Complex division, imag part

Percentage Accurate: 61.6% → 83.7%

Time: 7.4s

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: 61.6% 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: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{t\_0}, \frac{c \cdot b}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -4.2e+126)
     t_1
     (if (<= d -2.32e-73)
       (/ (- (* c b) (* d a)) (fma c c (* d d)))
       (if (<= d 2e-121)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 5e+146) (fma (- a) (/ d t_0) (/ (* c b) t_0)) t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -4.2e+126) {
		tmp = t_1;
	} else if (d <= -2.32e-73) {
		tmp = ((c * b) - (d * a)) / fma(c, c, (d * d));
	} else if (d <= 2e-121) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 5e+146) {
		tmp = fma(-a, (d / t_0), ((c * b) / t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -4.2e+126)
		tmp = t_1;
	elseif (d <= -2.32e-73)
		tmp = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(c, c, Float64(d * d)));
	elseif (d <= 2e-121)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 5e+146)
		tmp = fma(Float64(-a), Float64(d / t_0), Float64(Float64(c * b) / t_0));
	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[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.2e+126], t$95$1, If[LessEqual[d, -2.32e-73], N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2e-121], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 5e+146], N[((-a) * N[(d / t$95$0), $MachinePrecision] + N[(N[(c * b), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.1999999999999998e126 or 4.9999999999999999e146 < d

   1. Initial program 32.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 92.2

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

   5. Applied rewrites 92.2%

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

   if -4.1999999999999998e126 < d < -2.32e-73

   1. Initial program 90.4%

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

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 90.4

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

   4. Applied rewrites 90.4%

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

   if -2.32e-73 < d < 2e-121

   1. Initial program 69.7%

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

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

      7. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 95.2

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

   7. Applied rewrites 95.2%

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

   if 2e-121 < d < 4.9999999999999999e146

   1. Initial program 74.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. div-sub N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

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

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 82.3%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c \cdot b}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* d a)) (fma c c (* d d))))
        (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -4.2e+126)
     t_1
     (if (<= d -2.32e-73)
       t_0
       (if (<= d 2e-121)
         (/ (fma (* d a) (/ -1.0 c) b) c)
         (if (<= d 3.8e+38) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (d * a)) / fma(c, c, (d * d));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -4.2e+126) {
		tmp = t_1;
	} else if (d <= -2.32e-73) {
		tmp = t_0;
	} else if (d <= 2e-121) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else if (d <= 3.8e+38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(d * a)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -4.2e+126)
		tmp = t_1;
	elseif (d <= -2.32e-73)
		tmp = t_0;
	elseif (d <= 2e-121)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	elseif (d <= 3.8e+38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.2e+126], t$95$1, If[LessEqual[d, -2.32e-73], t$95$0, If[LessEqual[d, 2e-121], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 3.8e+38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.1999999999999998e126 or 3.7999999999999998e38 < d

   1. Initial program 41.2%

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

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if -4.1999999999999998e126 < d < -2.32e-73 or 2e-121 < d < 3.7999999999999998e38

   1. Initial program 85.8%

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

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -2.32e-73 < d < 2e-121

   1. Initial program 69.7%

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

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

      7. lower-*.f64 95.2

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

   5. Applied rewrites 95.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 95.2

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

   7. Applied rewrites 95.2%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.2 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.32 \cdot 10^{-73}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;d \leq 3.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{c \cdot b - d \cdot a}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d \cdot a, \frac{-1}{c}, b\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -65000000000.0)
     t_0
     (if (<= d 2.1e-81) (/ (fma (* d a) (/ -1.0 c) b) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 2.1e-81) {
		tmp = fma((d * a), (-1.0 / c), b) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 2.1e-81)
		tmp = Float64(fma(Float64(d * a), Float64(-1.0 / c), b) / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -65000000000.0], t$95$0, If[LessEqual[d, 2.1e-81], N[(N[(N[(d * a), $MachinePrecision] * N[(-1.0 / c), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5e10 or 2.0999999999999999e-81 < d

   1. Initial program 56.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -6.5e10 < d < 2.0999999999999999e-81

   1. Initial program 72.7%

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

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

      7. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac2 N/A

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

      7. div-inv N/A

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

      8. lower-fma.f64 N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. remove-double-neg N/A

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

      12. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

Alternative 4: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -65000000000.0)
     t_0
     (if (<= d 2.1e-81) (/ (- b (/ (* d a) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 2.1e-81) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 2.1e-81)
		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[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -65000000000.0], t$95$0, If[LessEqual[d, 2.1e-81], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5e10 or 2.0999999999999999e-81 < d

   1. Initial program 56.3%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. sub-neg N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -6.5e10 < d < 2.0999999999999999e-81

   1. Initial program 72.7%

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

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

      7. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

Alternative 5: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{a}{d}\\ \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d))))
   (if (<= d -65000000000.0)
     t_0
     (if (<= d 1.9e+35) (/ (- 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 <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 1.9e+35) {
		tmp = (b - ((d * a) / c)) / 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 <= (-65000000000.0d0)) then
        tmp = t_0
    else if (d <= 1.9d+35) then
        tmp = (b - ((d * a) / c)) / 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 <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 1.9e+35) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -(a / d)
	tmp = 0
	if d <= -65000000000.0:
		tmp = t_0
	elif d <= 1.9e+35:
		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 <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 1.9e+35)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / 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 <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 1.9e+35)
		tmp = (b - ((d * a) / c)) / 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, -65000000000.0], t$95$0, If[LessEqual[d, 1.9e+35], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5e10 or 1.9e35 < 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 80.7

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

   5. Applied rewrites 80.7%

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

   if -6.5e10 < d < 1.9e35

   1. Initial program 74.0%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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. *-commutative N/A

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

      7. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{a}{d}\\ \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d))))
   (if (<= d -65000000000.0)
     t_0
     (if (<= d 1.9e+35) (/ (- 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 <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 1.9e+35) {
		tmp = (b - (a * (d / c))) / 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 <= (-65000000000.0d0)) then
        tmp = t_0
    else if (d <= 1.9d+35) then
        tmp = (b - (a * (d / c))) / 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 <= -65000000000.0) {
		tmp = t_0;
	} else if (d <= 1.9e+35) {
		tmp = (b - (a * (d / c))) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -(a / d)
	tmp = 0
	if d <= -65000000000.0:
		tmp = t_0
	elif d <= 1.9e+35:
		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 <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 1.9e+35)
		tmp = Float64(Float64(b - Float64(a * Float64(d / c))) / 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 <= -65000000000.0)
		tmp = t_0;
	elseif (d <= 1.9e+35)
		tmp = (b - (a * (d / c))) / 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, -65000000000.0], t$95$0, If[LessEqual[d, 1.9e+35], N[(N[(b - N[(a * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -6.5e10 or 1.9e35 < 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 80.7

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

   5. Applied rewrites 80.7%

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

   if -6.5e10 < d < 1.9e35

   1. Initial program 74.0%

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

      1. lift-*.f64 N/A

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

      2. lower-fma.f64 74.0

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   6. 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. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 81.3

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

   7. Applied rewrites 81.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -65000000000:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{b - a \cdot \frac{d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{a}{d}\\ \mathbf{if}\;d \leq -57000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (- (/ a d))))
   (if (<= d -57000000000.0) t_0 (if (<= d 2.2e-104) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -(a / d);
	double tmp;
	if (d <= -57000000000.0) {
		tmp = t_0;
	} else if (d <= 2.2e-104) {
		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 <= (-57000000000.0d0)) then
        tmp = t_0
    else if (d <= 2.2d-104) 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 <= -57000000000.0) {
		tmp = t_0;
	} else if (d <= 2.2e-104) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -(a / d)
	tmp = 0
	if d <= -57000000000.0:
		tmp = t_0
	elif d <= 2.2e-104:
		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 <= -57000000000.0)
		tmp = t_0;
	elseif (d <= 2.2e-104)
		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 <= -57000000000.0)
		tmp = t_0;
	elseif (d <= 2.2e-104)
		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, -57000000000.0], t$95$0, If[LessEqual[d, 2.2e-104], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -5.7e10 or 2.20000000000000012e-104 < 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 74.6

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

   5. Applied rewrites 74.6%

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

   if -5.7e10 < d < 2.20000000000000012e-104

   1. Initial program 73.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.5

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

   5. Applied rewrites 69.5%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -57000000000:\\ \;\;\;\;-\frac{a}{d}\\ \mathbf{elif}\;d \leq 2.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;-\frac{a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% 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 64.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 41.3

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

5. Applied rewrites 41.3%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(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))))