Complex division, imag part

Percentage Accurate: 60.8% → 84.2%

Time: 5.2s

Alternatives: 10

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 60.8% 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: 84.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(\frac{c}{t\_0}, b, \left(-d\right) \cdot \frac{a}{t\_0}\right)\\ t_2 := \frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{if}\;c \leq -7.6 \cdot 10^{+129}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma (/ c t_0) b (* (- d) (/ a t_0))))
        (t_2 (/ (fma (/ d c) a (- b)) (- c))))
   (if (<= c -7.6e+129)
     t_2
     (if (<= c -5.5e-58)
       t_1
       (if (<= c 5.8e-83)
         (/ (- (/ (* c b) d) a) d)
         (if (<= c 1.3e+154) t_1 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 / t_0), b, (-d * (a / t_0)));
	double t_2 = fma((d / c), a, -b) / -c;
	double tmp;
	if (c <= -7.6e+129) {
		tmp = t_2;
	} else if (c <= -5.5e-58) {
		tmp = t_1;
	} else if (c <= 5.8e-83) {
		tmp = (((c * b) / d) - a) / d;
	} else if (c <= 1.3e+154) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(c / t_0), b, Float64(Float64(-d) * Float64(a / t_0)))
	t_2 = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c))
	tmp = 0.0
	if (c <= -7.6e+129)
		tmp = t_2;
	elseif (c <= -5.5e-58)
		tmp = t_1;
	elseif (c <= 5.8e-83)
		tmp = Float64(Float64(Float64(Float64(c * b) / d) - a) / d);
	elseif (c <= 1.3e+154)
		tmp = t_1;
	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[(N[(c / t$95$0), $MachinePrecision] * b + N[((-d) * N[(a / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]}, If[LessEqual[c, -7.6e+129], t$95$2, If[LessEqual[c, -5.5e-58], t$95$1, If[LessEqual[c, 5.8e-83], N[(N[(N[(N[(c * b), $MachinePrecision] / d), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 1.3e+154], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.60000000000000011e129 or 1.29999999999999994e154 < c

   1. Initial program 26.6%

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

   4. Applied rewrites 31.3%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 86.4

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

   7. Applied rewrites 86.4%

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

   if -7.60000000000000011e129 < c < -5.49999999999999996e-58 or 5.7999999999999998e-83 < c < 1.29999999999999994e154

   1. Initial program 74.6%

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

   4. Applied rewrites 84.5%

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

   if -5.49999999999999996e-58 < c < 5.7999999999999998e-83

   1. Initial program 78.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 91.9

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

   5. Applied rewrites 91.9%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.6 \cdot 10^{+129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -5.5 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \left(-d\right) \cdot \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 5.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{c \cdot b}{d} - a}{d}\\ \mathbf{elif}\;c \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}, b, \left(-d\right) \cdot \frac{a}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.6:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\left|d\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -1.9e+193)
   (/ a (- d))
   (if (<= d -4.7e-38)
     (/ (- (* b c) (* a d)) (* d d))
     (if (<= d 1.6) (/ (- b (/ (* d a) c)) c) (/ (- a) (fabs d))))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.9e+193) {
		tmp = a / -d;
	} else if (d <= -4.7e-38) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 1.6) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = -a / fabs(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 (d <= (-1.9d+193)) then
        tmp = a / -d
    else if (d <= (-4.7d-38)) then
        tmp = ((b * c) - (a * d)) / (d * d)
    else if (d <= 1.6d0) then
        tmp = (b - ((d * a) / c)) / c
    else
        tmp = -a / abs(d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -1.9e+193) {
		tmp = a / -d;
	} else if (d <= -4.7e-38) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (d <= 1.6) {
		tmp = (b - ((d * a) / c)) / c;
	} else {
		tmp = -a / Math.abs(d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -1.9e+193:
		tmp = a / -d
	elif d <= -4.7e-38:
		tmp = ((b * c) - (a * d)) / (d * d)
	elif d <= 1.6:
		tmp = (b - ((d * a) / c)) / c
	else:
		tmp = -a / math.fabs(d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -1.9e+193)
		tmp = Float64(a / Float64(-d));
	elseif (d <= -4.7e-38)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d));
	elseif (d <= 1.6)
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	else
		tmp = Float64(Float64(-a) / abs(d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -1.9e+193)
		tmp = a / -d;
	elseif (d <= -4.7e-38)
		tmp = ((b * c) - (a * d)) / (d * d);
	elseif (d <= 1.6)
		tmp = (b - ((d * a) / c)) / c;
	else
		tmp = -a / abs(d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -1.9e+193], N[(a / (-d)), $MachinePrecision], If[LessEqual[d, -4.7e-38], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[d, 1.6], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision], N[((-a) / N[Abs[d], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if d < -1.89999999999999986e193

   1. Initial program 32.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 89.9

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

   5. Applied rewrites 89.9%

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

   if -1.89999999999999986e193 < d < -4.69999999999999998e-38

   1. Initial program 69.8%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -4.69999999999999998e-38 < d < 1.6000000000000001

   1. Initial program 70.0%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if 1.6000000000000001 < d

   1. Initial program 49.8%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 65.0%

      \[\leadsto \frac{a}{-\left|d\right|} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;d \leq 1.6:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\left|d\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e-11)
   (/ b c)
   (if (<= c 2.3e-86)
     (/ (fma c b (* (- a) d)) (* d d))
     (if (<= c 9.5e+109) (* (/ c (fma d d (* c c))) b) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-11) {
		tmp = b / c;
	} else if (c <= 2.3e-86) {
		tmp = fma(c, b, (-a * d)) / (d * d);
	} else if (c <= 9.5e+109) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e-11)
		tmp = Float64(b / c);
	elseif (c <= 2.3e-86)
		tmp = Float64(fma(c, b, Float64(Float64(-a) * d)) / Float64(d * d));
	elseif (c <= 9.5e+109)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e-11], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.3e-86], N[(N[(c * b + N[((-a) * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.5e+109], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5e-11 or 9.49999999999999972e109 < c

   1. Initial program 37.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -7.5e-11 < c < 2.29999999999999996e-86

   1. Initial program 79.5%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

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

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

      15. lift-neg.f64 N/A

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

      16. lower-*.f64 70.7

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

   7. Applied rewrites 70.7%

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

   if 2.29999999999999996e-86 < c < 9.49999999999999972e109

   1. Initial program 81.6%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(c, b, \left(-a\right) \cdot d\right)}{d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -7.5e-11)
   (/ b c)
   (if (<= c 2.3e-86)
     (/ (- (* b c) (* a d)) (* d d))
     (if (<= c 9.5e+109) (* (/ c (fma d d (* c c))) b) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -7.5e-11) {
		tmp = b / c;
	} else if (c <= 2.3e-86) {
		tmp = ((b * c) - (a * d)) / (d * d);
	} else if (c <= 9.5e+109) {
		tmp = (c / fma(d, d, (c * c))) * b;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -7.5e-11)
		tmp = Float64(b / c);
	elseif (c <= 2.3e-86)
		tmp = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(d * d));
	elseif (c <= 9.5e+109)
		tmp = Float64(Float64(c / fma(d, d, Float64(c * c))) * b);
	else
		tmp = Float64(b / c);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -7.5e-11], N[(b / c), $MachinePrecision], If[LessEqual[c, 2.3e-86], N[(N[(N[(b * c), $MachinePrecision] - N[(a * d), $MachinePrecision]), $MachinePrecision] / N[(d * d), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 9.5e+109], N[(N[(c / N[(d * d + N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(b / c), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if c < -7.5e-11 or 9.49999999999999972e109 < c

   1. Initial program 37.3%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 72.8

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

   5. Applied rewrites 72.8%

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

   if -7.5e-11 < c < 2.29999999999999996e-86

   1. Initial program 79.5%

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

   3. Taylor expanded in c around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if 2.29999999999999996e-86 < c < 9.49999999999999972e109

   1. Initial program 81.6%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 66.2

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

   7. Applied rewrites 66.2%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{elif}\;c \leq 2.3 \cdot 10^{-86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{d \cdot d}\\ \mathbf{elif}\;c \leq 9.5 \cdot 10^{+109}:\\ \;\;\;\;\frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)} \cdot b\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38} \lor \neg \left(d \leq 0.00285\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.7e-38) (not (<= d 0.00285)))
   (/ (- (* b (/ c d)) a) d)
   (/ (fma (/ d c) a (- b)) (- c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.7e-38) || !(d <= 0.00285)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = fma((d / c), a, -b) / -c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.7e-38) || !(d <= 0.00285))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.7e-38], N[Not[LessEqual[d, 0.00285]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.69999999999999998e-38 or 0.0028500000000000001 < d

   1. Initial program 55.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 80.7%

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

   if -4.69999999999999998e-38 < d < 0.0028500000000000001

   1. Initial program 69.7%

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 87.7

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

   7. Applied rewrites 87.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38} \lor \neg \left(d \leq 0.00285\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 0.0028:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-b}{d}, c, a\right)}{-d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4.7e-38)
   (/ (- (* b (/ c d)) a) d)
   (if (<= d 0.0028)
     (/ (fma (/ d c) a (- b)) (- c))
     (/ (fma (/ (- b) d) c a) (- d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4.7e-38) {
		tmp = ((b * (c / d)) - a) / d;
	} else if (d <= 0.0028) {
		tmp = fma((d / c), a, -b) / -c;
	} else {
		tmp = fma((-b / d), c, a) / -d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4.7e-38)
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	elseif (d <= 0.0028)
		tmp = Float64(fma(Float64(d / c), a, Float64(-b)) / Float64(-c));
	else
		tmp = Float64(fma(Float64(Float64(-b) / d), c, a) / Float64(-d));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4.7e-38], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[d, 0.0028], N[(N[(N[(d / c), $MachinePrecision] * a + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(N[((-b) / d), $MachinePrecision] * c + a), $MachinePrecision] / (-d)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.69999999999999998e-38

   1. Initial program 59.8%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 74.1

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

   5. Applied rewrites 74.1%

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

   7. Applied rewrites 79.6%

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

   if -4.69999999999999998e-38 < d < 0.00279999999999999997

   1. Initial program 69.7%

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

   4. Applied rewrites 67.6%

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

   5. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 87.7

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

   7. Applied rewrites 87.7%

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

   if 0.00279999999999999997 < d

   1. Initial program 50.5%

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in d around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 81.7

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

   7. Applied rewrites 81.7%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{elif}\;d \leq 0.0028:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{d}{c}, a, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-b}{d}, c, a\right)}{-d}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38} \lor \neg \left(d \leq 0.00285\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4.7e-38) (not (<= d 0.00285)))
   (/ (- (* b (/ c d)) a) d)
   (/ (- b (/ (* d a) c)) c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.7e-38) || !(d <= 0.00285)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4.7d-38)) .or. (.not. (d <= 0.00285d0))) then
        tmp = ((b * (c / d)) - a) / d
    else
        tmp = (b - ((d * a) / c)) / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4.7e-38) || !(d <= 0.00285)) {
		tmp = ((b * (c / d)) - a) / d;
	} else {
		tmp = (b - ((d * a) / c)) / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4.7e-38) or not (d <= 0.00285):
		tmp = ((b * (c / d)) - a) / d
	else:
		tmp = (b - ((d * a) / c)) / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4.7e-38) || !(d <= 0.00285))
		tmp = Float64(Float64(Float64(b * Float64(c / d)) - a) / d);
	else
		tmp = Float64(Float64(b - Float64(Float64(d * a) / c)) / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4.7e-38) || ~((d <= 0.00285)))
		tmp = ((b * (c / d)) - a) / d;
	else
		tmp = (b - ((d * a) / c)) / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4.7e-38], N[Not[LessEqual[d, 0.00285]], $MachinePrecision]], N[(N[(N[(b * N[(c / d), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / d), $MachinePrecision], N[(N[(b - N[(N[(d * a), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.69999999999999998e-38 or 0.0028500000000000001 < d

   1. Initial program 55.2%

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

   3. Taylor expanded in c around 0

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

      1. +-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. *-lft-identity N/A

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

      7. div-sub N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.4

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

   5. Applied rewrites 76.4%

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

   7. Applied rewrites 80.7%

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

   if -4.69999999999999998e-38 < d < 0.0028500000000000001

   1. Initial program 69.7%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 86.9

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

   5. Applied rewrites 86.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4.7 \cdot 10^{-38} \lor \neg \left(d \leq 0.00285\right):\\ \;\;\;\;\frac{b \cdot \frac{c}{d} - a}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b - \frac{d \cdot a}{c}}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 1.5:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\left|d\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= d -4e-10) (/ a (- d)) (if (<= d 1.5) (/ b c) (/ (- a) (fabs d)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4e-10) {
		tmp = a / -d;
	} else if (d <= 1.5) {
		tmp = b / c;
	} else {
		tmp = -a / fabs(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 (d <= (-4d-10)) then
        tmp = a / -d
    else if (d <= 1.5d0) then
        tmp = b / c
    else
        tmp = -a / abs(d)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if (d <= -4e-10) {
		tmp = a / -d;
	} else if (d <= 1.5) {
		tmp = b / c;
	} else {
		tmp = -a / Math.abs(d);
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if d <= -4e-10:
		tmp = a / -d
	elif d <= 1.5:
		tmp = b / c
	else:
		tmp = -a / math.fabs(d)
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (d <= -4e-10)
		tmp = Float64(a / Float64(-d));
	elseif (d <= 1.5)
		tmp = Float64(b / c);
	else
		tmp = Float64(Float64(-a) / abs(d));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if (d <= -4e-10)
		tmp = a / -d;
	elseif (d <= 1.5)
		tmp = b / c;
	else
		tmp = -a / abs(d);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[d, -4e-10], N[(a / (-d)), $MachinePrecision], If[LessEqual[d, 1.5], N[(b / c), $MachinePrecision], N[((-a) / N[Abs[d], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if d < -4.00000000000000015e-10

   1. Initial program 59.2%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if -4.00000000000000015e-10 < d < 1.5

   1. Initial program 70.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

   if 1.5 < d

   1. Initial program 49.8%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 65.0

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

   5. Applied rewrites 65.0%

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

   7. Applied rewrites 65.0%

      \[\leadsto \frac{a}{-\left|d\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-10}:\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{elif}\;d \leq 1.5:\\ \;\;\;\;\frac{b}{c}\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{\left|d\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-10} \lor \neg \left(d \leq 1.5\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -4e-10) (not (<= d 1.5))) (/ a (- d)) (/ b c)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e-10) || !(d <= 1.5)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, c, d)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    real(8) :: tmp
    if ((d <= (-4d-10)) .or. (.not. (d <= 1.5d0))) then
        tmp = a / -d
    else
        tmp = b / c
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -4e-10) || !(d <= 1.5)) {
		tmp = a / -d;
	} else {
		tmp = b / c;
	}
	return tmp;
}

Python

def code(a, b, c, d):
	tmp = 0
	if (d <= -4e-10) or not (d <= 1.5):
		tmp = a / -d
	else:
		tmp = b / c
	return tmp

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -4e-10) || !(d <= 1.5))
		tmp = Float64(a / Float64(-d));
	else
		tmp = Float64(b / c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c, d)
	tmp = 0.0;
	if ((d <= -4e-10) || ~((d <= 1.5)))
		tmp = a / -d;
	else
		tmp = b / c;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -4e-10], N[Not[LessEqual[d, 1.5]], $MachinePrecision]], N[(a / (-d)), $MachinePrecision], N[(b / c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -4.00000000000000015e-10 or 1.5 < d

   1. Initial program 54.6%

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

   3. Taylor expanded in c around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -4.00000000000000015e-10 < d < 1.5

   1. Initial program 70.2%

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

   3. Taylor expanded in c around inf

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

      1. lower-/.f64 74.1

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

   5. Applied rewrites 74.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -4 \cdot 10^{-10} \lor \neg \left(d \leq 1.5\right):\\ \;\;\;\;\frac{a}{-d}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{c}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in c around inf

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

   1. lower-/.f64 44.4

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

5. Applied rewrites 44.4%

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

6. Final simplification 44.4%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

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

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 2024351 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

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

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