Complex division, imag part

Percentage Accurate: 61.6% → 83.9%

Time: 6.2s

Alternatives: 10

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: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{c}{t\_0}\\ \mathbf{if}\;c \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, b, \left(-d\right) \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{t\_0}, t\_1 \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ c t_0)))
   (if (<= c -8e+107)
     (/ (fma (/ d c) a (- b)) (- c))
     (if (<= c -2.1e-53)
       (fma t_1 b (* (- d) (/ a t_0)))
       (if (<= c 6.8e-67)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 4.3e+108)
           (fma (- a) (/ d t_0) (* t_1 b))
           (/ (fma d (/ a c) (- b)) (- c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = c / t_0;
	double tmp;
	if (c <= -8e+107) {
		tmp = fma((d / c), a, -b) / -c;
	} else if (c <= -2.1e-53) {
		tmp = fma(t_1, b, (-d * (a / t_0)));
	} else if (c <= 6.8e-67) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4.3e+108) {
		tmp = fma(-a, (d / t_0), (t_1 * b));
	} else {
		tmp = fma(d, (a / c), -b) / -c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(c / t_0)
	tmp = 0.0
	if (c <= -8e+107)
		tmp = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c));
	elseif (c <= -2.1e-53)
		tmp = fma(t_1, b, Float64(Float64(-d) * Float64(a / t_0)));
	elseif (c <= 6.8e-67)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4.3e+108)
		tmp = fma(Float64(-a), Float64(d / t_0), Float64(t_1 * b));
	else
		tmp = Float64(fma(d, Float64(a / c), Float64(-b)) / Float64(-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[(c / t$95$0), $MachinePrecision]}, If[LessEqual[c, -8e+107], N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, -2.1e-53], N[(t$95$1 * b + N[((-d) * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 6.8e-67], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.3e+108], N[((-a) * N[(d / t$95$0), $MachinePrecision] + N[(t$95$1 * b), $MachinePrecision]), $MachinePrecision], N[(N[(d * N[(a / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if c < -7.9999999999999998e107

   1. Initial program 37.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. lower-*.f64 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -7.9999999999999998e107 < c < -2.09999999999999977e-53

   1. Initial program 76.8%

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

   4. Applied rewrites 80.6%

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

   if -2.09999999999999977e-53 < c < 6.8000000000000002e-67

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 6.8000000000000002e-67 < c < 4.29999999999999996e108

   1. Initial program 87.5%

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

   4. Applied rewrites 96.6%

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

   if 4.29999999999999996e108 < c

   1. Initial program 27.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 77.1

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

   7. Applied rewrites 77.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -8 \cdot 10^{+107}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \left(-d\right) \cdot \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(-a, \frac{d}{t\_0}, \frac{c}{t\_0} \cdot b\right)\\ \mathbf{if}\;c \leq -6.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (fma (- a) (/ d t_0) (* (/ c t_0) b))))
   (if (<= c -6.4e+105)
     (/ (fma (/ d c) a (- b)) (- c))
     (if (<= c -2.1e-53)
       t_1
       (if (<= c 6.8e-67)
         (/ (- (/ (* b c) d) a) d)
         (if (<= c 4.3e+108) t_1 (/ (fma d (/ a c) (- 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(-a, (d / t_0), ((c / t_0) * b));
	double tmp;
	if (c <= -6.4e+105) {
		tmp = fma((d / c), a, -b) / -c;
	} else if (c <= -2.1e-53) {
		tmp = t_1;
	} else if (c <= 6.8e-67) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 4.3e+108) {
		tmp = t_1;
	} else {
		tmp = fma(d, (a / c), -b) / -c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(-a), Float64(d / t_0), Float64(Float64(c / t_0) * b))
	tmp = 0.0
	if (c <= -6.4e+105)
		tmp = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c));
	elseif (c <= -2.1e-53)
		tmp = t_1;
	elseif (c <= 6.8e-67)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 4.3e+108)
		tmp = t_1;
	else
		tmp = Float64(fma(d, Float64(a / c), Float64(-b)) / Float64(-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[((-a) * N[(d / t$95$0), $MachinePrecision] + N[(N[(c / t$95$0), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -6.4e+105], N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, -2.1e-53], t$95$1, If[LessEqual[c, 6.8e-67], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.3e+108], t$95$1, N[(N[(d * N[(a / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -6.4e105

   1. Initial program 37.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. lower-*.f64 9.6

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 12.1%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 91.2

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

   10. Applied rewrites 91.2%

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

   if -6.4e105 < c < -2.09999999999999977e-53 or 6.8000000000000002e-67 < c < 4.29999999999999996e108

   1. Initial program 82.3%

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

   4. Applied rewrites 88.1%

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

   if -2.09999999999999977e-53 < c < 6.8000000000000002e-67

   1. Initial program 63.8%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d} + \frac{b \cdot c}{{d}^{2}}} \]
   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. lower-*.f64 88.9

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

   5. Applied rewrites 88.9%

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

   if 4.29999999999999996e108 < c

   1. Initial program 27.9%

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 77.1

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

   7. Applied rewrites 77.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -6.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -2.1 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\right)\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 4.3 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{if}\;c \leq -3.05 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ d c) a (- b)) (- c))))
   (if (<= c -3.05e+77)
     t_0
     (if (<= c 3.4e-67)
       (/ (- (* c (/ b d)) a) d)
       (if (<= c 6e+26) (/ (fma (- a) d (* b c)) (fma d d (* c c))) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((d / c), a, -b) / -c;
	double tmp;
	if (c <= -3.05e+77) {
		tmp = t_0;
	} else if (c <= 3.4e-67) {
		tmp = ((c * (b / d)) - a) / d;
	} else if (c <= 6e+26) {
		tmp = fma(-a, d, (b * c)) / fma(d, d, (c * c));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c))
	tmp = 0.0
	if (c <= -3.05e+77)
		tmp = t_0;
	elseif (c <= 3.4e-67)
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	elseif (c <= 6e+26)
		tmp = Float64(fma(Float64(-a), d, Float64(b * c)) / fma(d, d, Float64(c * c)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]}, If[LessEqual[c, -3.05e+77], t$95$0, If[LessEqual[c, 3.4e-67], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6e+26], N[(N[((-a) * d + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.05000000000000016e77 or 5.99999999999999994e26 < c

   1. Initial program 42.1%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 12.8

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

   5. Applied rewrites 12.8%

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

   7. Applied rewrites 18.6%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 84.2

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

   10. Applied rewrites 84.2%

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

   if -3.05000000000000016e77 < c < 3.4000000000000001e-67

   1. Initial program 65.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 83.1

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

   5. Applied rewrites 83.1%

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

   7. Applied rewrites 84.4%

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

   if 3.4000000000000001e-67 < c < 5.99999999999999994e26

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 99.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 6 \cdot 10^{+26}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, b \cdot c\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -5.8e+152)
     t_0
     (if (<= d -5.1e-48)
       (/ (fma (- d) a (* b c)) (* d d))
       (if (<= d 5.5e+25) (/ (- 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 <= -5.8e+152) {
		tmp = t_0;
	} else if (d <= -5.1e-48) {
		tmp = fma(-d, a, (b * c)) / (d * d);
	} else if (d <= 5.5e+25) {
		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 <= -5.8e+152)
		tmp = t_0;
	elseif (d <= -5.1e-48)
		tmp = Float64(fma(Float64(-d), a, Float64(b * c)) / Float64(d * d));
	elseif (d <= 5.5e+25)
		tmp = Float64(Float64(b - Float64(Float64(a * 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, -5.8e+152], t$95$0, If[LessEqual[d, -5.1e-48], N[(N[((-d) * a + N[(b * c), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 5.5e+25], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -5.7999999999999997e152 or 5.50000000000000018e25 < 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

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if -5.7999999999999997e152 < d < -5.10000000000000011e-48

   1. Initial program 86.4%

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

   4. Applied rewrites 87.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-neg.f64 N/A

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

      11. div-add-rev N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 86.4%

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

   7. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 77.5

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

   9. Applied rewrites 77.5%

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

   if -5.10000000000000011e-48 < d < 5.50000000000000018e25

   1. Initial program 65.8%

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

   3. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b + -1 \cdot \frac{a \cdot d}{c}}{c}} \]
   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. lower-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -5.1 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-d, a, b \cdot c\right)}{d \cdot d}\\ \mathbf{elif}\;d \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.16e+48)
   (/ b c)
   (if (<= c 8.5e-55)
     (/ (- a) d)
     (if (<= c 2.1e+31) (/ (- (* b c) (* a d)) (* c c)) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.16e+48) {
		tmp = b / c;
	} else if (c <= 8.5e-55) {
		tmp = -a / d;
	} else if (c <= 2.1e+31) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if (c <= (-1.16d+48)) then
        tmp = b / c
    else if (c <= 8.5d-55) then
        tmp = -a / d
    else if (c <= 2.1d+31) then
        tmp = ((b * c) - (a * d)) / (c * c)
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.16e+48) {
		tmp = b / c;
	} else if (c <= 8.5e-55) {
		tmp = -a / d;
	} else if (c <= 2.1e+31) {
		tmp = ((b * c) - (a * d)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -1.16e+48:
		tmp = b / c
	elif c <= 8.5e-55:
		tmp = -a / d
	elif c <= 2.1e+31:
		tmp = ((b * c) - (a * d)) / (c * c)
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.16e+48)
		tmp = Float64(b / c);
	elseif (c <= 8.5e-55)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 2.1e+31)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (c <= -1.16e+48)
		tmp = b / c;
	elseif (c <= 8.5e-55)
		tmp = -a / d;
	elseif (c <= 2.1e+31)
		tmp = ((b * c) - (a * d)) / (c * c);
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.16e+48], N[(b / c), $MachinePrecision], If[LessEqual[c, 8.5e-55], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 2.1e+31], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.15999999999999992e48 or 2.09999999999999979e31 < c

   1. Initial program 41.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -1.15999999999999992e48 < c < 8.49999999999999968e-55

   1. Initial program 65.8%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   if 8.49999999999999968e-55 < c < 2.09999999999999979e31

   1. Initial program 99.2%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+48}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.05e+77) (not (<= c 9e-55)))
   (/ (fma (/ d c) a (- b)) (- c))
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.05e+77) || !(c <= 9e-55)) {
		tmp = fma((d / c), a, -b) / -c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.05e+77) || !(c <= 9e-55))
		tmp = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.05e+77], N[Not[LessEqual[c, 9e-55]], $MachinePrecision]], N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.05000000000000016e77 or 8.99999999999999941e-55 < c

   1. Initial program 49.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 17.9

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

   5. Applied rewrites 17.9%

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

   7. Applied rewrites 22.9%

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

   8. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 81.5

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

   10. Applied rewrites 81.5%

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

   if -3.05000000000000016e77 < c < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. 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. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.05e+77) (not (<= c 9e-55)))
   (/ (fma d (/ a c) (- b)) (- c))
   (/ (- (* c (/ b d)) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -3.05e+77) || !(c <= 9e-55)) {
		tmp = fma(d, (a / c), -b) / -c;
	} else {
		tmp = ((c * (b / d)) - a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -3.05e+77) || !(c <= 9e-55))
		tmp = Float64(fma(d, Float64(a / c), Float64(-b)) / Float64(-c));
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -3.05e+77], N[Not[LessEqual[c, 9e-55]], $MachinePrecision]], N[(N[(d * N[(a / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.05000000000000016e77 or 8.99999999999999941e-55 < c

   1. Initial program 49.7%

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. mul-1-neg N/A

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

      13. lower-neg.f64 81.5

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

   7. Applied rewrites 81.5%

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

   if -3.05000000000000016e77 < c < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. 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. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(d, \frac{a}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -3.05e+77) (not (<= c 9e-55)))
   (/ (- 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 <= -3.05e+77) || !(c <= 9e-55)) {
		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 <= (-3.05d+77)) .or. (.not. (c <= 9d-55))) 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 <= -3.05e+77) || !(c <= 9e-55)) {
		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 <= -3.05e+77) or not (c <= 9e-55):
		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 <= -3.05e+77) || !(c <= 9e-55))
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(Float64(Float64(c * Float64(b / d)) - a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -3.05e+77) || ~((c <= 9e-55)))
		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, -3.05e+77], N[Not[LessEqual[c, 9e-55]], $MachinePrecision]], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c * N[(b / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -3.05000000000000016e77 or 8.99999999999999941e-55 < c

   1. Initial program 49.7%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

   if -3.05000000000000016e77 < c < 8.99999999999999941e-55

   1. Initial program 65.6%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. 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. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 84.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -3.05 \cdot 10^{+77} \lor \neg \left(c \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{b}{d} - a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+48} \lor \neg \left(c \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.16e+48) (not (<= c 2.1e-45))) (/ b c) (/ (- a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.16e+48) || !(c <= 2.1e-45)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((c <= (-1.16d+48)) .or. (.not. (c <= 2.1d-45))) then
        tmp = b / c
    else
        tmp = -a / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.16e+48) || !(c <= 2.1e-45)) {
		tmp = b / c;
	} else {
		tmp = -a / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.16e+48) or not (c <= 2.1e-45):
		tmp = b / c
	else:
		tmp = -a / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.16e+48) || !(c <= 2.1e-45))
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.16e+48) || ~((c <= 2.1e-45)))
		tmp = b / c;
	else
		tmp = -a / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.16e+48], N[Not[LessEqual[c, 2.1e-45]], $MachinePrecision]], N[(b / c), $MachinePrecision], N[((-a) / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.15999999999999992e48 or 2.09999999999999995e-45 < c

   1. Initial program 49.4%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -1.15999999999999992e48 < c < 2.09999999999999995e-45

   1. Initial program 66.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 67.4

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

   5. Applied rewrites 67.4%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.16 \cdot 10^{+48} \lor \neg \left(c \leq 2.1 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.1% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 40.5

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

5. Applied rewrites 40.5%

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

6. Final simplification 40.5%

   \[\leadsto \frac{b}{c} \]
7. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024326 
(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))))