Complex division, real part

Percentage Accurate: 61.2% → 81.1%

Time: 5.2s

Alternatives: 8

Speedup: 1.6×

Specification

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{a \cdot c + b \cdot d}{c \cdot c + d \cdot d} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -7.5 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -7.8 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{d}{t\_0}, c \cdot \frac{a}{t\_0}\right)\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ (fma (/ b c) d a) c)))
   (if (<= c -7.5e+95)
     t_1
     (if (<= c -7.8e-134)
       (fma b (/ d t_0) (* c (/ a t_0)))
       (if (<= c 2.1e-21) (/ (fma (/ c d) a b) d) t_1)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -7.5e+95) {
		tmp = t_1;
	} else if (c <= -7.8e-134) {
		tmp = fma(b, (d / t_0), (c * (a / t_0)));
	} else if (c <= 2.1e-21) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -7.5e+95)
		tmp = t_1;
	elseif (c <= -7.8e-134)
		tmp = fma(b, Float64(d / t_0), Float64(c * Float64(a / t_0)));
	elseif (c <= 2.1e-21)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -7.5e+95], t$95$1, If[LessEqual[c, -7.8e-134], N[(b * N[(d / t$95$0), $MachinePrecision] + N[(c * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.1e-21], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5000000000000001e95 or 2.10000000000000013e-21 < c

   1. Initial program 45.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -7.5000000000000001e95 < c < -7.8000000000000002e-134

   1. Initial program 72.8%

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

   4. Applied rewrites 80.8%

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

   if -7.8000000000000002e-134 < c < 2.10000000000000013e-21

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. sqrt-pow1 N/A

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

      21. pow2 N/A

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

      22. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 2: 80.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{if}\;c \leq -4 \cdot 10^{+89}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -6.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{a \cdot c + b \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (/ b c) d a) c)))
   (if (<= c -4e+89)
     t_0
     (if (<= c -6.5e-134)
       (/ (+ (* a c) (* b d)) (fma c c (* d d)))
       (if (<= c 2.1e-21) (/ (fma (/ c d) a b) d) t_0)))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma((b / c), d, a) / c;
	double tmp;
	if (c <= -4e+89) {
		tmp = t_0;
	} else if (c <= -6.5e-134) {
		tmp = ((a * c) + (b * d)) / fma(c, c, (d * d));
	} else if (c <= 2.1e-21) {
		tmp = fma((c / d), a, b) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(b / c), d, a) / c)
	tmp = 0.0
	if (c <= -4e+89)
		tmp = t_0;
	elseif (c <= -6.5e-134)
		tmp = Float64(Float64(Float64(a * c) + Float64(b * d)) / fma(c, c, Float64(d * d)));
	elseif (c <= 2.1e-21)
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -4e+89], t$95$0, If[LessEqual[c, -6.5e-134], N[(N[(N[(a * c), $MachinePrecision] + N[(b * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.1e-21], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.99999999999999998e89 or 2.10000000000000013e-21 < c

   1. Initial program 45.6%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -3.99999999999999998e89 < c < -6.4999999999999998e-134

   1. Initial program 72.3%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. sqrt-pow1 N/A

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

      21. pow2 N/A

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

      22. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 72.3%

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

   if -6.4999999999999998e-134 < c < 2.10000000000000013e-21

   1. Initial program 70.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. sqrt-pow1 N/A

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

      21. pow2 N/A

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

      22. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 3: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 2.1 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -2.5e+54)
   (/ a c)
   (if (<= c 2.1e-21)
     (/ (fma (/ a d) c b) d)
     (if (<= c 2.05e+100) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -2.5e+54) {
		tmp = a / c;
	} else if (c <= 2.1e-21) {
		tmp = fma((a / d), c, b) / d;
	} else if (c <= 2.05e+100) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -2.5e+54)
		tmp = Float64(a / c);
	elseif (c <= 2.1e-21)
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	elseif (c <= 2.05e+100)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -2.5e+54], N[(a / c), $MachinePrecision], If[LessEqual[c, 2.1e-21], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 2.05e+100], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.50000000000000003e54 or 2.0500000000000001e100 < c

   1. Initial program 42.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.50000000000000003e54 < c < 2.10000000000000013e-21

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if 2.10000000000000013e-21 < c < 2.0500000000000001e100

   1. Initial program 81.1%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

Alternative 4: 64.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{a}{c}\\ \mathbf{elif}\;c \leq 9 \cdot 10^{-22}:\\ \;\;\;\;\frac{b}{d}\\ \mathbf{elif}\;c \leq 2.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, b, c \cdot a\right)}{c \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{a}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -1.8e+54)
   (/ a c)
   (if (<= c 9e-22)
     (/ b d)
     (if (<= c 2.05e+100) (/ (fma d b (* c a)) (* c c)) (/ a c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -1.8e+54) {
		tmp = a / c;
	} else if (c <= 9e-22) {
		tmp = b / d;
	} else if (c <= 2.05e+100) {
		tmp = fma(d, b, (c * a)) / (c * c);
	} else {
		tmp = a / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -1.8e+54)
		tmp = Float64(a / c);
	elseif (c <= 9e-22)
		tmp = Float64(b / d);
	elseif (c <= 2.05e+100)
		tmp = Float64(fma(d, b, Float64(c * a)) / Float64(c * c));
	else
		tmp = Float64(a / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -1.8e+54], N[(a / c), $MachinePrecision], If[LessEqual[c, 9e-22], N[(b / d), $MachinePrecision], If[LessEqual[c, 2.05e+100], N[(N[(d * b + N[(c * a), $MachinePrecision]), $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision], N[(a / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.8000000000000001e54 or 2.0500000000000001e100 < c

   1. Initial program 42.1%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -1.8000000000000001e54 < c < 8.99999999999999973e-22

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

   if 8.99999999999999973e-22 < c < 2.0500000000000001e100

   1. Initial program 81.1%

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

   3. Taylor expanded in c around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 69.6

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

   5. Applied rewrites 69.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 69.6

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 69.6

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

   7. Applied rewrites 69.6%

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

Alternative 5: 78.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{+54} \lor \neg \left(c \leq 2.1 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.25e+54) (not (<= c 2.1e-21)))
   (/ (fma (/ b c) d a) c)
   (/ (fma (/ c d) a b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.25e+54) || !(c <= 2.1e-21)) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = fma((c / d), a, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.25e+54) || !(c <= 2.1e-21))
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = Float64(fma(Float64(c / d), a, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.25e+54], N[Not[LessEqual[c, 2.1e-21]], $MachinePrecision]], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(c / d), $MachinePrecision] * a + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.24999999999999992e54 or 2.10000000000000013e-21 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -2.24999999999999992e54 < c < 2.10000000000000013e-21

   1. Initial program 71.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. sqr-abs-rev N/A

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

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

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

      5. rem-sqrt-square-rev N/A

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

      6. pow2 N/A

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

      7. sqrt-pow1 N/A

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

      8. metadata-eval N/A

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

      9. unpow1 N/A

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

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

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

      11. lift-*.f64 N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

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

      14. sqrt-pow1 N/A

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

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. remove-double-neg N/A

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

      18. unpow1 N/A

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

      19. metadata-eval N/A

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

      20. sqrt-pow1 N/A

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

      21. pow2 N/A

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

      22. rem-sqrt-square-rev N/A

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

   4. Applied rewrites 71.0%

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

   5. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. div-add N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{+54} \lor \neg \left(c \leq 2.1 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, a, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{+54} \lor \neg \left(c \leq 2.1 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.25e+54) (not (<= c 2.1e-21)))
   (/ (fma (/ b c) d a) c)
   (/ (fma (/ a d) c b) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.25e+54) || !(c <= 2.1e-21)) {
		tmp = fma((b / c), d, a) / c;
	} else {
		tmp = fma((a / d), c, b) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.25e+54) || !(c <= 2.1e-21))
		tmp = Float64(fma(Float64(b / c), d, a) / c);
	else
		tmp = Float64(fma(Float64(a / d), c, b) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.25e+54], N[Not[LessEqual[c, 2.1e-21]], $MachinePrecision]], N[(N[(N[(b / c), $MachinePrecision] * d + a), $MachinePrecision] / c), $MachinePrecision], N[(N[(N[(a / d), $MachinePrecision] * c + b), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.24999999999999992e54 or 2.10000000000000013e-21 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. *-lft-identity N/A

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

      3. metadata-eval N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. remove-double-neg N/A

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

      12. *-commutative N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

      16. mul-1-neg N/A

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

      17. mul-1-neg N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if -2.24999999999999992e54 < c < 2.10000000000000013e-21

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l/ N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-add N/A

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

      7. +-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 82.0

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

   5. Applied rewrites 82.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{+54} \lor \neg \left(c \leq 2.1 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{c}, d, a\right)}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{a}{d}, c, b\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+54} \lor \neg \left(c \leq 1.9 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -1.8e+54) (not (<= c 1.9e-21))) (/ a c) (/ b d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.8e+54) || !(c <= 1.9e-21)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    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 <= (-1.8d+54)) .or. (.not. (c <= 1.9d-21))) then
        tmp = a / c
    else
        tmp = b / d
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -1.8e+54) || !(c <= 1.9e-21)) {
		tmp = a / c;
	} else {
		tmp = b / d;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (c <= -1.8e+54) or not (c <= 1.9e-21):
		tmp = a / c
	else:
		tmp = b / d
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -1.8e+54) || !(c <= 1.9e-21))
		tmp = Float64(a / c);
	else
		tmp = Float64(b / d);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((c <= -1.8e+54) || ~((c <= 1.9e-21)))
		tmp = a / c;
	else
		tmp = b / d;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -1.8e+54], N[Not[LessEqual[c, 1.9e-21]], $MachinePrecision]], N[(a / c), $MachinePrecision], N[(b / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -1.8000000000000001e54 or 1.8999999999999999e-21 < c

   1. Initial program 47.8%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -1.8000000000000001e54 < c < 1.8999999999999999e-21

   1. Initial program 71.0%

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

   3. Taylor expanded in c around 0

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

      1. lower-/.f64 65.4

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

   5. Applied rewrites 65.4%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.8 \cdot 10^{+54} \lor \neg \left(c \leq 1.9 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{a}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.8

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

5. Applied rewrites 42.8%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (< (fabs d) (fabs c))
   (/ (+ a (* b (/ d c))) (+ c (* d (/ d c))))
   (/ (+ b (* a (/ 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (c / d))) / (d + (c * (c / d)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    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 = (a + (b * (d / c))) / (c + (d * (d / c)))
    else
        tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	} else {
		tmp = (b + (a * (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 = (a + (b * (d / c))) / (c + (d * (d / c)))
	else:
		tmp = (b + (a * (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(a + Float64(b * Float64(d / c))) / Float64(c + Float64(d * Float64(d / c))));
	else
		tmp = Float64(Float64(b + Float64(a * 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 = (a + (b * (d / c))) / (c + (d * (d / c)));
	else
		tmp = (b + (a * (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[(a + N[(b * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(c + N[(d * N[(d / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b + N[(a * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(d + N[(c * N[(c / d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024347 
(FPCore (a b c d)
  :name "Complex division, real part"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (fabs d) (fabs c)) (/ (+ a (* b (/ d c))) (+ c (* d (/ d c)))) (/ (+ b (* a (/ c d))) (+ d (* c (/ c d))))))

  (/ (+ (* a c) (* b d)) (+ (* c c) (* d d))))