Complex division, imag part

Percentage Accurate: 62.3% → 83.3%

Time: 3.7s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.3% 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

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 = ((b * c) - (a * d)) / ((c * c) + (d * d))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-a}{d}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- a) d (* c b)) (fma d d (* c c)))))
   (if (<= d -1.15e+98)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= d -9e-100)
       t_0
       (if (<= d 9.6e-123)
         (/ (fma a (/ d c) (- b)) (- c))
         (if (<= d 6.2e+77) t_0 (fma (/ b d) (/ c d) (/ (- a) d))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, d, (c * b)) / fma(d, d, (c * c));
	double tmp;
	if (d <= -1.15e+98) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (d <= -9e-100) {
		tmp = t_0;
	} else if (d <= 9.6e-123) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (d <= 6.2e+77) {
		tmp = t_0;
	} else {
		tmp = fma((b / d), (c / d), (-a / d));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-a), d, Float64(c * b)) / fma(d, d, Float64(c * c)))
	tmp = 0.0
	if (d <= -1.15e+98)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (d <= -9e-100)
		tmp = t_0;
	elseif (d <= 9.6e-123)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (d <= 6.2e+77)
		tmp = t_0;
	else
		tmp = fma(Float64(b / d), Float64(c / d), Float64(Float64(-a) / d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[d, -1.15e+98], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, -9e-100], t$95$0, If[LessEqual[d, 9.6e-123], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[d, 6.2e+77], t$95$0, N[(N[(b / d), $MachinePrecision] * N[(c / d), $MachinePrecision] + N[((-a) / d), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.15000000000000007e98

   1. Initial program 39.6%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 81.9

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 90.0

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

   11. Applied rewrites 90.0%

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

   if -1.15000000000000007e98 < d < -9.0000000000000002e-100 or 9.6e-123 < d < 6.19999999999999997e77

   1. Initial program 76.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.8

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if -9.0000000000000002e-100 < d < 9.6e-123

   1. Initial program 67.9%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 6.19999999999999997e77 < d

   1. Initial program 39.7%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. times-frac N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 75.7

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

   5. Applied rewrites 75.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b}{d}, \frac{c}{d}, \frac{-a}{d}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ t_1 := \frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-100}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+77}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (fma (- a) d (* c b)) (fma d d (* c c))))
        (t_1 (/ (fma c (/ b d) (- a)) d)))
   (if (<= d -1.15e+98)
     t_1
     (if (<= d -9e-100)
       t_0
       (if (<= d 9.6e-123)
         (/ (fma a (/ d c) (- b)) (- c))
         (if (<= d 6.2e+77) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(-a, d, (c * b)) / fma(d, d, (c * c));
	double t_1 = fma(c, (b / d), -a) / d;
	double tmp;
	if (d <= -1.15e+98) {
		tmp = t_1;
	} else if (d <= -9e-100) {
		tmp = t_0;
	} else if (d <= 9.6e-123) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (d <= 6.2e+77) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(fma(Float64(-a), d, Float64(c * b)) / fma(d, d, Float64(c * c)))
	t_1 = Float64(fma(c, Float64(b / d), Float64(-a)) / d)
	tmp = 0.0
	if (d <= -1.15e+98)
		tmp = t_1;
	elseif (d <= -9e-100)
		tmp = t_0;
	elseif (d <= 9.6e-123)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (d <= 6.2e+77)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[((-a) * d + N[(c * b), $MachinePrecision]), $MachinePrecision] / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]}, If[LessEqual[d, -1.15e+98], t$95$1, If[LessEqual[d, -9e-100], t$95$0, If[LessEqual[d, 9.6e-123], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[d, 6.2e+77], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if d < -1.15000000000000007e98 or 6.19999999999999997e77 < d

   1. Initial program 39.6%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 82.9

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

   11. Applied rewrites 82.9%

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

   if -1.15000000000000007e98 < d < -9.0000000000000002e-100 or 9.6e-123 < d < 6.19999999999999997e77

   1. Initial program 76.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. associate-*r* N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.8

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. pow2 N/A

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

      18. pow2 N/A

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

      19. +-commutative N/A

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

      20. pow2 N/A

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

      21. lower-fma.f64 N/A

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

      22. pow2 N/A

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

      23. lift-*.f64 76.8

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

   4. Applied rewrites 76.8%

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

   if -9.0000000000000002e-100 < d < 9.6e-123

   1. Initial program 67.9%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 89.5

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

   5. Applied rewrites 89.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.15 \cdot 10^{+98}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;d \leq -9 \cdot 10^{-100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{elif}\;d \leq 9.6 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;d \leq 6.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, d, c \cdot b\right)}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;c \leq -5.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -6.8 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* b (/ c (fma d d (* c c))))))
   (if (<= c -5.8e+114)
     (/ b c)
     (if (<= c -6.8e-10)
       (/ (fma c (/ b d) (- a)) d)
       (if (<= c -4.2e-105)
         t_0
         (if (<= c 1.4e-54)
           (/ (- (/ (* b c) d) a) d)
           (if (<= c 5.7e+84) t_0 (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b * (c / fma(d, d, (c * c)));
	double tmp;
	if (c <= -5.8e+114) {
		tmp = b / c;
	} else if (c <= -6.8e-10) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (c <= -4.2e-105) {
		tmp = t_0;
	} else if (c <= 1.4e-54) {
		tmp = (((b * c) / d) - a) / d;
	} else if (c <= 5.7e+84) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(b * Float64(c / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (c <= -5.8e+114)
		tmp = Float64(b / c);
	elseif (c <= -6.8e-10)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (c <= -4.2e-105)
		tmp = t_0;
	elseif (c <= 1.4e-54)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	elseif (c <= 5.7e+84)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -5.8e+114], N[(b / c), $MachinePrecision], If[LessEqual[c, -6.8e-10], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, -4.2e-105], t$95$0, If[LessEqual[c, 1.4e-54], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.7e+84], t$95$0, N[(b / c), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.8000000000000001e114 or 5.6999999999999997e84 < c

   1. Initial program 36.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -5.8000000000000001e114 < c < -6.8000000000000003e-10

   1. Initial program 58.2%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 55.8

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

   8. Applied rewrites 55.8%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 62.7

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

   11. Applied rewrites 62.7%

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

   if -6.8000000000000003e-10 < c < -4.2e-105 or 1.4000000000000001e-54 < c < 5.6999999999999997e84

   1. Initial program 80.4%

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

   3. Taylor expanded in a around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 74.3

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

   5. Applied rewrites 74.3%

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

   if -4.2e-105 < c < 1.4000000000000001e-54

   1. Initial program 70.7%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 87.2

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

   8. Applied rewrites 87.2%

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

Alternative 4: 65.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{if}\;c \leq -2.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq -1.02 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 4.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (* b (/ c (fma d d (* c c))))))
   (if (<= c -2.8e+148)
     (/ b c)
     (if (<= c -1.02e-134)
       t_0
       (if (<= c 4.4e-57) (/ (- a) d) (if (<= c 5.7e+84) t_0 (/ b c)))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = b * (c / fma(d, d, (c * c)));
	double tmp;
	if (c <= -2.8e+148) {
		tmp = b / c;
	} else if (c <= -1.02e-134) {
		tmp = t_0;
	} else if (c <= 4.4e-57) {
		tmp = -a / d;
	} else if (c <= 5.7e+84) {
		tmp = t_0;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(b * Float64(c / fma(d, d, Float64(c * c))))
	tmp = 0.0
	if (c <= -2.8e+148)
		tmp = Float64(b / c);
	elseif (c <= -1.02e-134)
		tmp = t_0;
	elseif (c <= 4.4e-57)
		tmp = Float64(Float64(-a) / d);
	elseif (c <= 5.7e+84)
		tmp = t_0;
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -2.8e+148], N[(b / c), $MachinePrecision], If[LessEqual[c, -1.02e-134], t$95$0, If[LessEqual[c, 4.4e-57], N[((-a) / d), $MachinePrecision], If[LessEqual[c, 5.7e+84], t$95$0, N[(b / c), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.7999999999999998e148 or 5.6999999999999997e84 < c

   1. Initial program 32.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -2.7999999999999998e148 < c < -1.02e-134 or 4.39999999999999997e-57 < c < 5.6999999999999997e84

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -1.02e-134 < c < 4.39999999999999997e-57

   1. Initial program 69.7%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.7

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

   5. Applied rewrites 68.7%

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

Alternative 5: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.8 \cdot 10^{+114}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 5.7 \cdot 10^{+84}:\\ \;\;\;\;b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -5.8e+114)
   (/ b c)
   (if (<= c 1.4e-54)
     (/ (fma c (/ b d) (- a)) d)
     (if (<= c 5.7e+84) (* b (/ c (fma d d (* c c)))) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -5.8e+114) {
		tmp = b / c;
	} else if (c <= 1.4e-54) {
		tmp = fma(c, (b / d), -a) / d;
	} else if (c <= 5.7e+84) {
		tmp = b * (c / fma(d, d, (c * c)));
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -5.8e+114)
		tmp = Float64(b / c);
	elseif (c <= 1.4e-54)
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	elseif (c <= 5.7e+84)
		tmp = Float64(b * Float64(c / fma(d, d, Float64(c * c))));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -5.8e+114], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.4e-54], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 5.7e+84], N[(b * N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.8000000000000001e114 or 5.6999999999999997e84 < c

   1. Initial program 36.5%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -5.8000000000000001e114 < c < 1.4000000000000001e-54

   1. Initial program 69.7%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 75.2

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

   11. Applied rewrites 75.2%

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

   if 1.4000000000000001e-54 < c < 5.6999999999999997e84

   1. Initial program 79.1%

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

   3. Taylor expanded in a around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 72.6

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

   5. Applied rewrites 72.6%

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

Alternative 6: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 2.1 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.55e-9) (not (<= d 2.1e+84)))
   (/ (fma c (/ b d) (- a)) d)
   (/ (fma a (/ d c) (- b)) (- c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.55e-9) || !(d <= 2.1e+84)) {
		tmp = fma(c, (b / d), -a) / d;
	} else {
		tmp = fma(a, (d / c), -b) / -c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.55e-9) || !(d <= 2.1e+84))
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	else
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.55e-9], N[Not[LessEqual[d, 2.1e+84]], $MachinePrecision]], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.55000000000000009e-9 or 2.10000000000000019e84 < d

   1. Initial program 47.7%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 81.8

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

   11. Applied rewrites 81.8%

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

   if -2.55000000000000009e-9 < d < 2.10000000000000019e84

   1. Initial program 69.0%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 77.3

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

   5. Applied rewrites 77.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 2.1 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 4.4 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.55e-9) (not (<= d 4.4e+39)))
   (/ (fma c (/ b d) (- a)) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.55e-9) || !(d <= 4.4e+39)) {
		tmp = fma(c, (b / d), -a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.55e-9) || !(d <= 4.4e+39))
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.55e-9], N[Not[LessEqual[d, 4.4e+39]], $MachinePrecision]], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.55000000000000009e-9 or 4.4000000000000003e39 < d

   1. Initial program 48.0%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 80.2

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

   11. Applied rewrites 80.2%

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

   if -2.55000000000000009e-9 < d < 4.4000000000000003e39

   1. Initial program 69.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 77.8

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

   5. Applied rewrites 77.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 4.4 \cdot 10^{+39}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 2.1 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -2.55e-9) (not (<= d 2.1e+84)))
   (/ (fma c (/ b d) (- a)) d)
   (/ (- b (* d (/ a c))) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -2.55e-9) || !(d <= 2.1e+84)) {
		tmp = fma(c, (b / d), -a) / d;
	} else {
		tmp = (b - (d * (a / c))) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -2.55e-9) || !(d <= 2.1e+84))
		tmp = Float64(fma(c, Float64(b / d), Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(d * Float64(a / c))) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -2.55e-9], N[Not[LessEqual[d, 2.1e+84]], $MachinePrecision]], N[(N[(c * N[(b / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(d * N[(a / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -2.55000000000000009e-9 or 2.10000000000000019e84 < d

   1. Initial program 47.7%

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

   3. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   6. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-/l* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lift-neg.f64 81.8

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

   11. Applied rewrites 81.8%

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

   if -2.55000000000000009e-9 < d < 2.10000000000000019e84

   1. Initial program 69.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. frac-2neg N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 76.0

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

   7. Applied rewrites 76.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -2.55 \cdot 10^{-9} \lor \neg \left(d \leq 2.1 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(c, \frac{b}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - d \cdot \frac{a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-37} \lor \neg \left(d \leq 2.35 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.4e-37) (not (<= d 2.35e+113))) (/ (- a) d) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.4e-37) || !(d <= 2.35e+113)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	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 ((d <= (-4.4d-37)) .or. (.not. (d <= 2.35d+113))) then
        tmp = -a / d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.4e-37) || !(d <= 2.35e+113)) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.4e-37) or not (d <= 2.35e+113):
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.4e-37) || !(d <= 2.35e+113))
		tmp = Float64(Float64(-a) / d);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.4e-37) || ~((d <= 2.35e+113)))
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.4e-37], N[Not[LessEqual[d, 2.35e+113]], $MachinePrecision]], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.40000000000000004e-37 or 2.3499999999999999e113 < d

   1. Initial program 49.0%

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

   3. Taylor expanded in c around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 68.2

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

   5. Applied rewrites 68.2%

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

   if -4.40000000000000004e-37 < d < 2.3499999999999999e113

   1. Initial program 67.9%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 61.0

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

   5. Applied rewrites 61.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.4 \cdot 10^{-37} \lor \neg \left(d \leq 2.35 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.4% 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

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 = 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 59.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 42.9

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

5. Applied rewrites 42.9%

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

Developer Target 1: 99.2% 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

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 = (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 2025061 
(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))))