Complex division, imag part

Percentage Accurate: 61.7% → 82.4%

Time: 6.7s

Alternatives: 12

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: 61.7% 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: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{if}\;d \leq -5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\frac{d}{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* d d) (* c c)))))
   (if (<= d -5e+124)
     (/ 1.0 (/ d (fma (/ c d) b (- a))))
     (if (<= d -2.6e-107)
       t_0
       (if (<= d 3.4e-137)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 4.5e+45) t_0 (/ (fma (* (/ -1.0 (- d)) c) b (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	double tmp;
	if (d <= -5e+124) {
		tmp = 1.0 / (d / fma((c / d), b, -a));
	} else if (d <= -2.6e-107) {
		tmp = t_0;
	} else if (d <= 3.4e-137) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 4.5e+45) {
		tmp = t_0;
	} else {
		tmp = fma(((-1.0 / -d) * c), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (d <= -5e+124)
		tmp = Float64(1.0 / Float64(d / fma(Float64(c / d), b, Float64(-a))));
	elseif (d <= -2.6e-107)
		tmp = t_0;
	elseif (d <= 3.4e-137)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 4.5e+45)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(-1.0 / Float64(-d)) * c), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -5e+124], N[(1.0 / N[(d / N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, -2.6e-107], t$95$0, If[LessEqual[d, 3.4e-137], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+45], t$95$0, N[(N[(N[(N[(-1.0 / (-d)), $MachinePrecision] * c), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -4.9999999999999996e124

   1. Initial program 21.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 85.7%

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

   9. Applied rewrites 86.8%

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

   if -4.9999999999999996e124 < d < -2.6000000000000001e-107 or 3.40000000000000014e-137 < d < 4.4999999999999998e45

   1. Initial program 86.6%

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

   if -2.6000000000000001e-107 < d < 3.40000000000000014e-137

   1. Initial program 73.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 4.4999999999999998e45 < d

   1. Initial program 43.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 83.7%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -5 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{\frac{d}{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{if}\;d \leq -6.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* c b) (* a d)) (+ (* d d) (* c c)))))
   (if (<= d -6.4e+122)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -2.6e-107)
       t_0
       (if (<= d 3.4e-137)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 4.5e+45) t_0 (/ (fma (* (/ -1.0 (- d)) c) b (- a)) d)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((c * b) - (a * d)) / ((d * d) + (c * c));
	double tmp;
	if (d <= -6.4e+122) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -2.6e-107) {
		tmp = t_0;
	} else if (d <= 3.4e-137) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 4.5e+45) {
		tmp = t_0;
	} else {
		tmp = fma(((-1.0 / -d) * c), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(Float64(d * d) + Float64(c * c)))
	tmp = 0.0
	if (d <= -6.4e+122)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -2.6e-107)
		tmp = t_0;
	elseif (d <= 3.4e-137)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 4.5e+45)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(-1.0 / Float64(-d)) * c), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(N[(d * d), $MachinePrecision] + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -6.4e+122], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -2.6e-107], t$95$0, If[LessEqual[d, 3.4e-137], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 4.5e+45], t$95$0, N[(N[(N[(N[(-1.0 / (-d)), $MachinePrecision] * c), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -6.40000000000000024e122

   1. Initial program 21.1%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 66.0

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

   5. Applied rewrites 66.0%

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

   7. Applied rewrites 85.7%

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

   if -6.40000000000000024e122 < d < -2.6000000000000001e-107 or 3.40000000000000014e-137 < d < 4.4999999999999998e45

   1. Initial program 86.6%

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

   if -2.6000000000000001e-107 < d < 3.40000000000000014e-137

   1. Initial program 73.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if 4.4999999999999998e45 < d

   1. Initial program 43.9%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   7. Applied rewrites 83.7%

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

   9. Applied rewrites 83.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -6.4 \cdot 10^{+122}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -2.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{elif}\;d \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d + c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -0.009:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* a d)) (* d d))))
   (if (<= d -2.8e+71)
     t_0
     (if (<= d -0.009)
       t_1
       (if (<= d -8.6e-106)
         (* (/ d (fma d d (* c c))) (- a))
         (if (<= d 1.8e-41) (/ b c) (if (<= d 1.85e+159) t_1 t_0)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (a * d)) / (d * d);
	double tmp;
	if (d <= -2.8e+71) {
		tmp = t_0;
	} else if (d <= -0.009) {
		tmp = t_1;
	} else if (d <= -8.6e-106) {
		tmp = (d / fma(d, d, (c * c))) * -a;
	} else if (d <= 1.8e-41) {
		tmp = b / c;
	} else if (d <= 1.85e+159) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(d * d))
	tmp = 0.0
	if (d <= -2.8e+71)
		tmp = t_0;
	elseif (d <= -0.009)
		tmp = t_1;
	elseif (d <= -8.6e-106)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * Float64(-a));
	elseif (d <= 1.8e-41)
		tmp = Float64(b / c);
	elseif (d <= 1.85e+159)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.8e+71], t$95$0, If[LessEqual[d, -0.009], t$95$1, If[LessEqual[d, -8.6e-106], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[d, 1.8e-41], N[(b / c), $MachinePrecision], If[LessEqual[d, 1.85e+159], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -2.80000000000000002e71 or 1.85e159 < d

   1. Initial program 27.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.80000000000000002e71 < d < -0.00899999999999999932 or 1.8e-41 < d < 1.85e159

   1. Initial program 78.3%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if -0.00899999999999999932 < d < -8.6000000000000004e-106

   1. Initial program 90.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -8.6000000000000004e-106 < d < 1.8e-41

   1. Initial program 76.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 71.8

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

   5. Applied rewrites 71.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -0.009:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+159}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ t_1 := \frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{if}\;d \leq -2.8 \cdot 10^{+71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+159}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)) (t_1 (/ (- (* c b) (* a d)) (* d d))))
   (if (<= d -2.8e+71)
     t_0
     (if (<= d -4.5e-7)
       t_1
       (if (<= d 1.4e-13)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 1.85e+159) t_1 t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double t_1 = ((c * b) - (a * d)) / (d * d);
	double tmp;
	if (d <= -2.8e+71) {
		tmp = t_0;
	} else if (d <= -4.5e-7) {
		tmp = t_1;
	} else if (d <= 1.4e-13) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.85e+159) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = -a / d
    t_1 = ((c * b) - (a * d)) / (d * d)
    if (d <= (-2.8d+71)) then
        tmp = t_0
    else if (d <= (-4.5d-7)) then
        tmp = t_1
    else if (d <= 1.4d-13) then
        tmp = (b - ((a * d) / c)) / c
    else if (d <= 1.85d+159) then
        tmp = t_1
    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 t_1 = ((c * b) - (a * d)) / (d * d);
	double tmp;
	if (d <= -2.8e+71) {
		tmp = t_0;
	} else if (d <= -4.5e-7) {
		tmp = t_1;
	} else if (d <= 1.4e-13) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 1.85e+159) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	t_1 = ((c * b) - (a * d)) / (d * d)
	tmp = 0
	if d <= -2.8e+71:
		tmp = t_0
	elif d <= -4.5e-7:
		tmp = t_1
	elif d <= 1.4e-13:
		tmp = (b - ((a * d) / c)) / c
	elif d <= 1.85e+159:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	t_1 = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(d * d))
	tmp = 0.0
	if (d <= -2.8e+71)
		tmp = t_0;
	elseif (d <= -4.5e-7)
		tmp = t_1;
	elseif (d <= 1.4e-13)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 1.85e+159)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	t_1 = ((c * b) - (a * d)) / (d * d);
	tmp = 0.0;
	if (d <= -2.8e+71)
		tmp = t_0;
	elseif (d <= -4.5e-7)
		tmp = t_1;
	elseif (d <= 1.4e-13)
		tmp = (b - ((a * d) / c)) / c;
	elseif (d <= 1.85e+159)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -2.8e+71], t$95$0, If[LessEqual[d, -4.5e-7], t$95$1, If[LessEqual[d, 1.4e-13], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 1.85e+159], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -2.80000000000000002e71 or 1.85e159 < d

   1. Initial program 27.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 74.9

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

   5. Applied rewrites 74.9%

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

   if -2.80000000000000002e71 < d < -4.4999999999999998e-7 or 1.4000000000000001e-13 < d < 1.85e159

   1. Initial program 76.6%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if -4.4999999999999998e-7 < d < 1.4000000000000001e-13

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 1.85 \cdot 10^{+159}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.7e+132)
   (/ b c)
   (if (<= c -2.5e-84)
     (* (/ c (fma d d (* c c))) b)
     (if (<= c 7.6e-65)
       (/ (- a) d)
       (if (<= c 9.8e+61) (/ (- (* c b) (* a d)) (* c c)) (/ b c))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.7e+132) {
		tmp = b / c;
	} else if (c <= -2.5e-84) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else if (c <= 7.6e-65) {
		tmp = -a / d;
	} else if (c <= 9.8e+61) {
		tmp = ((c * b) - (a * d)) / (c * c);
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.7e+132)
		tmp = Float64(b / c);
	elseif (c <= -2.5e-84)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	elseif (c <= 7.6e-65)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 9.8e+61)
		tmp = Float64(Float64(Float64(c * b) - Float64(a * d)) / Float64(c * c));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.7e+132], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.5e-84], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 7.6e-65], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 9.8e+61], N[(N[(N[(c * b), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.70000000000000013e132 or 9.8000000000000005e61 < c

   1. Initial program 40.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if -1.70000000000000013e132 < c < -2.5000000000000001e-84

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 79.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 79.5

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

   if -2.5000000000000001e-84 < c < 7.6000000000000003e-65

   1. Initial program 72.3%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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 68.0

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

   5. Applied rewrites 68.0%

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

   if 7.6000000000000003e-65 < c < 9.8000000000000005e61

   1. Initial program 92.4%

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

   3. Taylor expanded in c around inf

      \[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{{c}^{2}}} \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 7.6 \cdot 10^{-65}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 9.8 \cdot 10^{+61}:\\ \;\;\;\;\frac{c \cdot b - a \cdot d}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.5e-7)
   (/ (fma c (/ b d) (- a)) d)
   (if (<= d 1.4e-13)
     (/ (- b (/ (* a d) c)) c)
     (/ (fma (* (/ -1.0 (- d)) c) b (- a)) d))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.5e-7) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= 1.4e-13) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = fma(((-1.0 / -d) * c), b, -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.5e-7)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= 1.4e-13)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = Float64(fma(Float64(Float64(-1.0 / Float64(-d)) * c), b, Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.5e-7], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 1.4e-13], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(N[(-1.0 / (-d)), $MachinePrecision] * c), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.4999999999999998e-7

   1. Initial program 44.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

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

   7. Applied rewrites 82.8%

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

   if -4.4999999999999998e-7 < d < 1.4000000000000001e-13

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if 1.4000000000000001e-13 < d

   1. Initial program 54.4%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 78.4

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

   5. Applied rewrites 78.4%

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

   7. Applied rewrites 81.6%

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

   9. Applied rewrites 81.6%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-1}{-d} \cdot c, b, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -4.5e-7) t_0 (if (<= d 1.4e-13) (/ (- b (/ (* a d) 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 <= -4.5e-7) {
		tmp = t_0;
	} else if (d <= 1.4e-13) {
		tmp = (b - ((a * d) / 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 <= -4.5e-7)
		tmp = t_0;
	elseif (d <= 1.4e-13)
		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[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -4.5e-7], t$95$0, If[LessEqual[d, 1.4e-13], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -4.4999999999999998e-7 or 1.4000000000000001e-13 < d

   1. Initial program 49.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 82.2%

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

   if -4.4999999999999998e-7 < d < 1.4000000000000001e-13

   1. Initial program 79.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

Alternative 8: 75.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{if}\;c \leq -4.6 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (/ (* a d) c)) c)))
   (if (<= c -4.6e-39) t_0 (if (<= c 2.3e-64) (/ (- (/ (* c b) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a * d) / c)) / c;
	double tmp;
	if (c <= -4.6e-39) {
		tmp = t_0;
	} else if (c <= 2.3e-64) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (b - ((a * d) / c)) / c
    if (c <= (-4.6d-39)) then
        tmp = t_0
    else if (c <= 2.3d-64) then
        tmp = (((c * b) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a * d) / c)) / c;
	double tmp;
	if (c <= -4.6e-39) {
		tmp = t_0;
	} else if (c <= 2.3e-64) {
		tmp = (((c * b) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a * d) / c)) / c
	tmp = 0
	if c <= -4.6e-39:
		tmp = t_0
	elif c <= 2.3e-64:
		tmp = (((c * b) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a * d) / c)) / c)
	tmp = 0.0
	if (c <= -4.6e-39)
		tmp = t_0;
	elseif (c <= 2.3e-64)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((a * d) / c)) / c;
	tmp = 0.0;
	if (c <= -4.6e-39)
		tmp = t_0;
	elseif (c <= 2.3e-64)
		tmp = (((c * b) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4.6e-39], t$95$0, If[LessEqual[c, 2.3e-64], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.60000000000000016e-39 or 2.3000000000000001e-64 < c

   1. Initial program 57.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 72.5

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

   5. Applied rewrites 72.5%

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

   if -4.60000000000000016e-39 < c < 2.3000000000000001e-64

   1. Initial program 74.5%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 85.8

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

   5. Applied rewrites 85.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-39}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -1.5 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -1.5e+109)
     t_0
     (if (<= d -8.6e-106)
       (* (/ d (fma d d (* c c))) (- a))
       (if (<= d 1.9e-15) (/ b c) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -1.5e+109) {
		tmp = t_0;
	} else if (d <= -8.6e-106) {
		tmp = (d / fma(d, d, (c * c))) * -a;
	} else if (d <= 1.9e-15) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -1.5e+109)
		tmp = t_0;
	elseif (d <= -8.6e-106)
		tmp = Float64(Float64(d / fma(d, d, Float64(c * c))) * Float64(-a));
	elseif (d <= 1.9e-15)
		tmp = Float64(b / c);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -1.5e+109], t$95$0, If[LessEqual[d, -8.6e-106], N[(N[(d / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision], If[LessEqual[d, 1.9e-15], N[(b / c), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.50000000000000008e109 or 1.9000000000000001e-15 < d

   1. Initial program 42.4%

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

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{a}{d}} \]
   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 66.3

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

   5. Applied rewrites 66.3%

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

   if -1.50000000000000008e109 < d < -8.6000000000000004e-106

   1. Initial program 87.9%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if -8.6000000000000004e-106 < d < 1.9000000000000001e-15

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq -8.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot \left(-a\right)\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.7e+132)
   (/ b c)
   (if (<= c -2.5e-84)
     (* (/ c (fma d d (* c c))) b)
     (if (<= c 2.9e-54) (/ (- a) d) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.7e+132) {
		tmp = b / c;
	} else if (c <= -2.5e-84) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else if (c <= 2.9e-54) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.7e+132)
		tmp = Float64(b / c);
	elseif (c <= -2.5e-84)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	elseif (c <= 2.9e-54)
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.7e+132], N[(b / c), $MachinePrecision], If[LessEqual[c, -2.5e-84], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 2.9e-54], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.70000000000000013e132 or 2.90000000000000015e-54 < c

   1. Initial program 53.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if -1.70000000000000013e132 < c < -2.5000000000000001e-84

   1. Initial program 79.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-in N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. remove-double-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. neg-mul-1 N/A

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

      14. associate-/r* N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 79.5

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-*.f64 N/A

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

      20. lower-fma.f64 79.5

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

   4. Applied rewrites 79.5%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 54.7

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

   7. Applied rewrites 54.7%

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

   if -2.5000000000000001e-84 < c < 2.90000000000000015e-54

   1. Initial program 71.9%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 67.7

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

   5. Applied rewrites 67.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -2.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{elif}\;c \leq 2.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -9 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -9e-106) t_0 (if (<= d 1.9e-15) (/ b c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -9e-106) {
		tmp = t_0;
	} else if (d <= 1.9e-15) {
		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 <= (-9d-106)) then
        tmp = t_0
    else if (d <= 1.9d-15) 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 <= -9e-106) {
		tmp = t_0;
	} else if (d <= 1.9e-15) {
		tmp = b / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -9e-106:
		tmp = t_0
	elif d <= 1.9e-15:
		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 <= -9e-106)
		tmp = t_0;
	elseif (d <= 1.9e-15)
		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 <= -9e-106)
		tmp = t_0;
	elseif (d <= 1.9e-15)
		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, -9e-106], t$95$0, If[LessEqual[d, 1.9e-15], N[(b / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -8.99999999999999911e-106 or 1.9000000000000001e-15 < d

   1. Initial program 55.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if -8.99999999999999911e-106 < d < 1.9000000000000001e-15

   1. Initial program 77.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 69.0

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

   5. Applied rewrites 69.0%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9 \cdot 10^{-106}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;d \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.3% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 41.5

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

5. Applied rewrites 41.5%

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

Developer Target 1: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024331 
(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))))