.3e15 or 6.2e16 < c

   1. Initial program 49.5%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   7. Applied rewrites 78.5%

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

   if -1.3e15 < c < 6.2e16

   1. Initial program 73.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.3 \cdot 10^{+15} \lor \neg \left(c \leq 6.2 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-76} \lor \neg \left(d \leq 1650\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -1.9e-76) (not (<= d 1650.0)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e-76) || !(d <= 1650.0)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / 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 <= (-1.9d-76)) .or. (.not. (d <= 1650.0d0))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -1.9e-76) || !(d <= 1650.0)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -1.9e-76) or not (d <= 1650.0):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -1.9e-76) || !(d <= 1650.0))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -1.9e-76) || ~((d <= 1650.0)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -1.9e-76], N[Not[LessEqual[d, 1650.0]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -1.9000000000000001e-76 or 1650 < d

   1. Initial program 52.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   7. Applied rewrites 76.1%

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

   if -1.9000000000000001e-76 < d < 1650

   1. Initial program 73.9%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. fp-cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{-76} \lor \neg \left(d \leq 1650\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.75e+83)
     t_0
     (if (<= d -2.7e-59)
       (/ (fma (- a) d (* c b)) (* d d))
       (if (<= d 9.2e-9) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.75e+83) {
		tmp = t_0;
	} else if (d <= -2.7e-59) {
		tmp = fma(-a, d, (c * b)) / (d * d);
	} else if (d <= 9.2e-9) {
		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 <= -1.75e+83)
		tmp = t_0;
	elseif (d <= -2.7e-59)
		tmp = Float64(fma(Float64(-a), d, Float64(c * b)) / Float64(d * d));
	elseif (d <= 9.2e-9)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.75e+83], t$95$0, If[LessEqual[d, -2.7e-59], N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-9], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.74999999999999989e83 or 9.1999999999999997e-9 < d

   1. Initial program 43.9%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 73.1

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

   5. Applied rewrites 73.1%

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

   if -1.74999999999999989e83 < d < -2.6999999999999999e-59

   1. Initial program 82.1%

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

   3. Taylor expanded in c around 0

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{d}^{2}}} \]
   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 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

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

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 61.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 61.6

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

   7. Applied rewrites 61.6%

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

   if -2.6999999999999999e-59 < d < 9.1999999999999997e-9

   1. Initial program 74.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.75 \cdot 10^{+83}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.7 \cdot 10^{-59}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -4.8 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -4.8e+121)
     t_0
     (if (<= d -2.2e-64)
       (* (- a) (/ d (fma d d (* c c))))
       (if (<= d 9.2e-9) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -4.8e+121) {
		tmp = t_0;
	} else if (d <= -2.2e-64) {
		tmp = -a * (d / fma(d, d, (c * c)));
	} else if (d <= 9.2e-9) {
		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 <= -4.8e+121)
		tmp = t_0;
	elseif (d <= -2.2e-64)
		tmp = Float64(Float64(-a) * Float64(d / fma(d, d, Float64(c * c))));
	elseif (d <= 9.2e-9)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -4.8e+121], t$95$0, If[LessEqual[d, -2.2e-64], N[((-a) * N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 9.2e-9], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.8e121 or 9.1999999999999997e-9 < d

   1. Initial program 42.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 74.7

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

   5. Applied rewrites 74.7%

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

   if -4.8e121 < d < -2.2e-64

   1. Initial program 77.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   if -2.2e-64 < d < 9.1999999999999997e-9

   1. Initial program 74.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.8 \cdot 10^{+121}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;\left(-a\right) \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 9.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-59} \lor \neg \left(d \leq 9.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -3.5e-59) (not (<= d 9.2e-9))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -3.5e-59) || !(d <= 9.2e-9)) {
		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 <= (-3.5d-59)) .or. (.not. (d <= 9.2d-9))) 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 <= -3.5e-59) || !(d <= 9.2e-9)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -3.5e-59) or not (d <= 9.2e-9):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -3.5e-59) || !(d <= 9.2e-9))
		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 <= -3.5e-59) || ~((d <= 9.2e-9)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -3.5e-59], N[Not[LessEqual[d, 9.2e-9]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -3.5000000000000001e-59 or 9.1999999999999997e-9 < d

   1. Initial program 51.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 65.2

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

   5. Applied rewrites 65.2%

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

   if -3.5000000000000001e-59 < d < 9.1999999999999997e-9

   1. Initial program 74.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.6

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

   5. Applied rewrites 73.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -3.5 \cdot 10^{-59} \lor \neg \left(d \leq 9.2 \cdot 10^{-9}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+160} \lor \neg \left(d \leq 8.1 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -5.7e+160) (not (<= d 8.1e+192))) (/ a d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -5.7e+160) || !(d <= 8.1e+192)) {
		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 <= (-5.7d+160)) .or. (.not. (d <= 8.1d+192))) 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 <= -5.7e+160) || !(d <= 8.1e+192)) {
		tmp = a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -5.7e+160) or not (d <= 8.1e+192):
		tmp = a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -5.7e+160) || !(d <= 8.1e+192))
		tmp = 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 <= -5.7e+160) || ~((d <= 8.1e+192)))
		tmp = a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -5.7e+160], N[Not[LessEqual[d, 8.1e+192]], $MachinePrecision]], N[(a / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -5.6999999999999999e160 or 8.10000000000000019e192 < d

   1. Initial program 28.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 28.2

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

   4. Applied rewrites 28.2%

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

   5. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 83.2

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

   7. Applied rewrites 83.2%

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

   9. Applied rewrites 61.6%

      \[\leadsto \frac{-a}{\left|d\right|} \]
   10. Step-by-step derivation

   11. Applied rewrites 29.5%

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

   if -5.6999999999999999e160 < d < 8.10000000000000019e192

   1. Initial program 72.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 54.7

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

   5. Applied rewrites 54.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.7 \cdot 10^{+160} \lor \neg \left(d \leq 8.1 \cdot 10^{+192}\right):\\ \;\;\;\;\frac{a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 10.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lower-fma.f64 62.4

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

4. Applied rewrites 62.4%

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

5. Taylor expanded in c around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 45.0

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

7. Applied rewrites 45.0%

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

9. Applied rewrites 29.2%

   \[\leadsto \frac{-a}{\left|d\right|} \]
10. Step-by-step derivation

11. Applied rewrites 10.7%

    \[\leadsto \color{blue}{\frac{a}{d}} \]
12. 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 2024339 
(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))))