Complex division, imag part

Percentage Accurate: 61.6% → 84.9%

Time: 5.4s

Alternatives: 7

Speedup: N/A×

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: 61.6% 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: 84.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(d, d, c \cdot c\right)\\ t_1 := \mathsf{fma}\left(a \cdot \frac{d}{t\_0}, -1, b \cdot \frac{c}{t\_0}\right)\\ \mathbf{if}\;c \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+94}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (fma d d (* c c)))
        (t_1 (fma (* a (/ d t_0)) -1.0 (* b (/ c t_0)))))
   (if (<= c -7.2e+133)
     (/ (fma a (/ d c) (- b)) (- c))
     (if (<= c -8.2e-85)
       t_1
       (if (<= c 6.8e-141)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 7.5e+94) t_1 (fma (/ (- a) c) (/ d c) (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = fma(d, d, (c * c));
	double t_1 = fma((a * (d / t_0)), -1.0, (b * (c / t_0)));
	double tmp;
	if (c <= -7.2e+133) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (c <= -8.2e-85) {
		tmp = t_1;
	} else if (c <= 6.8e-141) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 7.5e+94) {
		tmp = t_1;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = fma(d, d, Float64(c * c))
	t_1 = fma(Float64(a * Float64(d / t_0)), -1.0, Float64(b * Float64(c / t_0)))
	tmp = 0.0
	if (c <= -7.2e+133)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (c <= -8.2e-85)
		tmp = t_1;
	elseif (c <= 6.8e-141)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 7.5e+94)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	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[(a * N[(d / t$95$0), $MachinePrecision]), $MachinePrecision] * -1.0 + N[(b * N[(c / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -7.2e+133], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, -8.2e-85], t$95$1, If[LessEqual[c, 6.8e-141], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 7.5e+94], t$95$1, N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -7.19999999999999956e133

   1. Initial program 30.3%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -7.19999999999999956e133 < c < -8.19999999999999987e-85 or 6.7999999999999997e-141 < c < 7.49999999999999978e94

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. mul-1-neg N/A

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

      10. associate-*r* N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. div-add-rev N/A

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

      14. associate-*r/ N/A

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

      15. +-commutative N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

   4. Applied rewrites 80.7%

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

   if -8.19999999999999987e-85 < c < 6.7999999999999997e-141

   1. Initial program 67.9%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 93.6

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

   5. Applied rewrites 93.6%

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

   if 7.49999999999999978e94 < c

   1. Initial program 30.4%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -8.2 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, -1, b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 7.5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \frac{d}{\mathsf{fma}\left(d, d, c \cdot c\right)}, -1, b \cdot \frac{c}{\mathsf{fma}\left(d, d, c \cdot c\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;c \leq 8.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (let* ((t_0 (/ (- (* b c) (* a d)) (+ (* c c) (* d d)))))
   (if (<= c -1.65e+146)
     (/ (fma a (/ d c) (- b)) (- c))
     (if (<= c -1e-84)
       t_0
       (if (<= c 8.4e-48)
         (/ (fma b (/ c d) (- a)) d)
         (if (<= c 6.8e+86) t_0 (fma (/ (- a) c) (/ d c) (/ b c))))))))

C

double code(double a, double b, double c, double d) {
	double t_0 = ((b * c) - (a * d)) / ((c * c) + (d * d));
	double tmp;
	if (c <= -1.65e+146) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (c <= -1e-84) {
		tmp = t_0;
	} else if (c <= 8.4e-48) {
		tmp = fma(b, (c / d), -a) / d;
	} else if (c <= 6.8e+86) {
		tmp = t_0;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	t_0 = Float64(Float64(Float64(b * c) - Float64(a * d)) / Float64(Float64(c * c) + Float64(d * d)))
	tmp = 0.0
	if (c <= -1.65e+146)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (c <= -1e-84)
		tmp = t_0;
	elseif (c <= 8.4e-48)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	elseif (c <= 6.8e+86)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	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[(N[(c * c), $MachinePrecision] + N[(d * d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+146], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, -1e-84], t$95$0, If[LessEqual[c, 8.4e-48], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], If[LessEqual[c, 6.8e+86], t$95$0, N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.65000000000000008e146

   1. Initial program 22.6%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -1.65000000000000008e146 < c < -1e-84 or 8.39999999999999954e-48 < c < 6.7999999999999995e86

   1. Initial program 78.6%

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

   if -1e-84 < c < 8.39999999999999954e-48

   1. Initial program 68.6%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 91.2

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

   5. Applied rewrites 91.2%

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

   if 6.7999999999999995e86 < c

   1. Initial program 30.4%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq -1 \cdot 10^{-84}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{elif}\;c \leq 8.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{elif}\;c \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (<= c -4.6e-32)
   (/ (fma a (/ d c) (- b)) (- c))
   (if (<= c 3.2e+81)
     (/ (fma b (/ c d) (- a)) d)
     (fma (/ (- a) c) (/ d c) (/ b c)))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if (c <= -4.6e-32) {
		tmp = fma(a, (d / c), -b) / -c;
	} else if (c <= 3.2e+81) {
		tmp = fma(b, (c / d), -a) / d;
	} else {
		tmp = fma((-a / c), (d / c), (b / c));
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if (c <= -4.6e-32)
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	elseif (c <= 3.2e+81)
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	else
		tmp = fma(Float64(Float64(-a) / c), Float64(d / c), Float64(b / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[LessEqual[c, -4.6e-32], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], If[LessEqual[c, 3.2e+81], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision], N[(N[((-a) / c), $MachinePrecision] * N[(d / c), $MachinePrecision] + N[(b / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -4.6000000000000001e-32

   1. Initial program 54.9%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -4.6000000000000001e-32 < c < 3.2e81

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

   if 3.2e81 < c

   1. Initial program 30.4%

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

   3. Taylor expanded in d around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. pow2 N/A

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

      4. times-frac N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 88.2

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

   5. Applied rewrites 88.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{elif}\;c \leq 3.2 \cdot 10^{+81}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{c}, \frac{d}{c}, \frac{b}{c}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-32} \lor \neg \left(c \leq 2.25 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -4.6e-32) (not (<= c 2.25e+82)))
   (/ (fma a (/ d c) (- b)) (- c))
   (/ (fma b (/ c d) (- a)) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -4.6e-32) || !(c <= 2.25e+82)) {
		tmp = fma(a, (d / c), -b) / -c;
	} else {
		tmp = fma(b, (c / d), -a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -4.6e-32) || !(c <= 2.25e+82))
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	else
		tmp = Float64(fma(b, Float64(c / d), Float64(-a)) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -4.6e-32], N[Not[LessEqual[c, 2.25e+82]], $MachinePrecision]], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(b * N[(c / d), $MachinePrecision] + (-a)), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -4.6000000000000001e-32 or 2.2499999999999998e82 < c

   1. Initial program 44.1%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 79.9

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

   5. Applied rewrites 79.9%

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

   if -4.6000000000000001e-32 < c < 2.2499999999999998e82

   1. Initial program 71.6%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 85.3

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

   5. Applied rewrites 85.3%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -4.6 \cdot 10^{-32} \lor \neg \left(c \leq 2.25 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, \frac{c}{d}, -a\right)}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{-32} \lor \neg \left(c \leq 1.4 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{b}{a} \cdot \frac{c}{d} - 1\right) \cdot a}{d}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= c -2.25e-32) (not (<= c 1.4e+22)))
   (/ (fma a (/ d c) (- b)) (- c))
   (/ (* (- (* (/ b a) (/ c d)) 1.0) a) d)))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((c <= -2.25e-32) || !(c <= 1.4e+22)) {
		tmp = fma(a, (d / c), -b) / -c;
	} else {
		tmp = ((((b / a) * (c / d)) - 1.0) * a) / d;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((c <= -2.25e-32) || !(c <= 1.4e+22))
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(b / a) * Float64(c / d)) - 1.0) * a) / d);
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[c, -2.25e-32], N[Not[LessEqual[c, 1.4e+22]], $MachinePrecision]], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision], N[(N[(N[(N[(N[(b / a), $MachinePrecision] * N[(c / d), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * a), $MachinePrecision] / d), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < -2.25000000000000002e-32 or 1.4e22 < c

   1. Initial program 49.2%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -2.25000000000000002e-32 < c < 1.4e22

   1. Initial program 70.1%

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

   3. Taylor expanded in d around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 88.8

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

   5. Applied rewrites 88.8%

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

   6. Taylor expanded in a around inf

      \[\leadsto \frac{a \cdot \left(\frac{b \cdot c}{a \cdot d} - 1\right)}{d} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. times-frac N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 84.0

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

   8. Applied rewrites 84.0%

      \[\leadsto \frac{\left(\frac{b}{a} \cdot \frac{c}{d} - 1\right) \cdot a}{d} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq -2.25 \cdot 10^{-32} \lor \neg \left(c \leq 1.4 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{b}{a} \cdot \frac{c}{d} - 1\right) \cdot a}{d}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+134} \lor \neg \left(d \leq 1.12 \cdot 10^{+170}\right):\\ \;\;\;\;b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (if (or (<= d -9.5e+134) (not (<= d 1.12e+170)))
   (* b (/ (- (/ c d) (pow (/ c d) 3.0)) d))
   (/ (fma a (/ d c) (- b)) (- c))))

C

double code(double a, double b, double c, double d) {
	double tmp;
	if ((d <= -9.5e+134) || !(d <= 1.12e+170)) {
		tmp = b * (((c / d) - pow((c / d), 3.0)) / d);
	} else {
		tmp = fma(a, (d / c), -b) / -c;
	}
	return tmp;
}

Julia

function code(a, b, c, d)
	tmp = 0.0
	if ((d <= -9.5e+134) || !(d <= 1.12e+170))
		tmp = Float64(b * Float64(Float64(Float64(c / d) - (Float64(c / d) ^ 3.0)) / d));
	else
		tmp = Float64(fma(a, Float64(d / c), Float64(-b)) / Float64(-c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_, d_] := If[Or[LessEqual[d, -9.5e+134], N[Not[LessEqual[d, 1.12e+170]], $MachinePrecision]], N[(b * N[(N[(N[(c / d), $MachinePrecision] - N[Power[N[(c / d), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(d / c), $MachinePrecision] + (-b)), $MachinePrecision] / (-c)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if d < -9.5000000000000004e134 or 1.1200000000000001e170 < d

   1. Initial program 32.3%

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

   3. Taylor expanded in d around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{d}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{\color{blue}{d}} \]

   5. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \mathsf{fma}\left(b, \frac{c}{d}, -b \cdot {\left(\frac{c}{d}\right)}^{3}\right)\right) - \left(-\frac{c \cdot c}{d} \cdot \frac{a}{d}\right)}{d}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \frac{b \cdot \left(\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}\right)}{\color{blue}{d}} \]
   7. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{\color{blue}{d}} \]

      2. lower-*.f64 N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{\color{blue}{d}} \]

      3. lower-/.f64 N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

      4. lower--.f64 N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

      5. lift-/.f64 N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

      6. cube-div N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

      7. lift-pow.f64 N/A

         \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

      8. lift-/.f64 30.3

         \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

   8. Applied rewrites 30.3%

      \[\leadsto b \cdot \color{blue}{\frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d}} \]

   if -9.5000000000000004e134 < d < 1.1200000000000001e170

   1. Initial program 70.1%

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

   3. Taylor expanded in c around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \leq -9.5 \cdot 10^{+134} \lor \neg \left(d \leq 1.12 \cdot 10^{+170}\right):\\ \;\;\;\;b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, \frac{d}{c}, -b\right)}{-c}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 18.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* b (/ (- (/ c d) (pow (/ c d) 3.0)) d)))

C

double code(double a, double b, double c, double d) {
	return b * (((c / d) - pow((c / d), 3.0)) / 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 / d) - ((c / d) ** 3.0d0)) / d)
end function

Java

public static double code(double a, double b, double c, double d) {
	return b * (((c / d) - Math.pow((c / d), 3.0)) / d);
}

Python

def code(a, b, c, d):
	return b * (((c / d) - math.pow((c / d), 3.0)) / d)

Julia

function code(a, b, c, d)
	return Float64(b * Float64(Float64(Float64(c / d) - (Float64(c / d) ^ 3.0)) / d))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b * (((c / d) - ((c / d) ^ 3.0)) / d);
end

Wolfram

code[a_, b_, c_, d_] := N[(b * N[(N[(N[(c / d), $MachinePrecision] - N[Power[N[(c / d), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / d), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.9%

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

3. Taylor expanded in d around inf

   \[\leadsto \color{blue}{\frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{d}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\left(-1 \cdot a + \left(-1 \cdot \frac{b \cdot {c}^{3}}{{d}^{3}} + \frac{b \cdot c}{d}\right)\right) - -1 \cdot \frac{a \cdot {c}^{2}}{{d}^{2}}}{\color{blue}{d}} \]

5. Applied rewrites 49.1%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, a, \mathsf{fma}\left(b, \frac{c}{d}, -b \cdot {\left(\frac{c}{d}\right)}^{3}\right)\right) - \left(-\frac{c \cdot c}{d} \cdot \frac{a}{d}\right)}{d}} \]

6. Taylor expanded in b around inf

   \[\leadsto \frac{b \cdot \left(\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}\right)}{\color{blue}{d}} \]
7. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{\color{blue}{d}} \]

   2. lower-*.f64 N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{\color{blue}{d}} \]

   3. lower-/.f64 N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

   4. lower--.f64 N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

   5. lift-/.f64 N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - \frac{{c}^{3}}{{d}^{3}}}{d} \]

   6. cube-div N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

   7. lift-pow.f64 N/A

      \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

   8. lift-/.f64 18.0

      \[\leadsto b \cdot \frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d} \]

8. Applied rewrites 18.0%

   \[\leadsto b \cdot \color{blue}{\frac{\frac{c}{d} - {\left(\frac{c}{d}\right)}^{3}}{d}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (a b c d)
  :name "Complex division, imag part"
  :precision binary64

  :alt
  (! :herbie-platform herbie20 (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))))