Complex division, imag part

Percentage Accurate: 61.2% → 84.4%

Time: 4.4s

Alternatives: 13

Speedup: 1.7×

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: 61.2% 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: 84.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_2 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -2.7 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-d}{t\_0}, a, \frac{b}{t\_0} \cdot c\right)\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+83}:\\ \;\;\;\;\frac{c}{t\_1} \cdot b - \frac{d}{t\_1} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma c c (* d d)))
        (t_2 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -2.7e+84)
     t_2
     (if (<= c -7.2e-99)
       (fma (/ (- d) t_0) a (* (/ b t_0) c))
       (if (<= c 5.2e-155)
         (/ (fma (/ c d) b (- a)) d)
         (if (<= c 4.5e+83) (- (* (/ c t_1) b) (* (/ d t_1) a)) t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma(c, c, (d * d));
	double t_2 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -2.7e+84) {
		tmp = t_2;
	} else if (c <= -7.2e-99) {
		tmp = fma((-d / t_0), a, ((b / t_0) * c));
	} else if (c <= 5.2e-155) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 4.5e+83) {
		tmp = ((c / t_1) * b) - ((d / t_1) * a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(c, c, Float64(d * d))
	t_2 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -2.7e+84)
		tmp = t_2;
	elseif (c <= -7.2e-99)
		tmp = fma(Float64(Float64(-d) / t_0), a, Float64(Float64(b / t_0) * c));
	elseif (c <= 5.2e-155)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 4.5e+83)
		tmp = Float64(Float64(Float64(c / t_1) * b) - Float64(Float64(d / t_1) * a));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -2.7e+84], t$95$2, If[LessEqual[c, -7.2e-99], N[(N[((-d) / t$95$0), $MachinePrecision] * a + N[(N[(b / t$95$0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 5.2e-155], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+83], N[(N[(N[(c / t$95$1), $MachinePrecision] * b), $MachinePrecision] - N[(N[(d / t$95$1), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -2.7e84 or 4.4999999999999999e83 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -2.7e84 < c < -7.2000000000000001e-99

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

      5. sub-flip N/A

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

      6. div-add N/A

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

      7. frac-2neg N/A

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

      8. distribute-neg-frac2 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

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

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

      13. distribute-neg-frac2 N/A

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

      14. lower-fma.f64 N/A

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < c < 5.20000000000000016e-155

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 5.20000000000000016e-155 < c < 4.4999999999999999e83

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{c \cdot b - d \cdot a}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-in N/A

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

      18. mult-flip N/A

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

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

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

   5. Applied rewrites 64.4%

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

Alternative 2: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(c, c, d \cdot d\right)\\ t_1 := \frac{c}{t\_0} \cdot b - \frac{d}{t\_0} \cdot a\\ t_2 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -1.3 \cdot 10^{+87}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 5.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 4.5 \cdot 10^{+83}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma c c (* d d)))
        (t_1 (- (* (/ c t_0) b) (* (/ d t_0) a)))
        (t_2 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -1.3e+87)
     t_2
     (if (<= c -7.2e-99)
       t_1
       (if (<= c 5.2e-155)
         (/ (fma (/ c d) b (- a)) d)
         (if (<= c 4.5e+83) t_1 t_2))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(c, c, (d * d));
	double t_1 = ((c / t_0) * b) - ((d / t_0) * a);
	double t_2 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -1.3e+87) {
		tmp = t_2;
	} else if (c <= -7.2e-99) {
		tmp = t_1;
	} else if (c <= 5.2e-155) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 4.5e+83) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(c, c, Float64(d * d))
	t_1 = Float64(Float64(Float64(c / t_0) * b) - Float64(Float64(d / t_0) * a))
	t_2 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -1.3e+87)
		tmp = t_2;
	elseif (c <= -7.2e-99)
		tmp = t_1;
	elseif (c <= 5.2e-155)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 4.5e+83)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(c / t$95$0), $MachinePrecision] * b), $MachinePrecision] - N[(N[(d / t$95$0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -1.3e+87], t$95$2, If[LessEqual[c, -7.2e-99], t$95$1, If[LessEqual[c, 5.2e-155], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 4.5e+83], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -1.29999999999999999e87 or 4.4999999999999999e83 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -1.29999999999999999e87 < c < -7.2000000000000001e-99 or 5.20000000000000016e-155 < c < 4.4999999999999999e83

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{c \cdot b - d \cdot a}}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-flip N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. distribute-rgt-in N/A

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

      18. mult-flip N/A

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

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

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

   5. Applied rewrites 64.4%

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

   if -7.2000000000000001e-99 < c < 5.20000000000000016e-155

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 3: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \frac{a}{-d}\\ \mathbf{if}\;d \leq -1.15 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{1}{\frac{d}{b}}, t\_1\right)\\ \mathbf{elif}\;d \leq -2.3 \cdot 10^{-151}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;d \leq 2.15 \cdot 10^{-29}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{elif}\;d \leq 2.35 \cdot 10^{+101}:\\ \;\;\;\;\frac{b}{t\_0} \cdot c - \frac{a}{t\_0} \cdot d\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{d}, \frac{b}{d}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c))) (t_1 (/ a (- d))))
   (if (<= d -1.15e+111)
     (fma (/ c d) (/ 1.0 (/ d b)) t_1)
     (if (<= d -2.3e-151)
       (/ (- (* b c) (* a d)) (fma c c (* d d)))
       (if (<= d 2.15e-29)
         (/ (- b (/ (* a d) c)) c)
         (if (<= d 2.35e+101)
           (- (* (/ b t_0) c) (* (/ a t_0) d))
           (fma (/ c d) (/ 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 = a / -d;
	double tmp;
	if (d <= -1.15e+111) {
		tmp = fma((c / d), (1.0 / (d / b)), t_1);
	} else if (d <= -2.3e-151) {
		tmp = ((b * c) - (a * d)) / fma(c, c, (d * d));
	} else if (d <= 2.15e-29) {
		tmp = (b - ((a * d) / c)) / c;
	} else if (d <= 2.35e+101) {
		tmp = ((b / t_0) * c) - ((a / t_0) * d);
	} else {
		tmp = fma((c / d), (b / d), t_1);
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = Float64(a / Float64(-d))
	tmp = 0.0
	if (d <= -1.15e+111)
		tmp = fma(Float64(c / d), Float64(1.0 / Float64(d / b)), t_1);
	elseif (d <= -2.3e-151)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)));
	elseif (d <= 2.15e-29)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	elseif (d <= 2.35e+101)
		tmp = Float64(Float64(Float64(b / t_0) * c) - Float64(Float64(a / t_0) * d));
	else
		tmp = fma(Float64(c / d), Float64(b / d), 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[(a / (-d)), $MachinePrecision]}, If[LessEqual[d, -1.15e+111], N[(N[(c / d), $MachinePrecision] * N[(1.0 / N[(d / b), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[d, -2.3e-151], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 2.15e-29], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[d, 2.35e+101], N[(N[(N[(b / t$95$0), $MachinePrecision] * c), $MachinePrecision] - N[(N[(a / t$95$0), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision], N[(N[(c / d), $MachinePrecision] * N[(b / d), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if d < -1.15000000000000001e111

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity 52.9

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

   6. Applied rewrites 52.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-unsound-/.f64 52.9

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

   8. Applied rewrites 52.9%

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

   if -1.15000000000000001e111 < d < -2.29999999999999996e-151

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

      7. distribute-rgt-neg-out N/A

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -2.29999999999999996e-151 < d < 2.1499999999999999e-29

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

   if 2.1499999999999999e-29 < d < 2.34999999999999985e101

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

   3. Applied rewrites 58.4%

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

   if 2.34999999999999985e101 < d

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. distribute-lft-neg-out N/A

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

      17. metadata-eval N/A

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

      18. *-lft-identity 52.9

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

   6. Applied rewrites 52.9%

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

Alternative 4: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ \mathbf{elif}\;c \leq 2.45 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(d, d, c \cdot c\right)}{c \cdot b - d \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -3.4e+86)
     t_0
     (if (<= c -7.2e-99)
       (/ (- (* b c) (* a d)) (fma c c (* d d)))
       (if (<= c 2.45e-115)
         (/ (fma (/ c d) b (- a)) d)
         (if (<= c 1.85e+50)
           (/ 1.0 (/ (fma d d (* c c)) (- (* c b) (* d a))))
           t_0))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.4e+86) {
		tmp = t_0;
	} else if (c <= -7.2e-99) {
		tmp = ((b * c) - (a * d)) / fma(c, c, (d * d));
	} else if (c <= 2.45e-115) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 1.85e+50) {
		tmp = 1.0 / (fma(d, d, (c * c)) / ((c * b) - (d * a)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -3.4e+86)
		tmp = t_0;
	elseif (c <= -7.2e-99)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)));
	elseif (c <= 2.45e-115)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 1.85e+50)
		tmp = Float64(1.0 / Float64(fma(d, d, Float64(c * c)) / Float64(Float64(c * b) - Float64(d * a))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.4e+86], t$95$0, If[LessEqual[c, -7.2e-99], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.45e-115], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.85e+50], N[(1.0 / N[(N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision] / N[(N[(c * b), $MachinePrecision] - N[(d * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -3.3999999999999998e86 or 1.85e50 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -3.3999999999999998e86 < c < -7.2000000000000001e-99

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

      7. distribute-rgt-neg-out N/A

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < c < 2.44999999999999994e-115

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 2.44999999999999994e-115 < c < 1.85e50

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 61.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. sqr-abs-rev N/A

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

      9. sqr-neg-rev N/A

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

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

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

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

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

      12. lift-*.f64 N/A

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

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

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

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

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

      15. sqr-neg-rev N/A

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

      16. sqr-abs-rev N/A

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

      17. lift-*.f64 N/A

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

      18. remove-double-neg N/A

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

      19. lower-fma.f64 61.0

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

   3. Applied rewrites 61.0%

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

Alternative 5: 82.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{\mathsf{fma}\left(c, c, d \cdot d\right)}\\ t_1 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -3.4 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -7.2 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 5.9 \cdot 10^{-117}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{elif}\;c \leq 1.85 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (fma c c (* d d))))
        (t_1 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -3.4e+86)
     t_1
     (if (<= c -7.2e-99)
       t_0
       (if (<= c 5.9e-117)
         (/ (fma (/ c d) b (- a)) d)
         (if (<= c 1.85e+50) t_0 t_1))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / fma(c, c, (d * d));
	double t_1 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -3.4e+86) {
		tmp = t_1;
	} else if (c <= -7.2e-99) {
		tmp = t_0;
	} else if (c <= 5.9e-117) {
		tmp = fma((c / d), b, -a) / d;
	} else if (c <= 1.85e+50) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / fma(c, c, Float64(d * d)))
	t_1 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -3.4e+86)
		tmp = t_1;
	elseif (c <= -7.2e-99)
		tmp = t_0;
	elseif (c <= 5.9e-117)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	elseif (c <= 1.85e+50)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(c * c + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -3.4e+86], t$95$1, If[LessEqual[c, -7.2e-99], t$95$0, If[LessEqual[c, 5.9e-117], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.85e+50], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.3999999999999998e86 or 1.85e50 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -3.3999999999999998e86 < c < -7.2000000000000001e-99 or 5.9000000000000003e-117 < c < 1.85e50

   1. Initial program 61.2%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-abs-rev N/A

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

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

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

      7. distribute-rgt-neg-out N/A

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

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

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

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

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

      10. lift-*.f64 N/A

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

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

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

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

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

      13. sqr-neg-rev N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower-fma.f64 61.2

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

   3. Applied rewrites 61.2%

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

   if -7.2000000000000001e-99 < c < 5.9000000000000003e-117

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 6: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -14500:\\ \;\;\;\;\frac{b - \frac{1}{\frac{c}{a}} \cdot d}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -14500.0)
   (/ (- b (* (/ 1.0 (/ c a)) d)) c)
   (if (<= c 7.5e-8) (/ (fma (/ c d) b (- a)) d) (/ (- b (* (/ a c) d)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -14500.0) {
		tmp = (b - ((1.0 / (c / a)) * d)) / c;
	} else if (c <= 7.5e-8) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = (b - ((a / c) * d)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -14500.0)
		tmp = Float64(Float64(b - Float64(Float64(1.0 / Float64(c / a)) * d)) / c);
	elseif (c <= 7.5e-8)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -14500.0], N[(N[(b - N[(N[(1.0 / N[(c / a), $MachinePrecision]), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.5e-8], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -14500

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. metadata-eval N/A

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

      4. lower-unsound-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-unsound-/.f64 53.0

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

   10. Applied rewrites 53.0%

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

   if -14500 < c < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 7.4999999999999997e-8 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 7: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -14500:\\ \;\;\;\;\frac{\mathsf{fma}\left(d, a \cdot \frac{-1}{c}, b\right)}{c}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -14500.0)
   (/ (fma d (* a (/ -1.0 c)) b) c)
   (if (<= c 7.5e-8) (/ (fma (/ c d) b (- a)) d) (/ (- b (* (/ a c) d)) c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -14500.0) {
		tmp = fma(d, (a * (-1.0 / c)), b) / c;
	} else if (c <= 7.5e-8) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = (b - ((a / c) * d)) / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -14500.0)
		tmp = Float64(fma(d, Float64(a * Float64(-1.0 / c)), b) / c);
	elseif (c <= 7.5e-8)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -14500.0], N[(N[(d * N[(a * N[(-1.0 / c), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision] / c), $MachinePrecision], If[LessEqual[c, 7.5e-8], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -14500

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. associate-*r/ N/A

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

      7. mult-flip N/A

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

      8. mul-1-neg N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

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

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

      12. mul-1-neg N/A

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

      13. distribute-rgt-neg-in N/A

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

      14. associate-*l* N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. distribute-lft-neg-out N/A

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

      18. metadata-eval N/A

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

      19. *-lft-identity N/A

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

      20. metadata-eval N/A

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

      21. frac-2neg-rev N/A

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

      22. lower-/.f64 53.0

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

   6. Applied rewrites 53.0%

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

   if -14500 < c < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

   if 7.4999999999999997e-8 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

Alternative 8: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{c}{d}, b, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -14500.0)
     t_0
     (if (<= c 7.5e-8) (/ (fma (/ c d) b (- a)) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -14500.0) {
		tmp = t_0;
	} else if (c <= 7.5e-8) {
		tmp = fma((c / d), b, -a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -14500.0)
		tmp = t_0;
	elseif (c <= 7.5e-8)
		tmp = Float64(fma(Float64(c / d), b, Float64(-a)) / d);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -14500.0], t$95$0, If[LessEqual[c, 7.5e-8], N[(N[(N[(c / d), $MachinePrecision] * b + (-a)), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -14500 or 7.4999999999999997e-8 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -14500 < c < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. distribute-lft-neg-out N/A

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

      11. metadata-eval N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lower-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. metadata-eval N/A

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

      16. *-lft-identity 54.5

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

   6. Applied rewrites 54.5%

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

Alternative 9: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{if}\;c \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{b \cdot c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- b (* (/ a c) d)) c)))
   (if (<= c -14500.0) t_0 (if (<= c 7.5e-8) (/ (- (/ (* b c) d) a) d) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -14500.0) {
		tmp = t_0;
	} else if (c <= 7.5e-8) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (b - ((a / c) * d)) / c
    if (c <= (-14500.0d0)) then
        tmp = t_0
    else if (c <= 7.5d-8) then
        tmp = (((b * c) / d) - a) / d
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = (b - ((a / c) * d)) / c;
	double tmp;
	if (c <= -14500.0) {
		tmp = t_0;
	} else if (c <= 7.5e-8) {
		tmp = (((b * c) / d) - a) / d;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = (b - ((a / c) * d)) / c
	tmp = 0
	if c <= -14500.0:
		tmp = t_0
	elif c <= 7.5e-8:
		tmp = (((b * c) / d) - a) / d
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(b - Float64(Float64(a / c) * d)) / c)
	tmp = 0.0
	if (c <= -14500.0)
		tmp = t_0;
	elseif (c <= 7.5e-8)
		tmp = Float64(Float64(Float64(Float64(b * c) / d) - a) / d);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = (b - ((a / c) * d)) / c;
	tmp = 0.0;
	if (c <= -14500.0)
		tmp = t_0;
	elseif (c <= 7.5e-8)
		tmp = (((b * c) / d) - a) / d;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]}, If[LessEqual[c, -14500.0], t$95$0, If[LessEqual[c, 7.5e-8], N[(N[(N[(N[(b * c), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if c < -14500 or 7.4999999999999997e-8 < c

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if -14500 < c < 7.4999999999999997e-8

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. distribute-lft-neg-out N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity 52.6

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

   6. Applied rewrites 52.6%

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

Alternative 10: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 2.75 \cdot 10^{+52}:\\ \;\;\;\;\frac{b - \frac{a \cdot d}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -3e+44) t_0 (if (<= d 2.75e+52) (/ (- b (/ (* a d) c)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3e+44) {
		tmp = t_0;
	} else if (d <= 2.75e+52) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-3d+44)) then
        tmp = t_0
    else if (d <= 2.75d+52) then
        tmp = (b - ((a * d) / c)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3e+44) {
		tmp = t_0;
	} else if (d <= 2.75e+52) {
		tmp = (b - ((a * d) / c)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -3e+44:
		tmp = t_0
	elif d <= 2.75e+52:
		tmp = (b - ((a * d) / c)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -3e+44)
		tmp = t_0;
	elseif (d <= 2.75e+52)
		tmp = Float64(Float64(b - Float64(Float64(a * d) / c)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -3e+44)
		tmp = t_0;
	elseif (d <= 2.75e+52)
		tmp = (b - ((a * d) / c)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -3e+44], t$95$0, If[LessEqual[d, 2.75e+52], N[(N[(b - N[(N[(a * d), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.99999999999999987e44 or 2.74999999999999998e52 < d

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 42.6

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

   7. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 42.6

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

   9. Applied rewrites 42.6%

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

   if -2.99999999999999987e44 < d < 2.74999999999999998e52

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. distribute-frac-neg2 N/A

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

      9. frac-2neg N/A

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

      10. lift-/.f64 51.8

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

   6. Applied rewrites 51.8%

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

Alternative 11: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-a}{d}\\ \mathbf{if}\;d \leq -3 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;d \leq 8 \cdot 10^{+89}:\\ \;\;\;\;\frac{b - \frac{a}{c} \cdot d}{c}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- a) d)))
   (if (<= d -3e+44) t_0 (if (<= d 8e+89) (/ (- b (* (/ a c) d)) c) t_0))))

C

double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3e+44) {
		tmp = t_0;
	} else if (d <= 8e+89) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -a / d
    if (d <= (-3d+44)) then
        tmp = t_0
    else if (d <= 8d+89) then
        tmp = (b - ((a / c) * d)) / c
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double t_0 = -a / d;
	double tmp;
	if (d <= -3e+44) {
		tmp = t_0;
	} else if (d <= 8e+89) {
		tmp = (b - ((a / c) * d)) / c;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	t_0 = -a / d
	tmp = 0
	if d <= -3e+44:
		tmp = t_0
	elif d <= 8e+89:
		tmp = (b - ((a / c) * d)) / c
	else:
		tmp = t_0
	return tmp

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(-a) / d)
	tmp = 0.0
	if (d <= -3e+44)
		tmp = t_0;
	elseif (d <= 8e+89)
		tmp = Float64(Float64(b - Float64(Float64(a / c) * d)) / c);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	t_0 = -a / d;
	tmp = 0.0;
	if (d <= -3e+44)
		tmp = t_0;
	elseif (d <= 8e+89)
		tmp = (b - ((a / c) * d)) / c;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := Block[{t$95$0 = N[((-a) / d), $MachinePrecision]}, If[LessEqual[d, -3e+44], t$95$0, If[LessEqual[d, 8e+89], N[(N[(b - N[(N[(a / c), $MachinePrecision] * d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if d < -2.99999999999999987e44 or 7.99999999999999996e89 < d

   1. Initial program 61.2%

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

   2. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 52.6

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 42.6

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

   7. Applied rewrites 42.6%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 42.6

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

   9. Applied rewrites 42.6%

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

   if -2.99999999999999987e44 < d < 7.99999999999999996e89

   1. Initial program 61.2%

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

   2. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 51.8

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

   4. Applied rewrites 51.8%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1 \cdot \frac{a \cdot d}{c}\right)\right)}{c} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(-1 \cdot \frac{a \cdot d}{c}\right)\right)}{c} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right)\right)}{c} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{a \cdot d}{c}\right)\right)\right)\right)}{c} \]

      7. distribute-neg-frac N/A

         \[\leadsto \frac{b - \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(a \cdot d\right)}{c}\right)\right)}{c} \]

      8. distribute-frac-neg2 N/A

         \[\leadsto \frac{b - \frac{\mathsf{neg}\left(a \cdot d\right)}{\mathsf{neg}\left(c\right)}}{c} \]

      9. frac-2neg N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      10. lift-/.f64 51.8

          \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{b - \frac{a \cdot d}{c}}{c}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{b - \frac{a \cdot d}{c}}{c} \]

      3. *-commutative N/A

         \[\leadsto \frac{b - \frac{d \cdot a}{c}}{c} \]

      4. associate-/l* N/A

         \[\leadsto \frac{b - d \cdot \frac{a}{c}}{c} \]

      5. *-commutative N/A

         \[\leadsto \frac{b - \frac{a}{c} \cdot d}{c} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{b - \frac{a}{c} \cdot d}{c} \]

      7. lower-/.f64 53.0

         \[\leadsto \frac{b - \frac{a}{c} \cdot d}{c} \]

   8. Applied rewrites 53.0%

      \[\leadsto \frac{b - \frac{a}{c} \cdot d}{c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.8 \cdot 10^{+60}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 1.25 \cdot 10^{+19}:\\ \;\;\;\;\frac{-a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.8e+60) (/ b c) (if (<= c 1.25e+19) (/ (- a) d) (/ b c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.8e+60) {
		tmp = b / c;
	} else if (c <= 1.25e+19) {
		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 (c <= (-4.8d+60)) then
        tmp = b / c
    else if (c <= 1.25d+19) 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 (c <= -4.8e+60) {
		tmp = b / c;
	} else if (c <= 1.25e+19) {
		tmp = -a / d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if c <= -4.8e+60:
		tmp = b / c
	elif c <= 1.25e+19:
		tmp = -a / d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.8e+60)
		tmp = Float64(b / c);
	elseif (c <= 1.25e+19)
		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 (c <= -4.8e+60)
		tmp = b / c;
	elseif (c <= 1.25e+19)
		tmp = -a / d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.8e+60], N[(b / c), $MachinePrecision], If[LessEqual[c, 1.25e+19], N[((-a) / d), $MachinePrecision], N[(b / c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if c < -4.8e60 or 1.25e19 < c

   1. Initial program 61.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{\frac{b}{c}} \]
   3. Step-by-step derivation

      1. lower-/.f64 42.0

         \[\leadsto \frac{b}{\color{blue}{c}} \]

   4. Applied rewrites 42.0%

      \[\leadsto \color{blue}{\frac{b}{c}} \]

   if -4.8e60 < c < 1.25e19

   1. Initial program 61.2%

      \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

   2. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{-1 \cdot a + \frac{b \cdot c}{d}}{d}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot a + \frac{b \cdot c}{d}}{\color{blue}{d}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d} \]

      4. lower-*.f64 52.6

         \[\leadsto \frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d} \]

   4. Applied rewrites 52.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \frac{b \cdot c}{d}\right)}{d}} \]

   5. Taylor expanded in a around inf

      \[\leadsto \frac{-1 \cdot a}{d} \]
   6. Step-by-step derivation

      1. lower-*.f64 42.6

         \[\leadsto \frac{-1 \cdot a}{d} \]

   7. Applied rewrites 42.6%

      \[\leadsto \frac{-1 \cdot a}{d} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{-1 \cdot a}{d} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(a\right)}{d} \]

      3. lower-neg.f64 42.6

         \[\leadsto \frac{-a}{d} \]

   9. Applied rewrites 42.6%

      \[\leadsto \frac{-a}{d} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.0% accurate, 4.9× 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 61.2%

   \[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d} \]

2. Taylor expanded in c around inf

   \[\leadsto \color{blue}{\frac{b}{c}} \]
3. Step-by-step derivation

   1. lower-/.f64 42.0

      \[\leadsto \frac{b}{\color{blue}{c}} \]

4. Applied rewrites 42.0%

   \[\leadsto \color{blue}{\frac{b}{c}} \]
5. Add Preprocessing

Developer Target 1: 99.2% accurate, 0.7× 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 2025155 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform c (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))))